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SO 4 — oF oY g * ee gm Vag KOPek COSC SYLOXS \ ye ~ eh we Sent Sa ET aN oS (ge i _— owes ee eetate fic eee THE ELEMENTS | OF GEOMETRITIE of the moft aunct- ent Philofopher EVCLIDE ca eg a 4 Raith sfubly( now far it } trane lated into the Englifhe toung by Hi. Billingfley,Citizea of London. IWhereunto are annexed certaine Scholies, Annotations and Inuenti- ons, ofthe bet? Mathematsts- ens, both of time past , and te this our age. Nee “te \} si \ f 7 ‘ FAAP AN IAN Ww de p fig f > cp (LP nf — / /f ff a tele "ee eb wy a ¢ Fon St me a e § ( he c ¥ - Lieder } f eats § phar tie aan (vg EA PE ~ - ’ alii rit Ath 4% a, hay, Pee) s un oo AN ‘ ics Wish Avery a or ie mah by M. 1. Dee, |} ae Ae the chiefe e*Lathematicalt Scieces, what they are,and wherunto commodsous: where,alfo, are dic! lofed certaine new Secrets Muthematicall and Mechanicall untill thefe our dates, greatly miffed. ee n+ a Vaan? da VAN fj Gus St ; ; : ~ 7 ole a 7 eye Fe “ey é ; te 1 ple SESS hy ; = \« 9; , 7 : 4 AH SO I~ ELBE OT —" ; Rn satis Speer Y 4 weg ae a sete Sao he eas NO ae > yy, & CH rye one oe Seed (¢ \\" sat iy ; $c Wr SF “ if oth 4 | “; Walt as 43 - of: t ths i pends sees ane. Ne kth As are ee wee SA Ha a , ate aeeneneeienmeitaeatl * TON COLLEGE LIBRARY. 48) eY oOorRDeR* or THE ; AY - (S PRESIDENT AND GOVSRNORS 1938, $a The Tranflator tothe Reader. [= Flere is (gentle Reader) nothing | bony font (the word of God onely ‘fet apart.) “<< va| which fo much beautifieth and a- 7 i. ee || dorneth the foule_. and minde. of pee Se : ¢ — a our eyes, the creatures of God, both in the heauens aboue,and inthe earth beneath: in which as inaglafe, we beholde the exceding maieftie and wifedome of God, in adorning and beautifying them as we fee : in geuing yn- to them [uch wonderfulland mantfolde proprieties, and natural workinges, and that fadiuerfly and in Juchvarietie : farther in maintatning and conferuing them continually, whereby to praife and adore him, as by S.Paule we are taught . The other tea- cheth vs rules and preceptes of vertue,how, in common life a monge/t mens, we ought to walke vprightly : what dueties per- , taine to our,felues, what pertaine to the gouernment or good or~ 4 der both of an houfholde, and alfo of a citie or common wealth. T he reading likewife of biftories ,conduceth not a litle,to the ad- orning of he foule minde of man., a ftudie of all men comen- ded : by itare feene and knowen theartes and doinges of infinite wife men gone before vs «In hiftories are contained infinite ex- amples of heroicall vertues to be of vs followed, and horrible ex- amples of vices to be of vs efchewed . Many other artes alfo there are which beautifiethe minde of man: but of all other none domore garnifhe (> beautifie it, then thofe artes which are cal- led Mathematicall.. Unto the knowledge of which noman can attaine, without the perfette knowledge and inftruttion of the principles, groundes,and Elementes of Geometrie . “But per- Ft. fettly Za) lofophie. Theonefettethbefore ~~ % TI rn $a) The Tranflator to the Reader. fet to be inftruéted inthem, requireth diligent fludie and reas ding of olde auncient authors. eAmongeft which none for be- ginner ts tobepreferred before the moft auncient ‘Philo opher Euclide of Megara. Forofall others he bath in a true, me- thode and infte order, gathered together whatfoeuer any before him had of thefe Elementes written: inuenting alfo and adding many thinges of his owne : wherby be hath in due forme accom- plifhed the artesfirft. geuing definitions, principles,@> groundes, wherof he deduceth his Propolitions or concluftons tn fuch won- der full wife, that that which goeth before , 1s of necefiitie requt- rvedto the proufes of that which followeth . So that without the diligent ftudie of Luchides Elementes,it is impofible to attaine unto the perfette knowledge of Geometrie, and confequently of any of the orber Mathematicall [ciences .. Wherefore confide- ring the want alfo abilitie to tranflate into our tounge and ro publifhe abroad fuch good authors,and bookes ( the thiefe in- ftrumentes of alltearninges ) : Jeng moreouer that many good wittes both of gentlemen and of others of all degrees much de- frrous and jtudions of thefe artes, and feeking for them as much as they can, [paring no pames, and yet fruftrate of their intent, by no meanes attaining to that which they feeke: Thane. for their fakes, with fome charge (> great trauaile, faithfully tran- flated into our yuleare touge,¢o fer abroad in Print , this booke of Euclide. Whereunto 1 bane added eafe and plaine decla- vations and examples by froures, of the definitions . Inwhich booke alfoyefhall in due place finde manifolde additions, Scho- lies, Annotations,and Inuentions: which Fbaue vathered out of many of the moft famous @chiefe Mathematicies , both of old time and in our ageras bydiligent reading it in courfe, ye /hall den well * epee SyThe Tranflater to the Reader. wellperceaue. The fruite and gaine which I require for thefe my pues and tranarle,fhall be nothing els, but onely that thon gentle eader , will gratefully accept the fame. :and that thon maye/t thereby receane fome ‘profite:and moreouer to excite and irre vp others learned, todo the like, ¢> to take paines in that bebalfe. By meanes wherof,our Englifhe tounge fball no leffe be enriched with good Authors , thenare other ‘firaunge tounges? as the Dutch, French, Ftalian , and Spanifbe : in which are red all good authors in a maner, found amongeft the Grekes or Latines. Which is the chiefeft caufe, that amongef the do flo» rifhe Jo many cunning and fkilfull men, in the inuentions of Jiraunge and wonderfull thinges, asin thefe our dates we fee there do. Which fruite and gaine if Lattaine vnto, it /hall encourage me hereafter, in fuch like Jort to tranflate , and fet abroad Some other good authors, both pertaming to religion ( as partly I bane already done) and alfo pertaining to the > eyes,and bend your myndesto that doctrine, which for our prefent purpofe ; my fimpletalentis hable to yeld you. : li thinges which are,& haue beyng, are found ynder a triple diueérfitie sénerall. For,either,they-are demed Supernaturall,Naturall,or,ofa third being. [hinges Supernatural, are immaterial, fimple, indiuifible incorruptible, & vnchangcable, Things Naturall, are materiall,compounded,diuifible,corruptible, and chaungea- ble, Thinges Supernaturall,are,of the minde onely,comprehended: Things Natu- rall.of the fenfe exterior,ar hable to. be perceiued.In thinges Naturall,probabilitie ». and conieaure hath place:Burin things Supernaturall,chief demoftration,& moft fare Science is to be had. By which properties. & comparafons of thefe two, more eafily may be defcribed,the ftare,condition, nature and property of thofe thinges, which,we beforetermed ofathird being: which,by a peculier name alfo,are called Thynges Mathematical ‘For,thefe,beyng(in a maner)middle, betwene thinges {u- pernaturall and naturall:are notio abfolute and excellent,as thinges {upernatural: Nor yetfo bafe and groffe,as things naturall: But are thingesimmateriall : and ne- | uerthelefle.by. material chings hable fomewhat to be fignified . And though their > particular Images», -by Art,areaggregable and diuiible,: yet the generall Formes,) notwithitandyng,are conftant,vn chaungeable,vntrafformable,and incorruptible, Neither of the feafe,can they,at any tyme,be perceiued or iudged, Nor yet.for alk that, in the royall mrynde of man,firft conceiued.But,firmountyng the impertectio of conie@ure,weenyng and opinion -and commyng fhort.ofhigh intellectuall co- ceptid,are the Mereunial fruite of Dianeticall difcourfe,in perfect imagination fub- fityng. A metuaylous newtraliineauc, thefe thinges Alathematicall : and allo a ftraunge participatio. betwene thinges fupernaturall immortal intellectual, fimple and indiuifible:and thynges naturall, mortall,fenfible,compounded and diuifible. Probabilitic and fenfible profe, may well ferne in thinges naturall:and is commen; dable:In Mathematicall reafoninges,a probable Argument, ts nothyng regardeds hit. credited: But onclya perfect demontftra, ’ nox yetthe teftimony offenfe,any w ! . tion, of rruthes certaine,neceflary,and inuincible:yniuerfally and neceflaryly con- | , cluded: * Tohn Dee his Mathemaaticall Preface. j cluded:is allowed as fuffiicient for an 'Argumentexactly and purely: Mathematicali +: Of Mathematicall thinges,are two.principall kindestnamely; Newber,and. Mag- Numbéi. Number.we define,to be,a certayne Mathematicall Samcjof Knits. Andan Note the verde, a Vuit, that thing Mathematicall,Indiuifible, by participation of fone likenes of “™#,to exprefse aa whofe praperty,any thing,which is indeede,oris counted One,may refonably be ° . - Nise oe ul called One. Weaccountan Fit, a thing. Mathematicall, though ithe no Nwmbery rie igs we bade and alfo indiuifible: becaufe,of itsmaterially, Number doth confaft swhich , princes: «4, commonly. pally, isa thing Mathematicall. Magnitudeis. athing Mathematically byparticipation sag of fome likenes of whofe nature; any thing is iudged long , broadesorthicke.A 4smtndes thicke Magnitude we calla Solide, oraBody.. What Magnitudefo.euer,is Solide.or, ” Thicke,is alfo broade,& long.A broade magnitude,wecall a Superficies or aPlaines Every playne magnitude,hath alfo length. Along magnitude, we terme a LinesA Line is neither thicke nor broade, buronely long: Euery certayne Line, hath two endes:The endes of aline,are Porntes called.A Point,isa thing Mathematicall,indi- A peint. uifible, which may haue a certayne;determined fituation. IfaPoyntmoue froma » determined fituation , the way wherein it moued, is allo a Lene: mathematically produced. whereupon,of theauncient Mathematiciens,a Line iscalledthe race or 4 Line. .courfe of a Point. A Poynt'wedefine, by the name ofa thing»Mathematicall: though itbe no Magnitude, and indiuifible : becaufe itis the propre endeyand bound of a Lines whichis a tnie Magnitude. And Magnitude we may define tobe Magnitude. tharthing Mathematicall,which is diuifible for euer,in partes diuifible,long,broade or thicke. Therefore though aPoynt be no Magautude, yet T erminatinely we xec- ; ken ita thing Mathematicall(as Lfayd)by reafon itis.properly theend, and bound of a line. ; Neither Number, nor (Magnitudehaue any Materialitie. Firlt,we will confides of Number,and of the Science Mathematicall ,to itappropriate,callecl Arithmetikes ° and afterward. of Mugnitude,and his Science, called Geometrie; But t/aat name con- tenteth me not: whereofa word.or two hereafter hall be fayd. How Immaterial . and free from all matter , Nwmber is,, who doth notperceaue ¢ yea; who dothnot wonderfully woderat it? For, neither pure Element, not Ariftoteles, Quinta Efventta, is hable to heuc for Numberyas his propre matter. Nor yetth¢ puritie and fimple- nésof Subftance Spirituall or Angelicall, will-be found propre ¢nough thereto. And therefore the great & godly Philofopher Austins Boetius, fayd:0 mnia quacung, aprimeua rerum natura confiructa, unt, Numerorinavidentur ratione formata. Hoc enim fuit principale in animo Conditoris Exemplar. Thaeis: © Al thin ges ( which from the very firSt original being of thinves';hane bene framed and made } do appeare to be Formed by the reafon of Numbers . For this ‘was the princtpall example or patterné 1a the minde of the Creator. O comifor- table allurement, O rauifhing periwafion, to deale with a Science, whofe Subiect, is fo Auncient,fo pure.fo exceifent,fo furmounting all creatures.fo vied of the Al- mighty and incomprehenfible wifdome of the Creator, in the diftiné creation of all creatures:in all their difting partes, properties, natures , andyertues, by order, and moftabfolute number,brougsht,from Nething,tothe Farmalitie of their being and ftate.By Numbers propertic cherefore,ofvs.by all poflible meanes,(to the per- fc&ion of the Science ) learned, we may both winde and draw ourfelues into the inward and deepe {earch and s:ew,of all.creatures diftinct vertues,naturcs, proper tiesand Formes: And alfo,fard er arife,clime,afcend,and mount vp (with Specula- tine winges ) in {pirit, to behold in the Glas.ofCreation, the Forme of Formes, the Exemplar Number of all thiages Numerable:both vifible and inuifible.: mortall and er cas ne immortali ~ >= , ai x. ee eee yh | 7, tad lohn Dee his Mathemiaticall Preface. immortall;€ orporall and Spirituall,Part of this profound and dinine Science,had Joachim'the Prophefier atteyned vnto:by Numbers Formall,Naturall,and Ratig-all, forfeyng;concludyng,and forfhewyng great particular euents ,long beforgzheir *‘ comming.His bookes yet remainyng,hereof,are good profe:And the nobie Earle | Of Mérandula, (befides that,)afuflicient witnefle:that loachim,in his prophesies, proce- Ano, 1488.0. debby no other way sthen by Numbers Formall.And this Earle hym felte,in Rome, ‘fet 3 ‘= vp 90o.Conclutions:in all kinde of Sciences.openly to be difputed of:and among * thereft, in his\C onclufions Mathematical; (in the eleuenth Conclufion ) hath in Eatin,this Engliflyfentence.By Numbersja wayis had , to the fearchyng out,aud vnder- fandyne of ewery thyneshable to be knowen .For-the verifying of which Conclufion , Fpro- mife to aun{were tothe'74. Queftions onder written by the way of Numbers. Which Co- clufions,f omit hereto rehearfe: afwell auoidyng {uperfluous prolixitie:as , by- caufe loannes Picusworkes jare commonly had. But,in any cafe,I would with that thofe Conclufions were red-diligently , and perceiued of fuchjas are earneft Ob- « feruers and Confiderers of the conftant law of nibers:which is planted in thyngs Naturall andSupernaturall:andis.prefcribed to all Creatures, inuiolably to be kept.For,fojbelides many other thinges: , in thofe Conclufions tobe marked,it would apeare}how fincerely;8 within my boundes,] difclofe the wonderfull my- fteries,by numbers,to be atteyned vnto. Of my former wordes,eafyit is to be gathered that Number hatha treble ftate: One,in the Creator:an otherin euery Creature(in refpect of his complete con{ti- tution: )aud the third,in Spirituall and Angelicall Myndes,and in the Soule of ma. Inthe firft and third :ftate, Nymber ,is termed Number Numbryne. But in all Crea- tures,otherwife,Number,is termed Naber Numbred. And in our Soule,Niberbeas reth fuch a fvaye;and hath fuch-an affinitie therwith: that fome of the old PAi/o/o- _ phers taught, Mans Soule,to bea Number monyne a felfe. And in dede,in vs, though it be avery Accident: yet {uch an Accident it is,that before all Creatures it had per- fe& beyng, inthe Creator,Sempiternally. Number Numbryng therfote,is the difcre- tion:difterning;and diftincting’ofthinges. Butin God the Creator, ‘This difcre- tionsin the bepinnynhg, produced orderly and diftinély all thinges .. Forhis Num- bryng,then,was his Creatyng ofall 'thinges. And his Continuall Numobryng , of all thinges,is the Conferuation of them in being: Aind,where and when he will lacke an. ¥ nit: there and then,that particular thyng fhalbe Défcreated.Here I ftay.But our Scuerallyng,diftin@yng,and Nembryngycreateth nothyng: but of Multitude con- fidered,maketh.cextaine and diftinGdetermination ,..And albeit thefe thynges: be waighty and truthes of great importance, yet ( ,by,the infinite goodnes of the Al, mighty. Termarie,) Artificiall Methods and ealy wayes are made , by which the ze- lous Philofopher,may wyn nere this Riuerifh Jda,this Mountayne of C qn tion:and more theh Contemplation And alfo,though Number, bea thyng fo Im- materiall,fo diuineand xtérnall-yet by degrees,by litle and litle, ftretchyng forth, and applying fomelikenes of it,as firft, to thinges Spirituall:and then,bryngyng it lower,to thynges {enfibly percciued:as ofa momentanyc founde iterated: then to the leaft thynges that may be {éeh numerable? And at length, (moft groffely,) toa multitude ofany corporall thynges feen,or felt:and fo,of thefe groffe and fenfible thynges,we are trayned to learne.acertaine Image or likenes of numbers : and to vfe Arte in them'to our pleafire atid proffit.So gtoffe is our conuerfation, and dull is our apprehenfion! while mortall Senfe, in vs ,raleth the common wealth of our litle world.Hereby we ite, ene three:ora Ternarie. ‘Three Egles,are (F three,ora Terwarie. Which* Ternartes’, are eche,the Vion, knot and V-niformitie,of three difcrete and diftin& Ynits. That is;we may in eche Fernarie , thrife , feuerally pointe,and fhew a part,o ne,One,and One. Where, in Numbryng,we flay sc itl ree, lohn Deé his Mathematicall Preeface.. Three. But how farre,thele vifible Ones do difterre from our Indiuifible Vnits (in Mure Avithmetike,principally confidered)no man is ignorant... Yetfromthefe grofl nd materiall thynges,may we be led vpward,by degrees, f o,informyng our \aginatto n,toward the coceiuyng of Numbers abfolutely ( :Not{uppofing, Peyng any thyng created, Corporall or Spirituall,to fupport,conteyne,or reprefent thole Numbers imagined: ) that atlength, we may be hable, to finde the number of our owne name , glorioufly exemplified and regiftred in the booke of the Trinitie moft blefled.and xternall. But farder vnderftand,that vulgar Practifers,haue Numbers , otherwife, in fun- dry C onfiderations:and extend their name farder,then to Numbers , whofe leaft partis an Vwit.bor the common Logift,R eckenmatter, or Arithmeticicn, in hysv- fing of N umbers:of an Vnit,imagineth lefle partes:and calleth them FracZrons. As ofan nit, he maketh.an halfe,and thus noteth it,~.and fo. of other, (infinitely di- uerfe) partes ofan Vit. Yeaand farder, hath; Fractions.of Fractions. &c. And,foraf- inuch,as, Addition , § ubftraction , Multiplication, Diusfion and Extraction of Rotes,are the chief,and fufficient partes of Arithmetike : whichis, the Science that demonfira- Arithmetiken teth the pro erties,of N. umbers,and all operatios ; in numbers to be performed: How often, »5 Ce fiue fundry fortes of Operations, do, forthe moft part,oftheirexe- », Note. cution,diffcrre from the fiue operations oflike generall property and name,in our » Whole numbers practifable,So often , (for amore diftinct doctrine ) we,vulgarly ; ? a accountand name it,an other kynde ot Arsthmetike. And by this reafon:the Con- y, ee fideration,doctrine,and working,in whole numbers onely:where, ofan /nz,is no B leffe part to beallowed:is named(as it were)an Arithmetike by itfelfe. And fo of the Arithmetike of Fractions.In lyke forte,the neceflary,wonderfull and Secret doc- trine of Proportion , and proportionalytie hath purchafed vnto itfelfea peculier 2, maner.of handlyng and. workyng:and fo. may feme an.other forme of Arithmetike. aes Moreouer,the Astronomers for {pede andmore commedious calculation, haue de- 3. ae uifed a peculier maner of orderyng ntibers,about theyr circular motions, by Sexa- genes,and Sexagelmes.By Signes, Degrees and Minutes &c. which commonly ts ‘Called the Arithmetike of Astronomical or Phificall Fractions. T hat, haue | briefly no- ted by the name of Avithmetike Circular: Bycaufe itis alfo.vfed in circlesnet Affros : nomicall.erc.Praifehath led N umbers farder., and hath framed them,to take vpon 4. . them; the thew. of Magnitudes properties Which is Incommen furabilitie and Irratio- | nalitie. (Fortin pure Arithmetike an Vait;is thé common Meafure of all Numbets.) ‘And:here;Niibers are become,as Lynes,Playnes and Solides: fome tymes Ratio- wall {ome tymes IrrationallyAnd haue propre and peculier characters, (asV3°.V Co. u arid {o of other. Which is'to fignifie Rote Square , Rote Cubik:and fo forth: )& propre A ‘and pedilier fathions in the fiue principall partes: Wherfore the practif er,eftemeth this.a ditierle Hrithmetike from the other .- Practife bryngeth in,here,diuerfe com- poundyng of Numberssas fome tyme,two,three,foure(or more) Radicall nibers, ditierfly knit, bv fignesjofM ore & Lefle:as thus V3 12+ WC 15.0 rthus VES 19 Lye raw 2. &c:Andfome tyme with whole numbers, or fractions of whole Numbet,amég them:3s 204V/F24.V £164 33-V F 10. VFS 44412 V9. 4 And fo. infinitely , may hap the varietie. “After this : Both the one and the other : hath factions incidént:and {6 is this 4vithmetike greately enlarged, by diuerfe ex- asi ‘hibityng and vfe of CompoOfitions and mixtynges. Confiderhow, I(beyng defi- # rous to deliuer the ftudentfrom error and Cauillation)do git to this Practife,the = nameéofthe Avithmetike of Radicall numbers: Not,of Ifrationall ox Surd, Numbers: 5 which otherwhile, are Rationall: though-they haue the Signe of a Rote before q Abt? ? Shia 574 Dit Sf] DH OTyY OB? IES CIO ' . : | lf. them, lohn Dee his Mathematical] Preface. them, which, Arithmetike of whole Numbers mioft vfuall , would fay they had no fuch Roote:and fo. account them Surd Numbers:which generally fpoké, is vyfcue: as Euclides tenth booke may teach you.:Therfore to call them , generally , Bidicall Nimbers, (by reafon of thé figne v.prefixed,)is 4 fare way:and a fufficientgenerall ditinG@ion from all other ordryng and vfing of Numbers: And yer whofe degrecs‘are * truely: x 7446 fame kadwen : Ofnéctiittie; citherthey steoF one Qirntitie atid waight, or of dinerle.’ part of Lullas oa. taht ofon¢é Qua dw a its. cleat Sega ai be antitic and waight? whethertheif formes, be Contrary'Qua- counfayle in tits, of OF oie Kinde (butt of ditterfe intentions ahd déprécs or a Temperare; and a’ bis books de C ontrary . Lhe forme refilting of their Mixture,is in the Middle betwene the dearees of Q.E fentia. 51310 88ros5 fos souBoro Mob ssresb ywsreig ort mon J etats S = sa << essai ; 4 e\ 4, c~~ ated ef bi Abs / : a ! ‘hfs’ i é. a fe 47855247 i John Dee his Mathematicall Preface. che formes mixt. As for example,let 4,be Moist in the firft degree : and B, Dry in the third degree. Adde a.and 3. that maketh 4: the halfe or middle of 4.Afi2« *Note. This 2.isthe middle, equally diftant from 4 and B( for the” Temperament is Ch un- tednone. And for it, youmuft put a Ciphre, if-at any time, it bein mix ae HOTE Cc v _ > > — > ” ~~ ® ~ 7 > 7 » ’ ~~? ~~ a | -—' a ‘ ré : od . J he 232 . > . , : ~ : mes : ree Pe ; = ; & > £> ry if . eo a + 4 ea & . : : ainit - Se va * « is LaF al a .3 . ie a : : -; -, 77 - ow 7 r a | +t as Se ee ae ee ” _ “Ve - . ee 7 oe 5 & a _ : a y whe a> . ol ae - 3 — Oe Se ee ‘ t. i] a Se Ses BUSTS see. fs ASHI SC i fd agte ; : c 3 ” : y +3 — $s; , wi A SHUOGR Si UY 26 VWASOL 2&8 10 TO* enol isd s bag galodick d- 5 ‘pmeeid Gounting then from 3, 2.degrees , toward «4 syou finde it.to,be Dry in the fit degree 150 isthe Farmer efulting of the Mixture of.4,and B,in our example, I will eeue You an otherexample. Suppofe,, ou have two thinges,as C,and D ; and.of C, the Heatese be inthe 4,degree:.and of,.D, the-Golde,.to beremiffe,euenwato the'T: emaper Ament « ‘Now,for C.you take.4; and. or D,youtake 2, Ci phr e iz wl i b, added-vnto 4, y¢ldeth onely4i TE hemiddle,or halfe, whereof, is 2.,, Wihetefore the Forme refulting of C yandiD, isHoxe in-shefecond degree’. for, ize degrees accoMns _-. red fron. C;roward D., endeiufte.in the 2. degree of heat, Of the third ma. , néx,I will geucalfoan exa mple: which let be this : Lhaue ata sid Medicine whofe ... andan-ether liquid Medicine.Phatte,-. whofe Qualitic, is heate, in the firit degrees . Of echeofthefe; I mixtalike quantitie-:.Subttact here,the leffe fr5 the more’ :.and » the refiduediuide intotwe-equall partes whereof, the one part, either added to the leffe,orfubrrated from oe higher degree, doth produce the degree of the Forme aT IG ON John Dée his Mathématicall Preface; ' Fortine réfulting.by this mixture of Cjand E. As,ifftom 4. ye abate r.theré refteth z.theMalfe of 3. is 1. + Adde to n.thisa +> youhaue 2 , Orfubvacfrom:4. this 1X + you haue likewife2+ remayning . Which declareth , the Forme'réful- ting, tO be Heste, in the middle ofthe third degree. : | ‘Butif the Quantities oftwo thihges Commixt, be diuerfe, and ‘the Intenfi- ons (of theit Formés Miftible ) bein diuérfe degrees’, and heigthes. ( Whether thofe Formes be of one Kinde, or of Contrary kindes, ot of'a Temperaré'and a Contrary , What proportion is of the leffe quantitie to the greater, the fame. fhall be of the difference which is betwene the decree of the Forme refultine, and the degree of the greater ier? Ades whee ork | “Iptadeee ae ee? quantitie of thething mifcible, to the difference, ‘which is betwéne the fame degree of the, Forme re{ulting,and the degree of the le(%e quantitie. As for example. Lettwo pound of Liquor be geuen, hotein the 4.degree:& one pound of Liquot be geuen, hote in the third degree . would gladly know the Forme refilting,in the Mixture of thefe two Liquors. Set downe your niibers in order, thus. Now by the rule of Algiebar, haue I deuifed avety eafie, | F 2. | Hote. 4. | briefe, and generall manér of working in this cafe . Letvs firft, fappofe that Middle Forme refulting , to be 17¢ : as that Ruleteacheth. And becaufe (by our Rule, here geuen) as | © 7 | Hote. 3. the waight of r.is to 2°: Sois the difference betwene 4.(the | degree of the greater quantitie ) and 1% : to the difference betwene 172 arid: 3- (the degreeof the thing, in leffe quatitie-And with all, 17¢ , being alwayes in a cer- taine middell, betwene the two heigthes or degrees) . For the firft difference, [fet 4—17 : and for the fecond,] fet 149 —3 .. And, now again¢, T fay,'as 1.15 to 2.fo is 4—12e to1ze—3. Wherfore, of thefe foure proportional numbers, the firftand the fourth Multiplied,one by the other,do make as much, as the fecond and the third Multiplied the oné by the other. Let thefe Multiplications be made accor dingly. And of the firftand the fourth,we haue 122 —3.and of the fecond & the third, 8—2%¢ .Wherfore} our Aquationis betwene 14 —3: and 8—2ze .Which may be reducéd,according to the Arte of Algicbar:as,here,adding 3.to eche part, geueth the Aquation,thus,176 —11—27%. And yet againe,contracting, of Redu- cing it: Adde to eche part, 2% : Then haue you 379, equall to 11 : thus reprefen- ted 3711. Wherefore, diuiding 11.by 3: the Quotient is 3— : the Valew ofout 12 ,Cofs,or.T hing, firlt{uppofed. And thatis the heigth, or Interifion of the Forme refulting : Which is,Heate, in two. thirdes of the fourth degree : And here T fet the fhew of the worke in conclufion, thus. The proufe hereofis eafie: by fubtracting 3-from 3-,refteth : | | ; ~. . Subtracte the | fame heigthof the Rcciul Bete. dies cA SOR | Forme refulting, = —s akht arabe heforme (which is 3:+-) fro : | re rE is ' 4, then refteth Bicker san| « Hote. 23. — You fee, that —. is double to“ —; a: | , , | as 2.P.is double to 1. &. So thould it be : by therule here geuen . Note. As you ad- ded to eche part of the Aquation, 3 : fo if ye firft added to eche’part 27¢ , it would ftand, 3x2 —3—8. And now adding to ide patt3 - you haue(as afore)3-@—=. And though I; here,fpeake onely of two thyngs Mifcible: and moft common- ty,mo then three, foure, fue or fix,(&c. )are to be Mixed: (and in one C ompound - * iii. to 323 T he Sé- »5 cond >» Rule. 2? Tax ix}. 39 3 John Dee his Mathematicall Preface. to be reduced.& the Forme refultyng of the fame, to ferue the turne)yet thefe R u-- les are fufficient: ducly repeated and iterated.In procedyng firft, with any oy, cand. then, with the Forme Refulting,and an other:& fo forth: For, the laft worke, con- cludeth the Forme refultyng of them all:I nede nothing to fpeake, of the Mixture (here.firppofed) what it is; Common Philofophie hath definedit, faying, mixtio eft mifcibilum, alteratorum., per minima coniunctorum,V nie. Euery word in the de- finition, is of great importance. I nede not alfo {pend any time;,to fhew,how,the other manner of diftributing of degtees,doth agree to thefe Rules. Neither nedel of the fardet vfe belonging to the Croffe of Graduation (before defcribed) in this.» place declare,vnto fuch as are capable of that,which I haue all ready fayd. Neither yet with examples {pecifie the Manifold varieties , by the forefayd two gene- rall Rules,to be ordered. The witty and Studious,here,haue fufiicient: And they which arenot hable to atteinc to this, without liuely teaching, and more in parti- cular: would haue larger difcourfing,then is mete in this place to be dealt withall: And other(perchaunce)with a proude {nuffe will difdaine this lide:and would be vnthankefull for much more’. T,therfore conclude : and with fuch as haue modeft and earneft Philofophicall mindes,to laude God highly forthis:and to Meruayle, that the profoundett and fubtileft point, concerning Mixture of Formes and Quali- ties Naturall,is fo Matchtand maryed with the motkfimple,eafte,and. fhort way of thenoble Rule of Algiebar. Who can remaine , therfore vnperfuaded,to louc,a- low,and honor the excellent Science of Avithmetike 2 For,here,you may perceiue that the litle finger of Arithmetike,is: of more might and contriuing,then a hun- derd thoufand mens wittes,of the middle forte , are hableto perfourme, or trucly to.conclude, with out helpe thereof. . Now will.we farder;by the wifeand valiant Capitaine,be certified; what helpe he hath, by the Rules of Arithmeteke-in one of the Artes to,him appertaining: And ofthe Grekes named Texte. Thatis,, the Skill of Ordring Souldiers in Battell ray after the beft maner.to all purpofes. This:Artfo much dependeth vppon Numbers vie,and.the Mathematical, that Alianus ( the beft writer therof, ).in his worke,to thie Emper our Hadrianus., by his perfection, in the Mathematicals,(beyng greater, then-other before him had,).thinketh his booke to paffe all other the excellent workes,written of that Art,vnto his dayes.For,ofit, had written Aimess : Cyneas of iz heffaly: Pyrrhus Epirota:and Alexander hisfonne:Clearchus: Paufanias: Euangelus: Polybius,familier frende to Sctpio - Eupolemus:1 hicrates , Poffidonius;and very many other worthy Capitaines , Philofophers and Princes of Immortall fame and me- mory: Whofe fayreft foure of their garland (in this feat ) was _4rithmetike : anda litle perceiuerance,in Geometricall Figures. Butin many other cafes doth /rith- metike {tand the Capitaine in great ftede. As in proportionyng ofvittayles, for the Army,cither remaining ata ftay :‘orfuddenly to be encreafed with acertaine nuinber of Souldiers:and for a certain tyme.Or by good Artto diminifh his com- pany,to make the viduals longer to ferue the remanent, & for a.certaine determi- ned tyme: ifnede fo.require. “And foin fundry' his other accountes, Recke- ninges,Meafurynges,and proportionynges,the wife,expert,and Circum{pect Ca- pitaine will affirme the Science of Arithmetike, to be one of his chief Counfaylors, _direétersand aiders, Which thing ¢by good meanes)was euident to the Noble, the Couragious , the loyall , and Curteous John , \ate Earle of Warwicke. Who was a yong Gentleman., throughly knowne to very few... Albeit his lufty valiant. nes,force,and Skill in Chiualrous feates and exercifes: his humblenes,and frende- lynes to all men, were thinges, openly ,of the world perceiued. But what rotes (otherwife,)vertue had fattened in his breft, what Rules of godly and honorable life ss blerequetisand initant Sol life he had framed'to him felfe:whatvicés;(m fometheh lining)norable, tie tooké fe care,to,efchew: what manly yertues , in.other noble men, ¢forithing before his cyes,)he Sythingly afpixed atter,; what prowefles he purpoled-and-ment to.a- with what feats and Artes, he began.to furnithand fraught him felfe ,for: Ne rferuice of his Kyng. and Countrey,both in peace} &iwartiecThefe (Lay). chicue;, the bette his, Heroicall Meditations.» forecaltinges and determinations ,,noltwayne , (J thinke)befide my. felfe,can to, perfecilyand truely, report... Aigd theifore.in-Con> {cience,| count it my part.forth briefly) tohaue put nis Name ,in the Regifter of Fame luemortalls To.our purpofe. This Johz,by one of his actes(befides many other both inka: gland and Fraunce,by.mé,in hun-noted, )..did difclofe his harty-Joue-to vertuous Sciences:and his noble intent,to.excellin Martiall prowefle: When he,with hurhe nett: iciting:got the beft Rules (citherin time patt by, Greke: or Romaine,orin our tune vfed:and new Stritagemes therin deuifed) for ordting: of all Companies, fummes and Numbers of mé;(Many,or few) with onekinde of weapdn,or mo, appointed: with Arullery,or without:on horfebacke, or on fortes: to.giue,or take onlet; to{eemimany, being few,:,tofeem few , beingmany, To. mardéhe in battaile orlornay: with:many {uch feates,to. Foughtren field,Skarmouth,, or Ambufhe appartaining: And of all thefe,liuely defignementes.(moft curioufly) to bein velame parchement defcribed: with Notes & peculier markes,as the Arte requireth:and all chefe Rules,and defcriptions Anthmeticall , inclofedina riche Cafe of: Gold, he vied to weare abouthis necke sas his Iuell moft, precious , and: ¢ honor, preferment, & procuring efvertue (thus,! This noble Earle. dyed Anno. 19 ¢ 4 ‘Counfaylourmoft trufty....,Thus,_4rithmetike,ot him,was fhrynedin gold: OF fkarfe os 4. “Numbers frute, he had good hope. _Now, Numbers therfore innumerable ini janine net ‘ hauing no if Numbers prayleshisthrynefhallfinde. Lyliicrsoatbad neers ready fie by his What nede 1, Gor farder profe.to you) of the-Scholemafters: of Iuftice;to, Wife: Daugh- require teftimony:how nedefull, how frurefull, howfkillfull a thing Arishmetike: ter to the Duke of So- is? | meane,the Lawyers ofall fortes.. Vndoubredly,the Ciuilians,can meruaylouf merfer, ly declare:how,neither the Auncient Romaine lawes , without good knowledge of Numbers art,can be perceiued.: Nor (Luftice in infinite Cafes )withoutdue pros portion, (narrowly confidered,)is hable to be executed. How Iuftly, & with great knowledge of Aste,did Papinianus infticute alaw of partition , and allowance, be= twetie mail and wife after'a ditiorce?But how Accur ins, Baldus, Bartolus,Jafon,Alex- ander andfinalby Alciatus; (being otherwife,notably well learned)do iumble.gefle, anderre,fromthe xquity art andIntent of the lawmaker : rithmetikecan detect, and conuince;,andiclerely, make the truthto.fhine., || Good Bartolus ,tyredin the examining:é proportioning of the matter:and with Accurfius Glofle, much cums: bred:burftout,and fayd:Nwl eff iz tota libro. hac gloffadifficilior : Cuins computation nem nec Scholaftict nec Doétores intelliguat. eyes, Thatis: In the-whole booke 9 there isno Gloffe harderthen cea accoumpt or -reckenyng , neither the Scho- lers mor the Dottours onderfland.tec. “What cart they fay of Iwhanus law , St ita Scripium.¢rc.Of the Teftators will iuftly performing, betwene the wife , Sonne. and datighter ¢ How can they perceiue the #quitie of Aphricanus , Arithmeticall Reckéning,where he treateth of Lex Falcidia? How.can they déliver him,from his Reprowers : and their maintainers : as oavnes , Accurfius Hypolitus and Alciatus? How Iuftly and artificially, was Africanus reckening made? Proportionating to the — Sommes bequeathed,the Contributions ofeche part“ Namely,for the hundred prefently receiued,17 — .”’ And for the hundred, receitied after cen monethes,12 Pareles el OF ?V als . OEM wilod .si! BY it . = which make the 30; which were to be cOstibuted by the legataries to the heire« 3 Bia SHOP UII OOD ra VO tae oad. | Ft Instice. “Ss & John Dee¢ his Marhematicall Preface. For, what proportion,106:hath:to.75:thefaune hath 17 + to12 4: Which is Sef- quitertia:thatis,as 4,to 3.which itiake 7. Wonderful many places, in the Giuile law require an'expert Arithméticien,to vnderftand the deepe Iudgemét,8¢1 ft de- terminatidof the: Atincient RGinaitie Lawmakers . ‘But much more expért ought he tobel who thould be fable, todeécide wich equitie,the infinite ‘Variette of Cafes; which do,6t may happen’, vader euery ond of thofe lawes and ordinances Giuile: Hereby,eaely, ye iiay tiow conieduire! thapin the’C anon law: ahd in the lawes of che Realme (which with vs , beare the chief Authoritie?) ; fafttce and ¢- quity mightbe greately preferred,and fkilfully executed, through dite {kill of A- rithmetike,atid proportions appertainyng. \° The worthy Philofophets , and pru- dent lawmakers(who haue written many bookts De Republica: How the bet ftate of Common wealthes might be procured and inainteined, ) haue very well deter- mined of luftice?!{ which , not onely’, is the Bafe ‘and foundacion of Common weales:buralfo the totall perfection of all out Workes, words, and thoughtes: )de- fining ito be that vertue,by which}to euery one,is reridred, thar to‘ him appertai- neth. ‘God challengeth rhis at our handeés,to be hondred'as Gods to beloued, as afathers to be feared as a Lord & matter.’ Our neighbours proportid,is alfo pref- ctibedofthe Almighty lawmaker-which is, todo to other, eutn as we would be done vnto: 'Phele proportions, are in Iuftice neceffary:in duety commendable: and of Comthon Wealthesothe life,ftrength, ftay and florifhing. | C4r7fforlein his . Ethikes(to fatcly the fede'of Tuftice,and light of direction, to yfeand execute the . fame) was fayiie to fly to the perfection, and power of Numbers : for proportions ‘ Arvirhmeticail and *“Geometricall. P/ato in his bookc called Ee (‘which boke, , in is the Threafwry ofall his doetrine) where, his purpofe is,to feke a Science; which, when a man had it,perfectly: he might feme,and fo be,in dede,W7fe. He briefly,ot other Sciensesdifcourfing,findeth them, not hable to bring itto paffe : But of the , |) = SeienetiofNumbers,he fayth: 1/a,qua numerim mortalinm generi dedit,id profecto ef- tied. Denwtnutentaliquem,magu quam fortunam, ad falutem nostram, hoc munus nobis arbisvorcontulifeeye . Naim ip{um Lonorum omninm Authorem, cur non maximt bont, Ppudcnsiedicd, cin[im arbitramare T hat Science ,verely which hath taught mane Rynide nihiber, (hall be able to bryng it to paffe. And,l thinke,a certaime God, rather then fortune,to bane giuends,this gt tt, for. our bliffe .... Fors why fhould ‘we not ladge bimwho is the Author ofall good things jto be alfoithe caufe-o the greatest. good thyng namely,Wifedome® °There;at length, he proueth Wifedome tobe atteyned , by good Skill of Numbers. With which great Teftimony, and the manifold profes , andreafons jbefore expreffed ; you may be fufficiently and fully perfuaded sof the perfect Science of Arithmetite,to make this accounte: That of ali Sciences,hext to Fheologie,it is moftdinine nok pure,moft ample and general, mot profeunde,, moft{ubtile, mof commodious and moft neceflary »\: Whole next Sifter, is the Abfolute Science of Magnitudes:of which (by the DireGtion and aide of him, whole Magnitude is Infiniteand of vs. Incomprehenfible ) Lnow.en- tend , {6 to write , that both with the catulitude,and alfo. with the Magnitude of Meruaylous and frutefull verities , you. ( my frendes and Countreymen ) may be ftird vp , atid awaked, to behold what certaine Artes and Sciences, (to.our.vn- fj seakable behofe)our heanenly father, bath forvs prepared, and reuealed,by fun- dry Philofophers and Mathematiciemse. dog : , Oth, Number and Magnitude , hauca certaine Paging fede, (as it were, ) ofan incredible property: and ofman, neuer hable, Fully, to bedeclared . Of. Number , an Vnit,and of Wtaenitude,a Poynte,doo feeme to be much like Origi- {8 nall lohn Dee his Mathematicall Preface. nail caufes : But the diuerfitie neuertheleffe,is great. We defined an Vwi, to beathing Mathematicall Indiuifible : A Point, likewife, we fayd to be a Ma- thematicall thing Indiuifible. And farder , thata Pointmay haue a certaine de- termined Situation: that is, that we may afligne,and prefcribe a Point,to be here, there, yonder. &c. Herein, ( behold ) our Vnit is free,and can abyde no bon- dage,or to be tyed to any place,or feat:diuifible orindiuifible. Agayne , by rea- - fon,a Point may haue a Situation limited to him: a certaine motion,therfore (to a place,and from a place) is to a Point incidentand appertainyng. Butan Ywit,can notbe imagined to haueany motion. A Point,by his motion, produceth ; Ma- thematically,a line: (as we fayd before which is the firft kinde of Magnitudes,and moft imple: An /#it,can not produce any number. A Line, though it be produ- ced of a Point moued,yet,it doth not confift of pointes : Number. though it be not produced of an Vit , yet doth it Confift of ynits,asa materiall caufe. But formally, Number,is the Vnion, and Vnitie of Vnits . Which vnyting and knit- ting,is the workemanfhip of our minde:which,of diftiné and difcrete Vnits , ma- keth a Number: by vniformitie,refulting of a certaine multitude of Vnits.And fo, euery number, nay haue his leaft part,giuen :nainely,an Vnit: But not of a Magni- tude,(no,not of a Lyne,)the leaft part can be giné:bycaufe,infinitly, diuifion ther- of, may be conceiued. All Magnitude,is either a Line,a Plaine, ora Solid. Which Line,Plaine,or Solid, ofno Senfe,can be perceiued, nor exactly by had (any way) reprefented:nor of Nature produced: But, as ( by degrees ) Number did cometo our perceiuerange: So,by vilible formes,we are holpen to imagine, what our Line Mathematicall, is. What our Point, is.So precife, are our Magnitudes , that one Line is no broader then an other:for they haneno bredth : Nor our Plaines haue any thicknes.Noryet our Bodies,any weight-be they neuer fo large of dimenfié. Our Bodyes,we can haue Smaller, then either Arte or Nature can produce a- ny : and Greater alfo, then all the world can comprehend. Our leaft Mag- nitudes,can be diuided into fo many partes, asthe greateft. As,a Line ofan inch long, ( withvs) may. be diuided into as many partes, as may the diame- ter of the whole world, as Eaftto Welt : orany way extended: What priui- ledges , aboue all manual Arte;and Natures might,hauie our two Sciences Ma- thematicall?to exhibite,and to deale with thinges of fuch power, liberty, fimplici- ty,puritie,and perfection? And in them,fo certainly,fo orderly,fo precifely to pro- cede:as,excellentis that workema Mechanical ludged ; who nereft can approche to the reprefenting of workes, Mathematically demonftrated? And ourtwo Sci- ences,remaining pure,and abfolute,in their proper termes,and in their owne Mat- ter:to haue,and allowe,onely fuch Demontftrations , as are plaine ,certaine , vni- uerfalljand ofan xternall veritye This Science of \Magnitude,his ptoperties,con- ditions,andappertenances : commonly,now is,and from the beginnyng , hath o all Philofophers,, beri called Geometrie.. But,veryly,witha name to bafe and {cant, fora Science of fuch dignitie and amplenes. . And,perchaunce, thatnamie,by c6- ‘mon and fecret confentjofall wifemen, hitherto hath ben fuffred to temayne:that it might carry with ita perpetuall memorye, of the firft and notableft benefite, by that Scién¢ée; to'common people fhewed : Which was’, when Boundes and meres of land and ground were loft, and confounded {as in Eeypt,yearely with the ouer- flowyng of Nilus,the greateft and longeft riuer in the world.) or , that ground be- queathed, were to be afligned: or, ground fold, were to be layd out: or (when dif- order preuailed)that Commés were diftributed into feueraltics.For, where, vpon thefe & fach like occafids,Some by ignorace; fome by negligéce, some by fraude, and fome by violence, did wrongfully limite,meaftre, en croach,or challenge ( by aij. -~pretence N; umber's E>) Geometri¢e lohn Dee his Mathematicall Preface. pretence of inft content, and:meafure) thofe landes and groundes : great loffe,dif- quictnes,murder,and warre did(full oft)enfue: Till, by Gods mercy, and mgfis In- duftrie, The perfect Science of Lines,Plaines, and Solides (like a diuine Iufticier,) gaue vnto eucry man, his owne. The people then,by this art pleafured,and great- ly relieued,in their landes iuft meafuring:& other Philofophers, writing Rules for land meafuring. betwene them both,thus,confirmed the name of Geometria,thatis, (according to the very etimologie of the word)Land meafuring. Wherin, the peo- ple knew no farder,of Magnitudes vie,but in Plaines:and the Philofophers,of thé, had no feethearers, or Scholers-farder to difclofe vnto , then of flar, plaine Geonze- trie. And though,thefe Philofophers, knew of farder vfe,and belt vnderftode the etymologye of the worde,yet this name Geometria,was of them applyed generally to all fortes of Magnitudes : vnleaft, otherwhile, of Plato, and Pythagoras : When *Plate.7.de they would precifely declare their owne doctrine, Then,was * Geometria, with Rep them, Stadium quod circa planum-verfatur. But, well you may perceiue by Euclides Elementes , that more ample is our Science , then to meafure Plaines:and nothyng leffe therin is tought(of purpofe)then how to meafure Land.An other name,ther- fore,mutt nedes be had, for our Mathematicall Science of Magnitudes : which re- gardeth neither clod,nor turff:neither hill,nor dale: neither earth nor heauen: but is abfolute «Afegethologia:notcreping on ground , and daffeling the eye, with pole 2» perchesrod or lyne-burtliftyng the hart aboue the heauens,by inuifible lines , and €# »> smmortall beames:meteth with the reflexions, of the light incomprehenfible: and 2» fo procureth Ioye,and perfeCtion vnfpeakable. Ofwhich true vie of our Wege- thica,or Megethologia, Diuine Platofecmed to haue good tafte,and indgement:and (by the name of Geometrie ) fo noted it:and warned his Scholers therof : as,in hys feuenth Dialog, of the Common wealth,may evidently be fene.. Where (in La- tin thus itis:right well tranflated: Profecto,nobis hoc non xegabunt, Quicung, vel pan- iulum quid Geormetria guftarunt, quin hac Scientia , contra, omnino fe habeat , quar de ea loquuntur , qui in ipfa-verfantur .In Englifh,thus. » Verely( fayth Plato }whofoener haue,( but enen very litle tasted of Geometrie will not denye vnto vs , this : but that this Science,is of an other condicion quite contrary to that , which they that are exercifed init , do Speake ofit. And there it followeth, of our Geometrie, Quod quaritur cognofcendi ilius gratia,quod femper eit non Cy eins quod eritur nando, Co interit.Geometria,cius quod es femper, Cognitio eft. Attollet igitur(d Generofe vir) ad V eritatemanimum:atg,ttaad Philofepbandew preparabit cogitationem,vt ad {upera con- uertAMmUs ‘ag ee gers quam decet,ad infertora deijcimus. Gc, Quam maximeigitur prasiptendum eft,vt qui praclari{simam hanc habitat Civitatem,nullo modo,Geometriam _ fperaant. Nam & que prateripfius propofitum, quodam modo effevidentur,paud exigua fuot.crc. It muft nedes be confefled(faith Plato) T hat ( Geometrieis learned : for the kyowyng of that , whichis ener:and not of that, which in tyme both is bred and is brought to an ende.¢7-c.Geometrieis the knowledge of that which is euers laftyng.It will lift vp therfare(O Gentle Syr_) our mynde to the Veritie: and by that meanes it will prepare the C-hought,to the P hilofophicall loue of wifdome: that we may turne or conuert toward heauenly thingestherh mynde and thought which now, otherwife then becommeth:ds ;Wwe cast down on bafe or inferior things.exc. Chiefly, therfore, Commaundement muft be ginen , that [uch as do inhabit this moft honorable ( itie by no meanes, despife Geometrie. For euen thofe thinges [dene by #* which in manner, Jeame to be , befide the purpofe of Geometrie; are of n6 John Dee his Mathematicall Preface, no fniallimportance . ¢x-c.And befides the manifold vfes of Geometrie, in matters oe eer to warre,he addeth more, of fecond vnpurpofed frute, and commo: ditye,arriling by Geometrie Saying :Scimus quin ctiam,ad Dife ciplinas ommes facilius per difcendas,wuterelfe omnino attigerit ne Geometriam aliquis,an non. ¢c. Hancergo Do- rinam, fecundo loco dif{cendam Iuuenibus flatuamus. That is. By ta Howe know, that for the more eafy learnyng of all Artes it importeth much , whether one haue any knowledge in Geometrie,orno.¢rc.. Let vs therfore make an ordte nance or decree , that this Science , of young men fhall be learned in the fecond place. This was Diwine Platohis ludgement,both of the purpofed , chief, and perfect vie of Geometrie: and of his fecond,dependyng , deriuatiue commodities. * And for vs,Chriften men,a thoufand thoufand mo occafions are, to haue nede of | the helpe of * Wegethologicall Contemplations : whetby,to trayne our Imagina- , 7+: r tions and Myndes,by litle and litle,to forfake and abandon, the groffe and corrup- —— i tible ObieGes,of our vtward fenfes:and to apprehend , by fure doctrine demon- fake e ss if ftratiue, Things Mathematicall. And by them, readily to be holpen and con- ‘egy bly SAN ducted to conceiue , difcourfe , and conclude of things Intelletual, Spirituall, of Geometrie. sternall,and fuch as concerne our Bliffe euerlafting : which, otherwife ( without | Speciall priuiledge of Illumination, or Reuelation fro heauen ) No mortall mans wyt(naturally )is hable to reach ynto,or to Compafle, _ And,veryly, by my {mall Talent(from aboue)I am hable to proue and teftihe,thar the litterall Textjand or- der of our diuine Law,Oracles,and Myfteries,require more fkill in Numbers,and Magnitudes - then(commonly) the expofitors haue vttered - but rather onely (at the moft){o warned : &fhewed their own want therin.(To name any, is nedeles: and to note the places,is,here,no place: But if be duely afked,my anfwere is rea- dy.) And without the litterall,Grammaticall,Mathematicall or Naturall verities of fuch places , by good and certaine Arte,perceiued,no Spirituall fenfe ( propre to | _ thofe places,by Abfolute 7 Aeologie)will thereon depend. No man,therfore,can ,, £y doute , but toward the atteyning of knowledge incomparable , and Heauenly Wifedome: Mathematicall Speculations,both of Numbers and Magnitudes: are meanes, aydes, and guides:ready, certaine, and neceflary. From henceforth,in , this my Preface,will I frame my talke,to Plato his fugitiue Scholers:or, rather, to fich, who well can, (and alfo wil,)vfe their veward fenfes,to the glory of God,the benefite of their Countrey,and their ownefecret contentation, or honeft prefer- ment, on this earthly Scaffold. To them,I will orderly recite, defcribe & declare a great Number of Artes , from our two Mathematical fountaines , deriuedinto the fieldes of Nature. Wherby, fuch Sedes, and Rotes ,as lye depe hydinthe . = grotid of Nature,are refrefhed,quickened,and prouoked to grow, fhote vp, floure, ee and giue frute,infinite,and incredible. And thefe Artes, fhalbe fuch as vpon Mag- | nitudes properties do depende,more,then vpon Number. And by good reafon we may call them Artes,and Artes Mathematical Deriuatiue : for ( atthis tyme)I 4s Arte. Define An Arte,to be a Methodicall coplete Doétrine,hauing abun- dancy of fufficient,and peculier matter to. deale with;by the allow- ance of the Metaphificall Philofopher: the knowledge whereof,to humaine {tate is neceffarye.Andthat Iaccount, .An Art Mathemati- sr _ call deriuatiue , which by. Mathematicall demonftratiue Method; ewer 3 in Nubers , or Magnitudes,ordreth and confirmeth his dodtrine, as much &as perfectly , as the matter fubiect will admit. And for that, ea 95: >> See ; bia a.tij. Tentend ae Sd lohn Dee his Maathemati¢all Preface. AMechani- Lentend tovfe the nameand propertie of a Mechanicien,otherwife,then (hitherto) $10M%e Geemetrie vulgar. it hath ben vfed,I thinke it good, ( for diftin@ion fake) to giue you alfoa brief def- ctiption, what I meane therby. A Mechanicien,or a Mechanicall work- man is he , whofe fkill is , without knowledge of Mathematicall demonftration , perfectly. to worke and finifhe any fenfible worke, by the Mathematicien principall or deriuatine, demonftrated or de- monftrable. Full well Iknow,thathe which inuenteth, or maketh thefe de- monftrations,is generally called 4 peculatine Mechanicien : which differreth no- thyng froma Mechanicall —Mathematicien. So,inrefpect of dinerleactions,one man may haue the name of fundry artés:as,fome tyme,ofa Logicien , {ome tymes (in the fame matter otherwife handled) ofa Rethoricien. Ofthefe trifles,I make, (as now, in refpect of my Preface, fall account: to fyle thé for the fine handlyng offubtile curious difputers.In other places , they may commaunde me,to giue good reafon:and yet,here,I will not be vnreafonable. Firft,then,from the puritie,abfolutenes,and Immaterialitie of Principall Geo- metrie; is that kinde of Geometrie deriued , which vulgarly is counted Geometrie : andis the Arte of Meafuring fenfible magnitudes, their iuft quatities and contentes . This, teacheth to meafure, either at hand: and the practifer, to beby the thing Meafured - and fo ,by due applying of Cumpafe, Rule, Squire, Yarde-Ell,Perch,Pole,Line,Gaging rod,(or fuch like inftrument)to the Length, . Plaine,or Solide meafured, “to be certified, either ofthe length, perimetry, or di- - ftancelineall : and this is called, ecometrie . Or*to be certified of the content of any plaine Superficies : whether it be in ground Surueyed, Borde, or Glaffe mea- . fured,or fuch like thing : which meafuring,is named Embadometrie.*Or cls to vn- derftand the Soliditie,and content of any bodily thing : as of Tymberand Stone, or the content of Pits,Pondes, Wells, Veffels. {mall & great,ofall fafhions. Where, of Wine,Oyle,Beere,or Ale veflells,fec,the Meafuring,commonly, hath a pecu- lier iame-and is called Gagimg . And the generall name of thefe Solide meafures, . is Stereometrie. Or els,this vulgar Geometric, hath confideration to teach the prac- .. tifer; how to meafure things,with good diftance betwene him and the thing mea- _ fired : and to vnderftand thereby, either *how Farre,a thing {eene(on land or wa-. " ter)iéfrom the meafurer: and this may be called Apomecometrie: Or,how High or depe, aboue or ynder theleuel of the meafiiters ftMding,any thing is,which is fene - on land orwater, called Hypfometrie.*Or, itinformeth the mealfurer , how Broad any thingis,which is in the meafurers vew: {0 itbe on Land or Water, fituated:and - may be called Platometrie. Though I vfe here to condition, the thing meafiired,to be on Land, or Water Situated: yet, know for certaine, that the fundry heigthe of Clondés, blafing Startes, and of the Mone, may(by thefe meanes)haue their di- ftances from the earth : and, of the blafing Starres and Mone,the Soliditie (afwell as diftarices to be meafured: But becaufe,neither thefe things are valgarly taught: nor ofa common practifer fo ready to be executed = L rather let fuch meafures be reckened incident to fome of our other Artes, dealing with thinges on highjmore purpofely, then this vulgar Land meafuring Geometrie doth : as in Per/pectine and _Astronomit, OC» F thefe Feates ( farther applied ) is Sprong the Feate of Geodefie , or Land Meafuring: more cunningly to meafure & Suruey Land, Woods, and Waters, a farre of: More'cunningly,I fay : But God knoweth (hitherto) in thefe Realmes of England and Ireland ( whether through ignorance or fraude > Lean not tell, in euery particular ) how great wrong and iniurie hath (in my time)bene _— | Y ™~% John Deéé his Matheiniandal Preface. by vntrue meafuring andfurireying of Land or Woods\any way 7And; this Tam furetthat the Value of the difference, berwene the trash and fiich Suirueyeswould haue beric hable'to haue foiid \(foreuer) in eche ofourtwo Vniverfities,an excel- lent Mathematicall Reader:to eche,allowing (yearly)ahundred Markes-of lawful * moneyot this:realme:whichjimdede,would{eme requifit,here,;to be had ( though by otherwayes prouided for)aswelljasithefamous Vniueriitieof Paris; Hath two Mathematical Readers ;.amdechejtwo hundreth French Crownes yearly, of the French Kinges magnificentliberalitie onelys .Now,againe,to dur purpofe retut- hing: Moreouer, of the formerknowledge Geometricall,are growen the Skills of Geographic , Choroeraphie.; H ‘ydrographie , and Stratarithmetrie.. BW 251 Ge ograp hie reacheth wayes,by which, in fadry form eS, (a5 Spharike,Plaine or other), the Situation of Cities, Townes, Villages, Fortes, Caftells, Mountaines, Woods,Hauens.Riuers, Crekes,& fuch other things,vpo.the outface ofthe earth- , ly Globe (either in the whole,or in fome oiiactoall rather and. portion therof co» 5, tayned )may be defcribed and defigned, in comenfurations Analogicallto, Nature ,, and veritie:and moft aptly to our vew,may be reprefented.Of this Arte how great ;, pleafure,and how manifoldecommodities do come vnto vs,daily and hourely: of moft men, is perceaued.. While,fome, to beautifie:their Halls,Parlers, Chambers, Galeries,Studies,or Libraries with: other fome.for thinges patft, as battels fought, — carthquakes,heauenly fyringes,& {uch occurentes,in hiftories mentioned: therby liuely,as it were,to vewe the place,the region adioyning,the diftance from vs: and fuch other circumftances.. Someother,prefently.ro.vewe the large dominion of the Turke : the wide Empire of the Mofchoutte. and the licle. morfell of ground, where Chriftendome(by profeffion)is certainly knowen. Litle, fay,in refpede of the reft, &c. Some,either for their owne iorneyes directing into farre landes: orto'viiderftand ofother mens tratiailes . To-conclude, fome, for oné purpofe : and-fome,for amother, liketh loueth,gettethjand vfech; Mappes, Chartes,& Geo- graphidall Globes. Ofwhofe-vfe, to fpeake fufficiently, would ‘require a booke ' ‘Chorographie feemeth to be an. vnderling,and a twig, of Geographie: and yet neuertheleffe, is in practife manifolde, and in-vfe very ample... This tea- », cheth Analogically to defcribe a {mall portion or circuite of ground, with the con- tentes; not regarding what commenfuration it hath tothe whole, or any parcell, withoutit, contained . Burinthe territory or parcell of ground which it taketh in ;, ‘hand to make defcription of,itleaueth out (or, vndefcribed) no-notable jor odde »s thing, zboue the ground vile. -Yeaand fometimes , of thinges ynder.ground, », gcueth fome peculier marke : or warning : as of Mettall mines, Cole pittes,Stone >. - piesa <¢,.: hus, a Dukedome,a Shiere,a Lordfhip, or lefle, may bedefcribed 2» iftincily.. | But marueilous pleafant, and profitable itis , in the exhibiting to our eye,and commenturation, the plat of a Citie, Towne, Forte, or Pallace, in true Symmetry.: notapprochingtoany of them : and out of Gunne fhot.&c.. Hereby, © the —drehitect may furnifhe him felfe,with ftore of what patterns heliketh : to his “ Ser rigcaeaay: »euen in thofe thinges which outwardly are proportioned: either unply in.them {elues : or re{pectiuely,to Hilles, Riuers, Havens, and Woods ad- vee ., Some alfo,terme this particular. defcription of places , Tepagraphie... ey ydrographie,delinereth to our knowledge’; on Globe ‘or iti Plaine; ,, the pérfeét Analogical defcription ofthe Océan Sea coaftes, through the whole » world : erin the-chiefe td sich partes thercof:: with thellés and chiefe >» 3 il}. paticular 33 3? “ 5? 43 John Deéehis Matheniandall Preface. particular places of daungersyconteyned withinthe boundes,and Seacoattes de deribed. 4s, of Quickfandes,Bankes,Pittes,R ockes,Races,C ountettidesWhrle- pooles.&e.).;,Thisy dealeth with the Element ofthe waterchiefy<.as Geographie did principally take the Element-of the, Harthesdefcription ( with: his. apperte- nances)), to -taske-».>/Andbefides \thys , Hydragraphie: requireth a particular “a ~ . Regifteroficertaine Landinatkes(wheremarkesmay be had) from the fea;well ha- bleto befkried, in what point oftheiSeacumpafelthey appeare,and what apparent forme,Situation,and:bignes they haue, in refpeite of any daungerous place in the {ea,ornere vnto.ityaflignedsAnd in all Coaftesyivhat Mone,maketh full Sea:and what way, the Tidésand\Bbbes; come and gosthe: 7 ydxographer oughtto recordés The Soundinges likewife,: andthe Chanels wayes: their number,and depthés\or- ' diniarily,atebbe and flud, ought the i hes by obferuation and diligence < Of Meafwying,to haue certainly knowen. \nd nrany other pointes are belonging : to sep i cake se andforts make'a Rutter, by’: of which, I nede not here Cc Speake? as 6Fthed «< the ng poet gat < Byeaufs the lines't fibing,in afy place, vpon Globe or Plaine, rhe 32.pointes of > (wherdF, fcarfly foure,in 5 ae > haueright knowledge: erof, areno firaightlines, nor Circles.) Of making due pro- jection of a Spliete in plaine.Ofthe Variacion ofthe Comipas , from true Northe: : Afid fochtike matters ( re oa BE ,all’) Fcaue'to fpeake of,in this place: bycaufeslinay feame(al ready)to hane enlarged the boundes,and duety ofan Hy- dographer, much tore,then'any man (to this day)hath noted, or prefcribed , Yet ait I welt hiable'to proue,all thefe thinges , to appt : : : , to appertaine, and alfo to be proper to theHydtogtapher. ThethieFvfe ahd ende of this Art, is the Art ofNauigation: Auitit harlvorher diuerfe vfes ¥ een by them to be enioyed, that neuer lacke fight 42 #7 RABAT * © and, : - ¥ ached Stratarithme trie, is the Skill, (appertainyngto the warre 5) by,whicha man,can fet inifigure,analogicall toany Geomefrirall figure appointed; any certaine pRBUSE or fumme'of men:of fucha figure capables (by reafon of the vfuall, {paces etwene Souldiers allowed : and for that , of men,can be made no Fractions. Yet; neuertheles,he can order the giuen fumme of men_.,. for the greateft{uch figure, : ° _ thar of then, @be ordred jand certifie, of the ouerplus: (if any beJand of the next « certaine fummegwhich,with the dtierplus, will admit afighreexacly proportional patheheuteahened. By which Skill alfo,ofany army or company of men ‘(the ~ figure & fides of whofe ordetly ftanding, or artay,is ktrowen jhe i$ able to expreffe T be aiffe- 3 rence be- >3 twene Strae 5s tarithme- trieana T aéticie, 32 a? 33 . theiuft riuinbei ofmen, within that figure conteined: or( orderly } able‘to be con- % Notes teined* And'this igureand fides therdf, heis Hable to know : either beyng by, ahdiat hatid:6t 4 fatre of.’ « Thuis farre, ftretcherh the deftiption and property of stvatarithinetriefothcient forthis tyme and place. ~ Tt differreth from the Feate Tactical, De atiebus infiruendis.bycaufe, there, is neceflary the wifedome and fore- fightito what parpofe he fo Ordreth thé men ; and Skillful hability , alfo’, for any oecafionzof purpole, to deuifé and vfe the apteft and moft neceflary order’ array aeabiphicest his Company and Summe of men . By figure,{ meane: as,cither ofa Per feet SqaresT riangle , Circle, O wale , long {quare , (of the Grekes it is‘called Erero- wckes:) Rhumbe, Rhomboid, Lunular, Ryng, $ erpentine, atid fuch other Geometrical feuresWhich,in wartes,hateben , and are tobe vied : for commodioufhes , ne- ceffityymndauauntage &c. Arid no fall fkill ought he to have’, that fhould make true report,or nere the truthjof the numbers and Summes,of footemen or horfe- * Men, in the Enemyes ordring . A farre of, to. make an eftimate , betwene.nere sermes of Mare and Leffe,is nota thyng very rife, among thofe thatgladly would gees ot) Shilton gh shed sie eo ‘ 2 ) Q a s ji + * 7 Ag et) eet y ei a4 » oe * - ae a lohn DeehisMathematicall?P tetace. do it. » Great pollicy,: may be vied ofthe Capitaines,(attymesfete,andin places zp, corenient)as to, vie Figures , which make greateft: thew ;offo many as he haths |, Frat. ; 7 ll find and.viing the aduauntage of the three kindes of viuall{paces: ( betwenefootemen ire peter ot horfemen)to take the largeft: or whien he wouldfeme to haue few, (beyng ma: we on xo ny: )contrarywile,in Figure,and fpace.,, The Herald,Purfeuant, Sergeant Royall, na sins 2 Capitaine, or who focuer is carefull tocome nere thetruth herein, befides the fr-nhamhecis Figures knowne (i ludgement of his expert eye,his {kill of Ordering Tactical sthe helpe of his Geo~ side: and ~angles) metricall inftrument:Ring, or Staffe Aftronomicall: ( commodioutly framed for vind whe, Repl cariage and vfe ) He may wonderfully helpe-him felfe,. by perfpectine.Glafles.Im te. bingy. 2 ‘ : ‘ : can ferue.drc».And which, (I truft) our pofterity, will proue more {killfulland expert , and togreater’ 30170" wh finse® : : range to deale thus urpofes , thenin thefe dayes, can(almoft) be credited;to be poffible. ; Sasorel with Ae Thus haue L lightly pafled ouer the Aruficiall Feates,chiefly dependyng vpon npr PM vulgar Geometrie : & commonly, and generally reckened vaderthe name of Geome- ats Me yon trie. . But there are other(very many) Wethodicall Artes, which, declyning from: like Geametrie the purity , fimplicitic,and Immateriality,of our Principal Science of Magnitudes: ps do yet neuertheles vie the great, ayde., direGtion: , and Method of the fayd rincipall Science, and haue proprenames, and diftinc :. both from the Science of Geometrie , ( from which they are deriued)and one from: the other... As .Per- fpectiue, Aftronomie’, Mufike, Cofmographie, Aftrologie,Statike; Anthropographie, Trochilike, Helicofophie, P neumatithmie, Me- nadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Archi- tecture, Mauiyanin , Vhaumaturgikeiand Archemattrie, 1 thinkeit A teh. < “213 V hue ' \ o matical which. demontftrateth the m aner,and.properti es of all Ra» *- +e 8 & | fofind rsphe our eye is deceiued,and abufedsas, while thie eye weeneth a rot és ; b,j. geth lohn: Deé his Mathematicall Preface. geth'a plaine Square, to berofid:fuppofeth walles parallels:to approche, farte of: rofe and floure parallels,the one to bend downward , the other torife vpward,at a little diftance from you: » Againe , of thinges being in like fwiftnes of mouing , to thinke the ncrer,to moue fafter:and the farder, much flower. Nay, of two thinges, wherofthe one(incomparably)doth mouefwifter then the other , to deme the flower to moue very fwiftj&the other to ftand:what an error is this,ofour eyes OF the Raynbow, both of his: Colours,of the order of the coloursjof the bignes of it, the place-and heith of it;(&c )to know the caufes demonftratiue;is it not pleafant, is irnotneceflaryzof two or three Sonnes appearing: of Blafing Sterres : and fuch like thinges': by naturall caufes , brought to pafle | (and yet ncuertheles , of fardet matter, Significatiue’) isit nor commodious forman to know the very true caufe, & occafiow Natural’? Yea,rather,is it not,eteatly, againft the Souerainty of Mans nature, to be fo ouerfhot‘andabufed , with’ thinges ( at hand’) before his eyes ¢ as with a Pecockes tayle3and a‘ Doues necke“: or a Whole ore, in water, hol- den to feme broken . “Thynges; farre ff to feeme nere: and nere, to feme A marueilous Glafse. (5 farre of. Small thinges , to feme' great’: arid great, to feme finalf .’ One man, to femean. Army: .°° Or a man t6' be curftly affrayed of his owne fhad- dow? .2/-Yea;fo much,to feare,thatjif you-being (alone) nerea certaine glafle , and proffer, with dagger orfword,to foyneattheglaffe , you fhallfuddenly bemoued to pine backe(in maner) by reafon of amImage ; appearing in the ayre, betwene you Sethe glaffe, with like hand, Ror One reer, & with like quicknes , foyning at your very tye likewife as you do atthe Glafle= ° Straunge,this is,to heare of: but more meruailous to béhold; then thefemy wordes can fignifie. And neuerthe- leffe’ by demonftration Optical, the order and caufe therof, 1s certified: even fo,as tlic effectis confequent.: Yea,chusmuchmore,dare I take vpon me,toward the fa- tiffying of the noble courrage,thatlongeth ardently for the wifedome of Caufes Natnitall:as.to let him vnderftand;thatjin‘London, he may with his owne eyes, hane profe of that, which I hauefayd hereins A’ Gentleman, (which, for his good feruice; done to his Countrey, is famous and honorable: and forfkill in the Ma. ticmaticall Sciences, and Languages, i$ the Od man of this land. '&c. ) euen-he,is hable:and(I am fure)will, Very willirigly;lecrhe Glaffe; andprofe be fene-and {fo I (here) requeft him ; for thie encreafe‘of wifedome.,in the honorable ; and for the {topping of the mouthes malicious ;, and reprefling the arrogancy of the ignorant. Ye fiiay eafily geffe , what | meane. - This Art of Per/pectine,is of that excellency, dtd nay be led,to the certifying,and executing offuch thinges ,as no man would qty Beleuc: without AGuall profe perceiued. I {peake nothing of Naturall Phi- bfopbie, which, without Per /pectine, can not be fully vnderftanded , nor perfectly at- teintdd imto.. Nor; of Afronomie: which, without Per/pectine,can not well be groun- dedaNoruAPrelogies naturally V etiht€d, and avouched. ‘Thar part hereofjwhich dealeth.with GlaffesGyhich name,Glafie,is a generall name, in this Arte;:for any thing, from which,a Bearne-reboundeth )is called Catoptrike - and hathfo many’ v- fes both merueilous,and proffirable: that,both,it would holdimeto long ;:to note therimthe principall conclufions,all ready knowne: Amd.alfo(perchaunce) fome thinges,mightlacke duecredite with you +. And, therby, to.leefe my labor:and you,toflip into lightIudgement*, Before youshaue leatned fufficiently the powte of Natureand Arte... lgslg vasse diiwsrigil oda bris.2ay9 10: Nlowsto procéde: Aftronomiey!s.an Arce Mathematical! ;which deitror ftraeth the diftance , magnitudes , and all narurall motions, apparences,and_palsions propre.to the Planets.and faxed Serres: for any lolin Dee his Mathematical Prieface. any time palt,prefent and to.comezinrefpe& of a certaine Horizon, or withoutrefpect of any Horizon. By'this Arte we até certified of the di- ftance of the Starry Skye,and of eche Planere from the Centre of the Earth: and of the greamnes of any Fixed flarre fene, or Planctesin relpect of the Earthes gteatnes. As , wearefure (by this Arte) thatthe Solidity , Maslines and Body of the Sonne, > conteineth the quantitie of the whole Earth and Sea,a hundred thre {core and” rwo times, leffe by +. one eight.parte of the earth. Butthe Body of the whole earthly globe and Sea,is bigger thémthe body of the Mone, three and forty times lefle by of the Mone. Wherfore the Sonneis bigger then the tone, 7600 + times, leile, by 59 that is, precifely 6940 % bigger then the Mone. Andyet the vnfkillfull man,would iudge them alike biggé. Wherfore,of Necefsity,the one is much farder from vs,then the other. The Sozxe,when heis fardeft from the earth(which,now,in ourage,is, when heis in the 8 -degree,of Cancer)is , 1179 Semidiamcters of the Earth,diftante. And the ~one when fhe is fafdeft from the earth,is 68 Semidiameters ofthe earth and a Thenereft , that the cA¢one com- meth to the earth,is Semidiameters 52+ _ The diftance of the Starry Skye is , fr6 vs,in Semidiameters of the earth2oo8i.<— Twenty thoufand fourefcore; one, and almofta halfe.. Subtra@ from this,che cores nereft diftance, from the Earth: and therof remaineth Semidiameters of the earth 20029. Twenty thoufand nine and twenty and a quarter. «So thicke is the'heauenly Palace , thatthe Péa- netes haueall theirexercifein,and moft‘meruailoufly perfourme the Commatide- ment and Charge to them giuen bythe omnipotent Maieftie of theking of kings. This is that, which \in Genefisiscalled Ha Rukia: Confiderit well. The Semidia- ». meter of the‘earth, coteineth of ourcommon miles 3436 -* three thoufand,foure hundred thitty fix and foure'eleuenth partes of one nyle:Such as the whole earth , and Sea, round about, is 21600. Oneand twenty thoufand fix hundred of our myles.Allowyng for euery degree of the grearelt circle, thre {Core myles. Now if you way well with yourfelfe. butthisJicle parcell’ of frure.. xfronomitall , as con- cerning the bignefle, Diftances of Sonne,Mone, Sterry Sky,and the huge maflines of H. Rakia, willyounot finde your Con{ciencesmoued, with the kingly Prophet, tofing the confeflion of Gods Glory,and fay,.—T he Heanens declare the glos xy of God,and the Firmament (H« Ratie] fheweth forth the workes of bis bandes. And fo forth, for thofe fiue firft taues,of that kingly Pfalme. Well,well,It is time for foie to lay hold oa wifedome,and to Iudge truly.of thinges: and notf{o to ex- pound the Holy word,all by Allegories : as to Neglect the wifedome, powreand . Goodnes-of Godin; and by his Creatures , and Creation to be feen.and learned. By parables and ‘Analogies of whofe natures and properties, the courfe of the Ho- ly Scriprure,alfo)declarechiigvs very many Myfteries, The whole Frame of Gods “Creatures, (which isthe whole world,)is to vs,a bright glaffe: from which, by re- flexion ,reboundcth to ourknowledge and perceiuerance, Beames ; and Radiati- ‘ons:reprefenting the Image of his Infinite goodnes, Omnipotécy,and wifedome. “And we therby , are taught and petfuaded to Glorifie our Creator,as God:and be thankefull therfore. ‘Could the Heatheniftes finde thefe vfes,of thefe moft purc, ‘beawtifull,and Mighty Corporall Creatures:and {hall we,after that the true Sonne oftightwifeneffaais rifen aboue the Horizon, of our temporall Hemifpherie,and hath fo abundantly {treamed into out hartes, the direét beames of his goodnes , mercy, _ and graces Whofeheat All Creatures feele : Spirituall and Corporall: se and Note, lohn Dee his Mathematical! Preface. Inuifible,S hall we(I fay)looke vpon the Heauen}Sterres,and Planets,as an Oxecnd an Afle doth: no furder carefull or inquifitiue,what they are: why were they Cre- ated, How do they execute that they were Created for?S eing,All Creatures, were for our fake created :_ and both we,and they,Created, chiefly to glorifie the Al- mighty Creator: and that, by all meanes,to vs poflible. Nolite ienorare({aith Plate in Epinomis) Aflronomiam, Sapientifsimi quiddameffe. Be ye not ignorant yA, {ftro» nomie.to be a thyng of excellent wifedome, » Affronomie,was to vsjtrom the be- pinning commended, and in maner commaunded by God him felfe.In afmuch as he made the Some, Mone,and Sterres,to be to vs,for Signes,and knowled se of Sea- fons,and for Diftinctions of Dayes,and yeares,.. Many word:snede not. Butl wifh,eucry man fhould way this word,Signes. And befides that, conferre italfo with the tenth Chapter of Hzeremre. And though Some thinke , that there,they haue found arod: Yet Modeft Reafon,will be indifferent ludge, who ought to be Beaten therwith,in refpect of our purpofe.. Leauing that : I pray you vnderftand this:that without great diligence of Obferuation , examination and Calculation, theirperiods and courfes (wherby Diflinction of Seafons,yeares,and New Mones miglit precifely be knowne)could not exactely be certified . Which thing to per- forme,is that art, which wehiere haue Defined to be Afroncmie. Wherby , we may-haue.the diftinct Courle of Times dayes,yeares, and Ages: afwell for Confi- deratid of Sacred Prophefies,accomplifhed in due time,foretold : as for hi gh My- fticall Solemnities holding: And for all other humaine affaires , Conditions , and couenantes, vpon certaine time , betwene man and man -: with many other great vics: Wherin, (verely) ,would be great incertainty, Confufionyntruth,and brit tith Barbaroufnes: without the wonderfull diligence'and fill of this Arte : conti- nually learning,and determining Times,and periodes of Time., by the Record of the heauenly booke., wherin all times are written - and.to be read with an.A/frono- omitall staffe,in ftede of afeftue. as ) 4 ogee Mautike,of Motion, hath his Original canfe : Therforé , after the motions moft {wift;and moft Slow,which are in the Firmament,of Nature performed:and vader the Aitronomers Confideration:now I-will S peake of an other kinde of Motion, | producing found,audible,and of Man numerable. Aujike! call here that Science, « Which of the Grekes is. called Harmonice. Not medling with the Controuerfie be- twene the auncient Harmanifles,and Canonistes.- Miarfike is a Mathematicall Science which teacheth by fenfe and teafon, perfectly to iudge,and order the diverfities of foundes,hye and low. Aifronomieand cUufike ave Sifters faith Plato.” As.for Affromomie, the eyes:So, for Harmonious Motion,the eares were made. “But as As#ronomie hath amore diuine Contemplation, and cé- modity,then mottall ¢ye can perceiue : So,is rm to be confidered,that the *‘Minde may be preferred, before the eare. And ffm audible found , we ought to'aleende, to the examimation : which numbers are Harmonious,and which not. And why,cither, the oneare:or the other are not. I could at large, in the heauenly * motions and diftances , defcribea meruailous Harmonie ,. of Pythagoras Harpe / with eight ftringes. ~Alfo,fomwhat might befayd of Mercurins* two Harpes, eche of foure Stringes Elemental. And ves ftraunge matter,might be alledged of the Harmonie,to our™ Spiritual! part appropriate. Asin Ptolomeus third boke,in the fourth and fixth Chapters may appeare..* And what is the caufe of the apt bondé,and frendly felowthip, of the Intellectuall and Mentall part of vs , with our erofle& corruptible body: buta certaine Meane, and Harmonious S — with lohn Dee his Mathematicall Preface. bothparticipatyng,e of both in a maner)refultyng: Inthe* Tune of Mans voytejand alfe * the [ound of Injlrument,whatmight be fayd, of Harmonie: No’ common Muficier. would lightly beleue.But of the dundry Mixture (as may terme it)andconcurle> 1'D, diuerfe collation,and Application of thele Harmonies: as of thre foure,fiue,or mos! Xead in - Maruailous haue the cheetes ben: and yet may befounde,and: produced the like: /¢"" his with fome proportional! confideration for ourtime,and being: : in refpect of the 4 a6 iJ State, of the thinges then : in which, and by which , the wondrous effets were | as re al wrought. DemocritusandT heophrastus afirmed, thgt,by Mufike, eriefes and di- ~ chap ins feafes of the Minde,and body might be cured,or inferred. And we findein Re- where you corde,that T expander, Arion l{menias, Orpheus, Amphion,Dauid, Pythagoras, Empedo- fhall hane cles, Afclepiades and T imotheus,by Harmonicall Confonacy,haue done,and brought ome occafion to pas,thinges,more then meruailous, to hereof, \. Of them then, making no far- f ee she der difcourfe,in this place : Sure I ap, that Common Mufike, commonly vied,is fike . us Mn found to the cAtuficiens and Hearers,to be fo Commodious and pleafant , That * Lapbese ly is I would {ay and difpute,but thus much; Thatit were to be otherwife vied , then it, shoyghe: is,I fhould finde more repreeuers, then I could finde priuy,or fkilfull of my mea- ning. Inthingestherfore euident,and better knowen,then I can exprefle:and fo allowed and liked of, (as I would with,fome other thinges,had the like hap) I will fpare to enlarge my lines any farder,but confeq uendly follow my purpofe. Ot Cofmographie,1 appointed briefly in this place, to geuc youfome intelligence. Cofmographie,is the whole and perfect defcription of the heauerily and alfo elementall parte of the world , and their ho- mologall application , and mutuall collation neceffarie. This Art, requireth Aifronomie , Geographie , Hydrographic and Mufike Therfore jit is no {mall Arte,nor fo fimple,, as in common practife , itis (flightly)confidered. This matcheth Heauen, and the Earth, in onc frame,and aptly applieth parts Correfp6- dent:So as, the Heauenly Globe, may (in practife) be duely defribed ypon the Geographicall ,and Hydrographicall Globe. And there, for vs to confider an Aiquinoctiall Circle , an Ecliptike line, Colures, Poles, Sterres in their true Longitudes, Latitudes, Declinations,and Verticalitie:alfo Climes;and Parallels:and by an Ho- rion annexed,and revolution of the earthly Globe(as the Heauen,is, by the Prz- mouant,caried about in 24.zquall Houres) to learne the Rifinges and Settinges of Sterres (of Virgil in his cha ner of Hefiod: of Hippocrates in his Medicinal Sphare, to Perdicca King of the Macedonians: of Diocles,to King Antigonus ,.and of other fa- mons Philofophers prefcribed)a thing neceflary, for due manuring of the earth, for Nauigation,for the Alteration ofmans body:being,whole,Sicke,wounded,or bru- fed. By the Reuolution , alfo , or mouing of the Globe Cofmographicall , the Rifing and Setting of the Sonne: the Lengthes,of dayes and nightes : cm oures and times (both night and day )are knowne : with very many other pleafantand neceflary vies : Whetof, fome are knowne:but better remaine,for fuch to know and vie: who of a fparke of true fire,can make'a wonderfull bonfire, by applying of sy due matter,duely. eed Se | ) Of Aftrologie » here I makean Arte, feuerall from 4ffronomie: not by new deuife, but by good reafon and authoritie : for, Aftrologie,is an Arte Mathematicall , which reafonably demonftrateth the operations and effectes, of the naturall beames, of light, and fecrete influence: of the Sterres and Planets :-in euery-element and elementall body: 3! | b.iil. at lohn Dee his Mathematicall Preface. atall times 3 imany Horizon afligned , This Arte is furnifhed wi th-ma- ny other great Artes and experiences: As with perfecte Perfhectine, Astronomic, Cofmographie; Nuturall P hilofophie of the 4.Elementés,the Arte of Grad vation,and ” fome good vnderftading in catufike : and yet moreouer, with an other great Arte, »\ hereafter! following, though, here, fer this before, for fome confiderations me -mouing. Sufficient (you fee) is the ftuffe, to make this rareand fecrete Arte,of: and hard enough to frame'to the 'Conclufion Syllogifticall . Yet both the mari- folde and continuall trauailes of the moft auncient and wife Philofo phers.for the atteyning of this Arte : and by examples of effectes , to’ confirme the fame : hath leftwnto vs fufhicient proufeand witnefle : and we,alfo,daily may perceane, That mans body, and all other Elemental! bodies, are altered, difpofed, ordred, pleafu- red, and difpleafured, by the Influentiall working of the Sune, Moxe,and the other Starres and Planets . And therfore fayth “7iiotle, in the firt of his Meteorologicall bookes, inthe fecond Chapter: E/? autem neceffario Mundas iste, fupernis lationibus Sere continuus . Vt, inde, viseiusvminer{a regatur. Ea fiquidem Caja prima putanda omuibus eft; vade motus principium existit. Thatis: T his [ Elemental World is of | necefsitie, almoft, next adioyning,to the heanenly motions: T hat,from thence / all his vertne or force may begouerned. For that is to be thought the firft Caufe ynto all : from which, the begmning of motion,is.. And againc,in the tenth Chapter. Oporter igitur & horum principia [umamns , & caufas onninim fimiliter. Principium igitur vt WLOUENS PF kcip using, cy omninm primum , Circulus tlle eftsin Guo manifefe Solis latio, &c~ And fo forth . His Meteorological bookes, até full of arg. mentes, and cftectuall demonftrations,of the vertue, operation, and power-of the heauenly bodies, inand vpon the fower Elementes, and other bodies,of them (either perfectly, or vhpertedtly ) compofed. And in his fecond booke, De Genera- tione & Corruptione ,in the tenth Chapter. Quocirca ¢ prima latio, Ortus Interte tus cau{anon eft: Sed obliqui Circuli latio : ea namy, cy continua efter duobus motibus fit: In Englifhe, thus. Wherefore the vppermost motion, 1s not the caufe of Genes ration and Corruption, but the motion of the Zodiake: for, that, both, is cons tinuall., ands caufed of twomouinges. And in his fecond booke,and fecond Chapter of hys Phyfikes. Homo namg, cenerat hominem, atg, Sol. Foy Man (fayth he) and the Sonne, are caufe of mans generation. Authorities may be brought, very many : both of 1000.2000.yea and 3000. yeares Antiquitie : of great Philo. Sophers, Expert, Wife, and godly menfor that Conclution: which, daily and houre- ly, we men,may difcerne and perceaue by fenfe and reafon: All beaftes do feele, atid fimply.{hew, by their actions and paffions, outward and. inward: ‘All Plants, Herbes, Trees, Flowers, and Fruites . And finally, the Elethentes,and all thinges of the Elementes compofed, do geue Teftimonie (as Ariftotle fayd’) that theyr Whole Difpofitions , vertues, and naturall motions , depend of the Attinitie of the heauenly motions and Influences. Whereby, befide the fpecificall order and » forme; due to enery feede: and befide the Nature, propre to the Indiuiduall Mg. trix, of the thing produced: What fhall be the heanenly Imprefsion’, the perfec? and circum|peéte AStrologien hath to Conclude. Not onely (by Apotelefimes) 3 5: butby Naturall and Mathematicall demonftration +s diér%. ~=Whereunto ; What Sciences are requifite ( without Exception ) I partly hatie here warned: And inmy Propedewmes ( betides other matter there difclofed ) I haue Mathematically furni- fhed vp the whole Method : To this our age,not fo carefully handled by any,that cuer lohii Dee his Matheniaticall: Preface! et I faw,or heard of. ‘Iwas, (for:*2i-yearesago) by certaine eatneft difputaté *Ayno.15 48 ontof the Learned-Gerardus Mercator,and Antonius Gogana, (and other, )therto fo and 1549.#s prouoked:and(by my-conftantand invincibtezeale totheveritic)inobferuations 4o2yn- of Heauenly Influencies(to the Minute of time,)than, fo diligent: And: chiefly by the Supernaturalhinfluence,from the Starte of lacob;foditected: That any Modett « and Sober Studént, carefully and diligently fcking forthe: Trarh, will both finde &ccéfefle, therin,to be the Veritie,of thefe my wordes:And alfo become a Reafo- nable'Reformer;of three Sortes of people + about thefe Influentiall Operations, « greatly feat from the truth. Wherof, the one, is Light Beleuers;theother, * Light Detpilers,and the third Light Pradtifers.*" ‘The firlt, & moft cémon Sort, thinke the Heauien and Sterres,to be anfwerable to any theirdgutesorde- —* fires: which is not fo:and, in dede,they,to much,ouerreache. The Second forte thinke no Influentiall vertue ( fro the heavenly bodies ) to beare any Sway in Ge- " 2: neration and Corruption,in this Elementall world. And to the Sunne , Mone and Sterres(being fo many,fo pure,fo bright , fo wonderfull bigge , fo farrcin diftance, _ fo manifold in their motions , fo conftant in their periodes ..&c .) they affigne a _ fleight,fimple office or two,and fo allow vnto thé(according to theifcapacities)as much vertue,and power Influentiall,as to the Signe of the Sanne,Mone, andfeucn Sterres, hanged vp uF Siones )in London, for diftinétion of hhoufes , dcfuch erofle helpes,in our wordly affaires: And they vnderftand not(or will not vnderftand) of the other workinges,and vertues ofthe Heauenly Suume,Mone, and Sterres : not {fo much,as the Mariner,or Hufband man ; no ,not fo much,as the Elephant doth, as the Cynocephalus , as the Porpentine doth ; nor will allow thefe perfedi, and incor- ruptible mighty bodies, fo much vertuall Radiation, & Force, as they fee in alice peece of a Magnes stone:which, at great diftance,{heweth his operation . “And per- chaunce they thinke,the Sea & Riuers ( a5 the Thames ) to be fome quicke thing, atid fo to ebbe.andAow, tun in and outsof themfelues,attheir owne fintafies. God helpeGod helpe.. Surely,thefe men,come to fhort: and cithes are to dull: or willfully blirid:or,perhaps,to malicious. “Yhe third man, is the comrfion and vulgare _4/rologien,or Practifer : who, being not duely,artificiallyjandperfectly = 3. furnifhed:yet,either forvaine glory,or gayne:orlikeafimple dolt, & blinde Bay- rd, both-in matter and maner,erreth:to, the difcreditof the Wary. and modett 4- firob jen:and to therobbing of thofé molt noble corporal Creatures oftheir Na- tutall Vertne: being’ molt mighty’: mol? beneficial 'to all clem€ntall Generation, Corruption andthe appartenances Sand moft Harmonious imtheir: Monarchie: Fox which thinges,being knowen,and modeltly vied:wemight highly,and cont nually glorifie God,with the princely Prophet, faying,-, J he Fdeauens declare the Glorie of God: who made the Heaués in his wifedome: who made the Sonne, for to haue dominion of the day : the Mone and Sterres to hauc dominion of the = whereby, Daytodayviterct/talke: andmightyto night declareth knows lye. Prayfebim cll ye Sterves;aind Light. Amen.) oss.’ ® ’ = [Noidét’, show foliivert , of Dbatthke sfomewhiar-es: fay j what we meane by ‘that namefand what ¢ommodity;dothjon tactvArt depend. . Statike , is an Arte Mathernaticall evhich'demonttrareth the caufes of heauynes, and lightnes of all thynges : and of motions and properties ,to hea- uynes aabbahen selonging. ... And for afinuch as, by the Bilan, or Bax a | oy 78 7 - Z >. _ } =) 4 oe sid lbiseang ~ Aes Ji JAAS C4446 lance(as the chief fenfible Inftrument ) Experiénce of thele demonftrations may a - * b » elilj. be SS SOS lohndDee his Mathematicall Preface. fcoroe* behad:wécall this Art,Statike:that isthe Experimentes of the Balance. Oh, that men i . witt;what proffit, (all maner of wayes)by this:Arte might grow, to the hable txa- 9 miner,and diligent pracifers Thowotelysknowertall thinges precifely (O God) » who haftmade weightand Balance,thy Iudgement: who.haft created all thinges >» in' Naber; Waicht,and Meafure:and haftwayed the mountaines and hils ina Ba- »> Jance: who haft peyfed in thy hand’; both Heauen and earth... .: We therfore war »> ned by the Sacred wotd,to Confider thy: Creatures:and by that confideration, to » wynne a glyms (asitwere, ) or fhaddow of perceiuerance, that thy wifedome, _ » might, and. goodhegisinfinite,and vnf{peakable,in thy Creatures declared: And » being farder aduertifed., by thy mercifull goodnes , that ,three principall wayes; | _ 2» were,of the,vfed in Creation ofall thy Creatures , namely , Number, Waightand »> < Meafure,And for as much as,of Number and Meafure,the two Artes( auncient, fa- _ » mous,and to humaine vies moft neceflary, ) aresall ready, fufficiently knowen and a9 extant: This third key, we befeche thee (through thy accuftomed goodnes,) » that it may come to the nedefull and fufficient knowledge,offuch thy Seruauntes, » ‘as inthy workemanfhip, would gladly finde,thy true occafions (purpofely of the »» vfed ) whereby we fhould glorifie thy name,and thew forth (to the weaklinges in » faith) thy wondrous wifedome and Goodnes. Amen. i ~~ Meruaile nothing at this pang(godly frend,you Gentle and zelous Student.) An other day, perchaunce,you will perceiue , what occafion moued me. Here, as now,] will giue you fome ground, and withall fome thew , of certaine commodi- ties,by this Arte arifing.’ And bycaufe this Arte is rare , my'wordes and practifes might be to darke : vnleaft you had fome ligh t,holden before the matter:and that; beft will be,in giuing you,out of Archimedes demonftrations,a few principal Con- clufions,as foloweth. -. . | — ~ ———— 4 _ : a sas — A EA ee ee - a - * ; : — = == a — ~~ ie ee fhe Siiperficies of euery Liquor, by it felfe confiftyng , and in SHY Stas SPIER AN * ‘the centre whereof, is thefame , which is the centre of the Earth. . |. of | tre Ss “ at FP J a Qe és i ee ee 5% | a > | If Solide Magnitiides, being of the fz ame bignes, or quatitie, that any Li a is,and hauyng alfo thefame, Waight: be let dovynein- ~ > ~ - ae ; tothefame Li norsthey-will fettledowneward,fo,thatno parte of then fthall be aboue the Superficies of the Liquor':and ‘yet nevier- theles they willinot finke'vtterly downe,or drowne, © ~~ = QUTOS SA4 SHH OAM ieBS INT 70h 1) TBHG 4d SATS OAG BOL) 15 sey Sak Sch LO MOLICIKOD SES OF WTHIG HRD SM ot: ens 344 to srorsscols Stl OF 04 «wolf any: Solide Magnitude pr.’ Lighter ‘then. a Liquor, be det downe into the fame Liquor , inwvilb fertle down; fofarreinto the famediquorsthat fo greatis quantitic ofthat Liguor,asis the parte of the Solid Magnitude, fertled downedtitothefame Liquor: isin Waighs,¢quall;ro she waight of he whole Solid Magniaides 2 oe te ~~ scota | a no ate = “e Fs 1 OF, <5 : 4 ae 5 f CJ EF ur a2 4h 2Se © Ow, ¥ i244 $¢ eeass ; ere 2s FRR | RAE OB are NO SS Rens PO ee Taglar eele e ee “he : . Ped : lid 4 4 a ot itud ’ Li tht Sri igi, Li 35 of, ata e , +e eee Ss 5 Any Solide Magnitude , Lighter thena Liquor’, forced downe __AANY SOLE WIA DILUGE , Lule er ee en a eG, COW NE Wa! Mik O12) 4 5 ERPS S LO LSaset whet as art? #56.#24.- “4 . \ sae Litt eet 23 \ SASS Pats ADR SY Jiae Mi : ee STs 3 | — | ‘into r a3 20 ow qhed.se\s ~ John Dee his Mathematicall Preface. oreata power, into the fame Liquor , will mouevpward , with fo ¢ bihow much, the Liquor hauyng xquall quantitie to the whole Magnitude,is heauyer then the fame Magnitude. 5. Any Solid Magnitude,heanyer then a Liquor, beyng let do wne into the fame Liquor, will finke downevtterly : And wilbe in that Liquor , Lighter by fo much, as is the waight or heauynes of the Liquor,hauing bygnes or quantitie,zquall to the Solid Magnitude. 6. If any Solide Magnitude , Lighter then a Liquor , belet downe into the fame Liquor, the waight of the fame Magnitude, will be, to the Waight of the Liquor. ( Which is zquall in quantitie to the whole Magnitude,)in that proportion , that the parte , of the Mag- 3 nitude fettled downe,is to the whole Magnitude. BY thefe verities , great Errors may be reformed, in Opinion ofthe Naturall Motion of thinges, Light and Heauy. Which efrors,are in Natural! Philofophie (alinoft ) of all mé allowed:to much trufting to Authority:and falfe Suppofitions. As,Of any two bodyes, the heauyer, to moue downward fafter then thelishter, . This error,is not firftby me,Noted: butby one iohu Baptist de Be- nediitu;- The chief of his propofitions,is this: which feemeth a Paradox. If there be two bodyes of one forme, and of one kynde, xquall in quantiti¢or vnzquall , they will moue by xquall fpace, in xquall tyme:So that both theyr mouynges be in ayre , or both in water : or in any one Middle. Hereupon’, in the feate of Gun nyng,cettaine good difcourfes ( otherwife) may rec¢iue greatamendement,andfurderance. In the entended purpofe, alfo, allowing {omwhat to the imperfection of Nature: notaunfwerable to the preci- fevies of demonftration.... Moreotier,by the forefaid propofitions ( wifely vied.) The Ayre;thewater,the Earth, the Pire, may be nerély,knowen,how light or hea- uy they are. (Naturally ) in their affigned partes: orin the whole. And then,to thinges Elementall,turning your practife: you may deale for the proportion of the Elementes;inthe thinges Compounded . Then, to the proportions of the Hu- mours in Mane their waightes: and the waight ofhis bones,and Ach. &c. Than, by waipht,to haue confideration of the Force of man,any maner of way: in whole or in part. Phen,may you, of Ships water drawing’, diuerfly,in'the Sea and in frefh water, haue pleafant confideration : and of waying vp of any thing , fonkenin Sea orinfrefhwater.&c. ~ And (toliftvp your heada loft : ) by waight,you may,as precifely.as by any inftrumentels,meafure the Diameters of Sonne and Mone.cre, Frende, I pray you , way thefe thinges,with theiuft Balance ofReafon. And you willfinde:Meruailes vpon Meruailes :. And efteme one Drop of Trath ( yea in Naturall Philefophic)more worth,then whole Libraries of O pinions, vndemon- ftrated:or notaunfwering to Natures La : =F apey Waid ycurexperience. Leauing thefe C.J. thinges, 4 ; . r.D. The Cutting of @ Sphere according te any proportion aj- frgned may by rhtg prepofition be done Mechanrcally by tempering Li- uortoa certayne watght 17 refpedt of the waight of the S phare rheretm Swymming. A common error noted. A paradox, N. T. The wonders ful ufe of thefe Propofie S10KS~ T he prattife Staticall, to kiiow the pro- portion, be- tryene the Iohn Dee his Mathematicalt P i Gike ; thinges,thus:I will giue you two or three, light practifés, to great purpofe : and fo finifh my Annotation Staticall. In Mathematicallmatters , by the Mechanicidis ayde , we will behold, here, the Commodity of waight.. Make.a Cube,of any one Vniforme : and through like heauy ftufte: ofthe fame Stuffe,make a Sphere or Globe,precifely,of a Diameter aquall to the Radicall fide ofthe Cube. Your {tufte,ymay be wood, Copper, Tinne, Lead,siluer.&c. (being,as I fayd,of like na- ture , condition,and like waight throughout. ) And you may, by Say Balance, Cube, andthe haue prepared a great number of the {malleft waightes : which, by thofe Balance Sphere. “LD; can be difcerned or tryed-and fo,haue proceded to inake you a perfect Pyle, com- pany & Number of waightes: to the waight, of fix,eight,or twelue pound waight: mott diligently tryed,all.And of euery one , the Contentknowen, in your leaft waight,thatis wayable. They thatcan not haue thefe waightes of precifenes: may, by Sand, Vniforme,and well dufted,make them a number of waightes,{ome- what nere precifenes : by halfing euer the Sand: they {hall at length come toa Jeaft common waight. Fherein,! leaue the farderimatter,to their difcretion,whom’ nede fhall pinche.3. ‘The Yenetians confideration of waight , ‘may feme precile enough:by eight defcentes progrefsionall,* halting , froma grayne. Your Cube, *For; fo, have Sphere,apt Balance,and conuenient waighteés,being ready :fall to worke.g. Purtt, you, 56. partes of a Graine. *The proportion of the Sauare tothe Circle infcribed. *The Squaring of the Circle, Mecha- nically, *To any Square even, to ceuca firele, equal. vay your Cube. Note the Number of the waight . Way, after that, your Sphere. Note likewife,the Naber of the waight.If you now find the waight of your Cube, to be to the waight of the Sphere,as 21 . is to 11; Then you fee, how the Mechani- cien and Experimenter , without Geometric and Demontftration, are (as nerely in. efect) tought the proportion of the Cube to the Sphere : as I haue demonftrared it,in the end of the twelfth boke of Ewclide.. Often, try with the fame Cube and Sphere. Then,chaunge,your Sphere and Cube,to an other matter: or to an other bignes : till youhaue made a perfec vniuerfall Experience ofit! Pofsible itis, that youfhall wynne to nerer termes,in the proportion, » pi3 31 -- When you haue found this one certaine/Drop of Naturall veritie,procede on, to Inférré,and dueély to make aflay,of matter depending. As, bycaufe it is well de- monitrated, that a Cylinder, whofe heith, and Diameter of his bafe.is zquall to the Diameter of the Sphere , is Sefquialter to the fame Sphere (that is,as 3. to 2:) Tothenumber ofthe waight ofthe Sphere,addehalfe fo much,as.t is: and fo haue you the number ofthe .waight of that Cylinder... Whichisalfo Compre- hended of our former Cube:So,that the bafeofthat Cylinder, is.a.Circle defcri- bed in the Square , which is the bafe. of our Cube. , But the Cube and the Cy- linder,being both of one heith , haue their Bafesin thefame proportion , in the which,they are, one to an, other, in their Mafsines or Soliditic. » But,before,we haue two numbers, exprefsing their, Mafsines , Solidities , and Quantities , by waight:wherfore,we haue * the proportion of the $quare,to the Circle, inferibed in thefame Square. And fo are we fallen into the knowledge fenfible, and Expe- rimentall of Archimedes great Secret: ofhim by great trauaile of minde, fought and found.’ Wherforesto.any Circle giuen,-you can giue a Square xquall: *as Thaue raught,in my: Annotation,vpon the firft. propofition of the twelfth boke, And likewife,to any Square giuen,you may gite poe ane ee *If you defcribe a Gircle,which fhall be in that proportion, to your Circleinicribed, as the Square is to the fame Circle: This, you may do,by my.Annotations,vpon thefecond pro- pofition of the twelfth boke of Euclde,, in my-third Probleme there.» Your dili- gence may come to a proportion, of the Square to the Circle infcribed’, nerer the truth, then is the proportion of 14.to.11... And confider, that you.may begyn at the Circleand Square, and fo come to conclude of the Sphire,& the a leir 7 nee zz lohn Dee his Mathematicall Preface. their proportion is:as now , you came fromthe Sphere to the Circle. For,of Sil uergor Gold,or Latton Lamynsor plates(thorough one hole drawé,as the maner is)if youmakea Square figure -8oway it:and theri;defcribing theron, the Circle in {cribed:& cut of, & file away, precifely (to the Circle) the ouerplus of the Square: you {hall then,waying your Circle , fee, whether the waight of the Square , be to your Circle; as14. to 11. As Lhaue Noted, inthe beginning of Euctides twelfth boke.&c.after this refort to my laft propofition,vpon the laft of the twelfth. An d there,helpe your felfe,to the end. And,here, Note this, by the way. Thatwe may Square the Circle , without hauing knowledge of the proportion,of the Cur- cumference to the Diameter: as you haue here perceiued. Andotherwayes alfo, Lcan demoniftrate it.So that,ymany haue cumberd them {eles fuperfuoutly, by trauailing in that point firft, which was not of necefsitie,firft : and alfo very in- tricate. And eafily,you may, (and that diuerfly) come to the knowledge of the Circumference:the Circles Quantitic , being firftknowen. Which thing,Ileane to your confideration:making haft to defpatch an other Magiftrall Probleme: and to bring it,nerer to your knowledge,and readier dealing with,then the world (be- fore this day, )had it for you,that I can tell of.And thatis, 4 Mechanicall Dubblyng of the Cube:erc. Which may, thus,be done: Make of Copper plates,or Fyn plates a fourfquare vpright Pyramis,or a Cone: perfectly fafhioned in theholow,within . Wherin, let great diligence be vied , to ap- proche (as nere as may be) to the Mathematicall perfection of thote fioures. Attheir bafes,let them be all open:euery where,.els, moit clofe,andiuftto. From the vertex, to the Circumference of the bafe of the Cone:& to the fides of the bafe of the Pyramis: Let 4.ftraight lines be drawen,in the infide of the Cone and Pyramis : makyng at their fall on the perimeters of the bafes , equall angles on both fides them felues , with the fayd perimeters. Thefe 4.lines (in the Pyra- mis;and as many ,in the Cone)diuide: one,in 12. equall partes : and an other,in 24.an other,in 60, and an other, in 100 . (reckenyng vp from the vertex.) Or vie other numbers of diuifion , as experience fhallteachyou. Then,“ fetyour Cone or Pyramis, with the vertex downward , perpendicularly , in refpect of the Bate.(T hough it be otherwayes, it hindreth nothyng.) So let the moft ftedily be flayed. Now, ifthere be a Cube,which you wold haue Dubbled.Make youa prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube,or Cubik coffen, of Copper, Siluer, Tynne,or Wood &c. Thefe,you may fo proportid in refpeét of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in water, 3. or 4. times:at the leaft: what ftuff fo euer they be made of,Let not your Solid angle, at the vertex,be to fharpe: but that the water may come with eafe,to the very vertex,of your hollow Cone or Pyramis.Put one of your Solid Cubes in a Balance apt: take the waight therof ex- adily in water . Powre that water, ( without loffe ) into the hollow Pyramis or Cone,quietly.. Marke in your lines, what numbers the water Cutteth: Take the waight of the fame Cube againe : in the fame kinde of water , which you had be- fore : put that *alfo, into the Pyramis or Cone,where you did put the firft. Marke now againe, in whatnumber or place of the lines, the water Cutteth them. Iwo C.1j. wayes Noté S quaring of the Circle Without bnows ledge of the pre portion ben twene C1r- chmfperence and Diame- ter. To Dubble the Cube re= dsly:by Art Mechanicall: depending vp- pen Demor- stration Mae thematicall, I. D» The g.fides of this “Pyramis muft be He Lfofceles Triangles a like and equal. ) *In all workinges with this Pyramis or Cone, Let their Situations be in all Pointes and Condi- tions, alike,or all one: while you are about one worke.Els you will erre. ID. * Confider well whan ou mu/? put your waters togyther: and whan ,you mut emp- ty your first water, out of your Pyrarnats or Cone. Els you will 477 é. * Vitruurns. L1b.9.Cap+3« God be tban- ked for tists Inmention <> the fruite en- faing. * Note. Note,as Cou- cerning t be Spharicall q Y 5 hell -. Superficies 0 OH DET IL es }j the Water. CF 2? >? 3»? >? 3? »? ? Notethts A- bridztement oF Dubbling the Cube.cec. lohn Dee hisMathematicall Preface. wayes you may conclude your purpofe : it is to:wete , either by numbers or lines. By numbers : as,if you°diuide the fide-of your Fundamentall Cube int fo many zquall partes, as it is capable ofjconueniently, with your eafe, and pre- cifenes of thediuifion . For,as the number of your firft and leffeline (in your hollow -Pyramis or Cone;) is to the fecond or greater ( both being counted fromthe vertex ) fo fhall the number of the fide of your Fundamental! Cube, betothe niiber belonging tothe Radicall fide,of the Cube,dubble to your Fun- damental! Cube: Which being multiplied Cubik wife,will fone fhew it elfe,whe- ther it be dubbleor no, to the Cubik number of your Fundamental! Cube. By lines,thus: As your lefleand firft line,(in your hollow Pyramis or Cone,)is to the fecond or greater,fo let the Radical fide of your Fundamétall Cube,be to afourth proportionallline , by the 12. propofition,of the fixth boke of Euclide . Which fourth line, fhall be the Rote Cubik,or Radicall fide of the Cube, dubbic to your Pundamentall Cube : which is the thing we defired’.’ For this,may I (with ioy) fay,eypHKa, EYPHKA, EypHKA: thanking the holy and glorious Trinity : hauin; greatercaufetherto , then* CAérchimedes had (forfinding the fraude vied in Kinges Crowne, of Gold):as all men may eafily Iudge : by the diuerfitieof the frute following of the one,and the other. Where.I {pake before, ofa hollow Cu- bik Coften-the like vie,is of it:and without waight.Thus. Fill it with water, preci- fely full,and poure that water into your Pyramis or Cone.And here note the lines cutting in your Pyramis or Cone. Againe,fill your coffen,like as you did before. Put that Water,alfo,to the firft. Marke the fecond cutting of your lines. “Now, as you proceded before, fo muftyou here procede.* And ifthe Cube, which you fhowld Double, be neuer fo great - you haue, thus, the proportion (in {mall ) be- twene your two litle Cubes: And then,the fide,of that great Cube(to be doubled) being the third , willhaue the fourth, found, to itproportionall : by the 12.0f the fixth of Ruclide. | Nogeschar all this while,I forget not my firft Propofition Staticall here rehear- fed: that, the Superficies of the water,is Sphericall. Whcerein,vfe-your difcretion: to the firit Jine,adding a final heare breadth,more:and to the fecond,halfea heare breadth more,to hislength. For,vou will cafily perceaue, that the difference can be no crearer, inany Pyramis or Cone, of youto be handled. Which you fhall thus ttve’. F or finding the [welling of the water aboue lenell. Square the Semidiame- ter, from the Centre of the earth,to your firlt Waters Superficies. Square then, halfe rhe Subtendent of that watry Superficies ( which Subtendent muft haue the equall partes of his ineafure, all one, with thofe of the Semidiameter of the earth to your watry Superficics ) : Subtracte this {quare, from the firft : Of the refidue, take the Rote Square. That Rote,Subtracte from your firft Semidiameter of the earth to your watry Superhicies : that, which remaineth, is the heith of the water, in the middle, abone the leuell. Which,you will finde, to be a thing infenfible, And though it were greatly fenfible,* yet, by helpe ofmy fixt Theoremevpon the laft Propofition of Euclides twelfth booke, noted : you may reduce all,to atrue pee and & MAP os ee John Déeéhis*Mathematrcall Preface} andall,vpon'one Mathematicall ‘Demonfiration depending. Take water (as muehas conueniently willferue. your rurne :.as L warned before of your Funda- ; mentall Cubes bignes): Way it precifely «) Pur thar water,into your Pyramis or > Cone. Of the fame kinde of water; then takeiagaine,the fame waight you had before : putthat likewife into the Pyramis or Cone. For, in eche time, j king of the lines, how the Waterdoth cur them, {hall geue you the proportion be- twen. the Radicall fides,of any two Cubes, wherof the one 1s Doubleto the other: working as before I haue taught you:*fauing that for you Fundamentall Cube his Radicall fide: here,you may take a right line, at pleafure: Yet farther proceding with our droppe of Naturall truth : you may (now) eure Cubes,one to the other, in any proportio gené: Rational or Ir- rationall ; on this maner.Make a hollow Parallelipipedon of Copper or Tinne: with one Bafe wating, or open:as in our Cubike Coften. Fro the bottome of that Parallelipipedon,raife vp,many perpendiculars,in euery of his towerfides. Wow if any proportion be affigned you, in right lines: Cut one of your perpendiculars(or ,, a line equall to it, or leffe then it ) likewife: by the 1o.of the fixth of Euclide. And , thofe two partes , fetin two fundry lines of thofe perpendiculars (or you may {et them both,in one line ) making their beginninges,to be, atthe bafe: and fo their lengthes toextend vpward .. Now, fet your’ hollow Parallelipipedon, vpright, erpendicularly,fteadie. Poure in water, handfomly, to the heith of your thorter bias -Poure that water, into the hollow Pyramis or Cone. \Marke the place of the rifing. . Settle your hollow Parallelipipedon againe . Poure water into tt: ynto the heith of the fecond line , exactly » Poure that water* duely into the hollow Pyramis or Cone: Marke now againe, where the water cutteth the fame line which you marked before. For, there, as the firft marked line, is to the fe- cond : So fhallthetwo Radicall fides be, one to the other, of any two Cubes: which, in their Soliditie, fhall haue the fame proportion, which was at the firft af- fined : wereit Rationall or Irrationall. Thus,in fundry waies you may furnifhe your felfe with fuch fraunge and pro- fitable matter:which;long hath bene withed for. And though.it be Naturally done and Mechanically : yet hath ita good Demontftration Mathematicall . Which is this.;Alwaies ,you haue two Like Pyramids: ortwo Like Cones,in the proporti- ons affigned ::and like Pyramids or Cones,are in proportion,one to the other, in the proportion of their Homologall fides (or lines) tripled. | Wherefore, if to the firlt, and fecond lines,found in your hollow Pyramis or Cone, you ioyne a third andafourth, in continual! proportion : that fourth line, fhall be to the firft, as the greater Pyramis or Cone, is to the leffe : by the 33.0fthe eleuenth of Euclide . If Pyramis to Pyramis, or Cone to Cone, be double, then fhall* Line ro Line, be alfo double,&c. But,as our firft line, is to the fecond,fo is the Radicall fide of our Fundamentall Cube,to the Radicall fide of the Cube to be made; or to be dou- bled: and therefore,to thofe ewaine alfo, athitd and a fourth line jin continuall propottion, ioyned : will geue the fourth line in that proportion to the firft,as our fourth Pyramidall, or Conike line, was to his firft: but that was’ double, or tre- ble,&c.as the Pyramids or Cones were, one to an other(as we hane proued) ther- fore,this fourth, fhalbe alfo double or treble to the firft,as the Pyramids or Cones were one to an other ; But our made Cube,is deftribed of the fecond in proporti- on, ofthe fower proportional lines : therfore * as the fourth line, is to the firft, fo is that Cube,to the firft Cube sand we haue proued the fourth line, to be to the firft, as the Pyramis or Conc,is to the Pyramis or Cone : Wherefore the see 1s Clif. to the Your Mat- : 3) - wu ew Note x Fy } Nate.* £3 * q Ri Pon) ] : ] O7tHe HOES Ss ; CHELOLNE Or a = lACr:i71 Any proportio7, ; i ) a ef tee Ratienall or Hy ‘ . i} i? yational : . »» Lmpty- 5 Ing the a firft The demonftrations of this Dubbing of the Cube,and uf the reft. ILD. * Hereby helpe your Self ta become a pre- ctfe prattifer. And fo confider,haw.no- thing at all, you are hindred( fenfibly )by the Conuegitis of the Water. *By the 33.0f the e- lewenth booke of Euclida, ee ee ee : | | : : FD. ©. And your diligence on pradijé,can [0 (12 waight of wa- fer )pecforme tts There/ore,nowsyou are able to gene good realan of your whole desng. * Note this Corollary. *T he great Commodities following of theft new In- uentions. cz* Such is the Fruite of the Mathematt- call Sciences ana Artes, lohn Dee his Mathematicall Preface, to the Cube,as Pyramis is to Pyramis,or Cone is to Cone. But we* Suppofe Py- ramis to Pyramisjor Cone to Cone, to be double or treble.&c. Therfore Cube,is to Cube,double,or treble,8ec. Which was to bedemonfttated.And of the Paralle- lipiped6,it is euidét , chat the water Solide Parallelipipedons,are one to the other, as their heithes are,feing they haue one bafe. Whertore the Pyramids or Cones, made of thofe water Parallelipipedons,are one torive Other; as the lines are(one to the other)betwene which, our proportion was affigned . But the Cubes made of lines,after the proportio of the Pyramidal or Conik omologal/ lines,are one to the other,as the Pyramides or Cones are , one to the other (as we before did proue) therfore,the Cubes made; fhalbe oneto the other,as the lines affigned,are one to the other: Which was to be demonftrated.Note. *This,my Demonftratio is more generall,then onely in Square Pyramis or Cone: Confiderwell. Thus, haueI, both Mathematically and Mechanically,ben very long in wordes:yet (I truft)no- thing tedious to them,who, to thefe thinges , are well affected. And verily Tam forced (auoiding prolixitie to omit fundry fuch things, eafie to be practifed: which to the Mathematicien,would be a great Threafure : andtothe Mechanicien,no finall gaine.*N ow may you, Betwene two lines given finde two middle proportionals,in Continuall proportion ; by the hollow Paralleli- pipedon, and the hollow Pyramis, or Cone. Now,any Parallelipipedon rectangle being giuen:thre right lines may be foundsproportionall in any propor- tion affigned,of which, fhal be produced a Parallelipipedon, equall to the Paralle- lipipedon giuen.Hereof,| noted fomwhat,vpon the 36.propotition,of the 11.boke of Euclide. _Now,allthofe thinges,which Vétraums in his Architecture fpecified hable to be done; by dubbling of the Cube:Or, by finding of two middle propor- tionall lines, betwene two lines giuen,may eafely be performed. Now, that Pro- bleme, which I noted vnto you,in the end of my Addition, vpon the 3q.of the 11. boke of Euclide, is proued pofsible. Now,may any regular body, be Tranftormed into another,&c. Now, any regular body:any Sphere, yea any Mixt Solid : and (that more is)Irregular Solides, may be made(in any proportié afligned)like vnto the body, firft einen. Thus,ofa Manneken,(as the Dutch Painters terme it)in the {fame Symmetrie , may a Giant be made. and that,with any gefture,by the Manne- ken vfed: and contrary wife. Now, may you , of any Mould, or Modell ofa Ship, make one,of thefame Mould (in any affigned proportion) bigger orlefler. Now, may you,of any*Gunne,or little peece of ordinatice,make an other, with the fame Symmetrie (inall pointes) as great,andas littleas you will.Marke that:and thinke onit. Infinitely, may you apply this, fo long fought for,and now fo eafily: concluded : and withall fo willingly and frankly communi- cated to fuch,as faithfully deale with vertuous f{tudies. Thus,can the Mathematicall minde,deale Speculatinely in his own Arte: and by good meanes, Mount aboue the cloudes and fterres :- And thirdly,he can, by order,Defcend,to frame Naturall thinges, to wonderfull vfes:and when he lift, retire home into his owne Centre: and there,prepare more Meanes, to Afcend or Defcend by : and, all,to the glory of God, and our ho sk delectation in earth. Although, the Printer , hath looked for this Praface,a day or two , yetcould I not bring my pen from the paper , before I had giuen you comfortable warning, and brief inftru@tions,offome of the Commodities, by Statzke,hable to be reaped: In the reft,I will therfore,be as brief,as it is pofsible-and with all,defcribing them, fomwhataccordingly. And that,you fhall perceine,by this, which in order com- meth lohn Dee his Mathematicall Preface. meth next. . For,wheras, it is fo.ample and woriderfull thar, one ight finde fruitfull matter therin,to f peake of:and alfo i fure endeles:yet will I glanfe ouerit,with wordes very few. THis dol call Anthrop ographie. Which isan Artreftored , and of my pteferment to your Seruice. — I pray you, thinke of It , as of oné of the chief pointes,of Humane knowledge. Although it be, but now, firtt Cofirmed, with this new name ; yet the matter, hath from the beginning, ben in confideration of all an whole yeare long, n practife,is a Threa- perfect Philofophers.. Anthropographie,is the defi cription of the Num- » ber,Meafure, Waight, figure, Situation, and colour of euery diuerfe thing, conteyned in the perfea body.of MAN : with certain know- ledge of the Symmetrie , figure , waight , Characterization, and due local] motion,of any parcell of the fayd body, afsigned: and of Ni- bers,to the fayd parcell appertainyng. This,is the one partofthe Dcfini- tion,mete for this place*Sufficient to notifie, the particularitie, and excellen cy of the Arte:and why itis, here , afcribed to the Mathematicals. Yfthe defer ption of the heauenly part of the world,had a peculier Art,called Aftronomie : lf the de- {cription of the earthly Globe, hath his peculier arte,called Geographie. If the Mat- ching of both,hath his peculier Arte,called Cofmographie: Which is the Def criptiO of the whole,and vniuerfall frame of the world: Why fhould not the defcription of him,who is the Lefle world:and,fr6 the beginning, called (Microcofmus(thatis. The Lefs eWorld. )And for whofe fake, and feruice, all bodily creatures els, were created : Who,alfo,participateth with Spirites,and Angels:and is made to the I- mage and fimilitude of God:haue his peculier Artzand be called the w4te of Artes: rather, then, either to wanta name,or to haue to bafe.and improprea name? Yon mutt of fundry profeffions,borow or challenge home, peculier partes hereof:and farder procede:as,God, Nature, Reafon and Experience fhall informe you. The Anatomiftes will reftore to you,fome part: The Phyfiognomittes,fome: The Chy- romantiftes fome. The Metapofcopiftes,fome: The excellent » Albert Durer,a good part:the Arte of Perfpectiue,will fomwhat,for the E ye,helpe forward : Pythagoras, Hipocrates,P lato,Galenus, Meletius,& man y other (in certaine thinges ) will be Con- tributaries. And farder,the Heauen,the Earth,and all other C reatures, will eche fhew,and offertheir Harmonionsfervice , to fill vp,that,which wanteth hereof: and with your own Experience, concluding: you may Methodically regifter the whole.for the pofteritie':. Whereby, good profe will be had, ofour Harmonious, and Microcofinicall conftitution. The outward Image,and vew hereof: to the Art « Of Zographie. and Painting, to Sculpture ,and Archite@ure : (for Ch MrAN_% the Leffe World. * urch,Houle, Aricroco/ Fort,or Ship) is moft neceffary and profitable : for that, itis the chiefe bafe and mus. - foundation ofthem’.’ Lookein * Vitruuius, whether I deale behoufe, orno. Lookein Albertus Durerus,De : in the 27.and 28. Chapters,ofthe fecond booke, De occulta Philofophia . .. Confi- der the Arke'of Noe. And by that, wade farther ... Remember the Delphicall Oracle NOSCE TEIPSVYM ( Knowe thy felfe ) fo long agoe pronounced: of fo many a Philofopher repeated : and of the Wifes attempted : And then »you will perceaue, how long agoe, you haue bene called to the Schole, where this Arte mightbe learned, Well. Iam ‘nothing affraydejofthe difdayne of fome fiich, as thinke Sciences and Artes, to be but Seuen. Pethaps;thofe Such,may, with igno- rance, and fhame enough, come fhort of them Seuen alfo: and yet neuertheleffe sane Colilj. they fincerely for your *£76.3. SymmetriahumaniCorporis. Looke ©4P+ oa é —_— - — ~- . : = . are ery = - = : ————— = Pres . oT ———— nee gee —— TT = at SA Gee ke - Zz 5 ieee —- = « - > + = a S ~ = om Se a, aden —. Se a < eeee - 3 se Neer << = ae S o, . =-—-—-— TET 8 = at ners veer ee f- — A eee yo, nd me oe ee ‘ aS en a a een ~~ rs am r a ~ : - = — —- ——_ " ‘ _ - _ — /- ~t— ~ es > os i ‘ ~ = - ae moter ge ai - es Tem pemrse tr —S —_ Siete == - wae ae) ae a % ——t ——s te re ~- os ee eee " 7 rt we . a8 ne 2 ie = S a rer — Pe a Ss ~ See 3Oee Gee as. | 2 = es Sot Se eo a ron ‘ Reece — * S ‘Sia r immo “tpi ~ a . oe pegs ee Pe > or ee aaa ; ae . = = ‘ . === < re - 2 iB “4 ee ee ee ak. = od .™— we A eins pp Pu i i y " A ; Ay a4 ae ee Hf), ; ai } Pie > ibe 4 fy P| Say 4h ah $ al re ai he y! ae aS 4 J ‘| ar ual bers be hy taf 44 «J iY Pe 4" \e Bi? Bi th. ad Boh, ck, 3Fhe) Pit Hy it): eg = r= 3 > ~ Pm : S2>, Se et oe ee > . ey '€ Saw Milles, *~ Fiheneus Lib.5.cap.8. Proclur. Pag. 8. Iohn Dee his Mathematicall Preface. they can not prefcribe a certaine number of Artes:and in eche,certaine vnpaflable boundes,to God,Nature,and mans Induftrie. New Artes, dayly rife vp: and hiere was no fuch order taken,that,All Artes, fhouldin one age,or in one land,or ofone man,be made knowen to the world.Let vs embrace the giftes of God , and wayes to wiledome , in this ume of grace , from aboue, continually beftowed on them, who thankefully will receiue them: £t bonis Omnia Cooperabuntur in bonum. Trochilike, is thar Art Mathematicall wwhich ‘demonftrateth the properties ofall Circular motions, Simple and Compounde. And bycaufe the frute hereof,vulgarly receiued,is in Wheles, it hath the name of Trochilike: asaman would fay,Whele Art.By this art,a Whele may. be geuen which fhall moue ones about, in any tyme afligned . Two Wheles may be giuen, whofe turnynges about in one and the fametyme, ( or equail tyes) , fhall haue, one to tie other, any proportion appointed. By Wheles, may a ftraight line be defcribed : Likewife,a Spirall linein plaine,Conicail Section lines,and other irre- gular lines, at pleafure, may be drawen. Thele, and fuchlike, are principall Con- clufions of this Arte : and helpe forward many pleafant and profitable Mechani- call workes: As Milles,to Saw greatand very long Deale bordes , no man being by. Such hauelfeenein Germany : andin the Citieof Prage : in the kingdome of Bohemia : Covning Milles,Hand Milles for Corne grinding: Andail maner of Milles,and Whele worke: By Winde, Smoke, Water, Waight, Spring, Man or Beaftmoued. Takein your hand, and is, An Arte Mathema- ticall, which demonftrateth the defigning of all ‘Spirall lines in Plaine, on Cylinder, Cone , Sphere , Conoid , and Sphzroid, and cheir properties appertay ning ; ihewe hereof, in Architecture , and di- uerfe Inftrumentes and Enginés,1s moft necefiary. For,in many thinges, the Skrue worketh the feate, which, els,could not be performed. By helpe hereof, itis *tecorded, that,where all the power of the Citie of Syracufa,was not hable to moue acertaine Ship(being on grotind)mightie Archimedes fetting to, his Skruifh Engine, caufed Hzerothe king, by him fel, at eafe,toremoueher, as he would, Wherar,the King wondring : Aw THUT HE. THs HLuNpac, wept navras, Apxined AEyorrt Teseutéow. From this day, forward (faid the King .) Credst ought to be ginen to _dr chimedes, what focner he fayth. be | pve , ; Pheumatithmie demonftrateth by clofe hollow Geometri- call Figures,(regular and irregular ) the ftraunge -properties( in mo- tion or ftay)of the Water, Ayre,Smoke:, and Fire,in theyr eg an ‘ ’ yy f mS ¥ y Tohn Deé his Mathematical Prafice. and as they areioyned to'the Elemientes next them. This Arte, to the N2turall Philofopher,is véty profitable: to proue,that Vacuum , or Emptinesis not in the world. And that,all Nature, abhorreth it much: that, contrary to ordi- nary lawjthe Elementés will moue or ftand: As, Water to afcend:rather then be- twene him and Ayre,Space or place fhould be left, more then (naturally )that qua- titic of Ayre requireth,or¢an fill: Againé, Water to hang, and not defcend: rather then by defcending,to leatie Emptines at his backe . The like,is of Fire and Ayre: they will defcend:when, either; their Cétinuitie fhould be diffolued: or their next Element forced from them. ~ And as they will notbe extended,to difcontinuitie: | So,will they not,nor yet of mans force,can be preft or pent,in ff pace , not{ufficient and aunfwerable to their bodily fubftance.Great force and violence will th cy vie, to enioy their naturall rightand libertie. creupon, two or three men together, by keping Ayre vnder a great Cauldron . and forcyng the fame downe, orderly, may without harme defcend to the Sea bottome : andcontinue there atyme &c. Where, Note,how the thicker Element(as the Water) giveth place to the thynner (as,is the ayre: )and receiueth violence of the thinner,in maner. &c, Pumps and all maner of Bellowes, haue their ground of this Art: and many other ftraungede- uiles.As, Hydrasulica,Organes goyng by water. &c. Of this Feat, (called common- ly Preumatica, ) goodly workes are extant ,bothin Greke,and Latin. Withold and learned Schole men, it is called Scientia de pleno cy vatuo.. Menadrie, isan Arte Mathematicall, which demonftrateth how , aboue Natures vertue and powerlimple: Vertueland. force may be multiplied : and fo, to direét,to lift, to pullto.,and to put or caft fro , any multiplied or fimple , determined Vertue , Waight.or Force: naturally not,fo., dire@tible or moueable. Very much is this Art furdred by other Artes - as, in fome pointes, by Per/pectiue: in fome, by Statike: in fome,by Trochilike-and in other,by Helicofophie:and Pneumatithmie. By this Art, all Cranes, Gybbettes,& Ingines to lift vp , orto force any thing,any maner way, are ordred: and the certaine caufe of their force,isknowne: © As,the force which one nian hath with the Duche waghen Racke: therwith,to fet vp agayne,a mighty waghen laden,being ouerthrowne. The force of the Croflebow Racke, is certain- ly,here,demonftrated. The reafon, why one ma, doth with a leauer, lift that,which Sixemen, with their handes‘onely, could not; fo eaftly do. ~ By this Arte,in our common Cranesin London’, where powre isto’ Crane vp, the waiglit of 2000, pound:by two Wheles more (by good otderadded ) Arte concludeth, that there inay be Craned vp 200000.pound waight&c.So well knew Archimedes this Arte: that he alone, with his deuifes and engynes,(twife or th rife) {poyled and difcomé- ted the whole Army and Hofte‘of the Romaiiies, belieging Syracu[a, Marius Mar- cellus the Conful, being their General Capitaine. Such huge Stones, fo many, with fuch force , and fo farreS didtheswith his engynes hayle among them, out of the Cite. And by Sea likewiie : though their Ships might come to the ‘walls of Syra- eujas yet hee vtterly confounded theRomaine Nauye. What with his mighty Stones hurlyng:what with Pikes of * 8 fote long,made like fhaftes: which he for- ced almofta quarter of a myle. What, with his catchyng hold oftheirShyps ,and hoyfing them vp.aboue the water , andfi uddenly letting thein fall ihtothe Sea a- gaine:what with his* Burning Glafles:by which he fired their other Shippes afar- of: what, with his other polliciés,deuifes, and engines, he fo manfully acquit him felte : that allthe Forée,couragéjand pollicie of the Romaines (for a great en) shake could / _—- Ce ——_——— To goto the bottom of the Sea Without daunger. Plutarchus in Mar- co Marcello. Synefius in Epifto- lis. Polybius. Plinius, Quintilianus. T. Liwius. * Athenaus, * Galenus, Anthemiu:. ba ln =. John Dee his Mathematicall Preface. could nothing preuaile,for the winning of Syracufa., Wherupon , theRomanes named Archimedes,Briareus,and Centimanus. Zonarasmaketh mention ofonePre- IN clus, who fo well had perceiued Archimedes Arte of AMenadrie, and had fo well in- | uented of his owne, that with his Burning Glafles, being placed vpon ithe walles of Byfance , he multiplied fo the heate of the Sunne,and directed the beames of the {ame againft his enemies Nauie with fuch force. and fo fodeinly, (like lighte- ning)that he burned and deftroyed both manand thip... And Dven f{pecifieth of Prifcus,a Geometricien in Byfance,who inuented and vied fondry Engins, of Force multiplied : Which was caufe, thatthe Evperour Senerus pardoned him, his life,af- rer he had wonne Byfance: Bycaufe he honored the Arte, wytt, and rare indultrie of P7ifcus. But nothing inferior to the inuention of thefe engines of Force,was the inuention of Gunnes. Which, from.an Englifh man,had the occafion and order of firftinuenting: though in an other land, and by other men,it was firft executed. . _ And they that fhould fee the record, where the occafion and order generall, of »» Gunning, is firft difcourfed of, would thinke:that,fmall thinges,flight,and cOmon: , comming to wife mens confideration,and induftrious mens handling , may grow to be of force incredible. Hypogciodie, is an Arte Mathematical, demonftratyng,how, vnderthe Sphericall Superficies of the earth, at any depth , to any perpendicular linc afsigned (whok e diftance from the perpendicular oftheentrance: and the Azimuth, likewife,in refpect of the {aid en- trance; is knowen)certaine way may be pre{cribed and gone : And how) any way aboue the Superiicies of the earth defigned , may vn- der earth at any depth limited , be kept : goyng alwayes , perpendi- cularly,vnderthe way, on earth defigned : And, contrarywile,Any way, (ftraight or croked , )vnderthe earth, beyng giuen : vppon the vtface,or Superficies of the earth,to Lyne out the fame: So,as, from ‘the Centre of the earth., perpendiculars drawen to the Sphericall Suiperficies of the earth’, {hall precifely fall in the Correfpondent pointes of thofetwowayes . “This, with all other Cafes and cir- cumftances herein, and appertenances , this Arte demonftrateth . This Arte, is very ample in vanietie.of Conclufions: and very profitable fundry wayés to the Common Wealth:.: The occafion, of my Inuenting this Arte,was at the requeft oftwo Gentlemen,who hada certaine worke(of gaine)vnder ground: and their groundes did ioyne quer the worke ; and by reafon of the crokednes, diuers depthes, and heithes of the way vnder ground, they were in doubt, and at controuerfie, vnder whofe ground, as then, the worke was . The name onely (be- forethis ) was of me publifhed, De Itinere Subterraneo: The reft,beat Gods will. - For Pioners, Miners, Diggers for Mettalls,Stone, Cole, and for fecrete paflages vnder ground,betwene place and place ( asthis land. hath diuerfe ) and for other purpofes,any man may eafily perceaue, both the great fruite of this Arte, and alfo in this Arte, the great aide of Geometnie, Hydragogie, demonftrateth the poflible leading of Water, by Natures lawe®, ‘and by artificial helpe , from any head (being a - Spring, ftanding, or running Water ) to any other place afligned, Long lohn Dee his Mathematicall Priefacei Long,hath this Arte bene in vie : and muchthereof written sand very marucilous wagkes thercin,performed : as may yet appeare,in Italy:by.the Ruynes remaining ofthe Aqueductes..In.other places,of Riuers leading through the Maine land, Nauigable many a Mile., And in other places,of the marueilous forcinges of Wa- terto Afcend . which alljdeclare the great {kill,to be required ofhim,who fhould in this. Arte be perfecte, for all occafions of waters poffible leading. Tolpeake ofthe allowance of the Fall, for euery hundred foote: or of the Ventillsafthe wa- ters labour be farre,and great) Incede not: Seing, at hand (about vs)many expert men can {ufficiently teftific, in effecte, the order: though the Demonttration of the Neceffitic thereof ;they know not : Noryet,if they fhould be led, vp and downe, and about Mountaines, from the head of the Spring:and then,a place be- ing affigned : and of them, to be demaunded, how low or high; that laft place is, in refpecte of the head, from which (fo crokedly,and.vp anddowne ) they be come: Perhaps,they would not, or could not, very redily,or nerely afloyle that queftion. Geometrie therefore, is neceflary to Hydragogie. -Ofthe fundry wayes to force wa- ter to.afcend , eyther by Tympane, Kettell mills, Skrue, Ctefibike, or fuch like : in Vi- truuius, Agricola, (and other,) fully,the maner may appeare . And{o,thereby,alfo be moft enident, how the Artes, of Pxenmatithmie;Helicofophie, Statike , Trochiltke, and cMenadrie, come to the furniture of this,in Speculation, and to the Commo- ditie of rhe Common Wealth,in practife. Horometrie,.is an Arte Mathematicall, which demGftrateth, how,at all times appointed, the precife vfuall denominati6 of time, may be knowen,for.any place aliigned . Thefe wordes,are {moth and laine eafie Englifhe, but the reach of their meaning,is farther, then you woulde lightly imagine. Some patt of this Arte, was called in olde time, Guomonice: and. of late,Horologiographia: and in Englifhe,may be termed,Dialing . Auncientis the vie, and more auncient,is the Inuention.. The vie,doth well appeare to hane bene (at the leaft) aboue two thoufand and three hundred yeare agoe +. in* King ~Achaz Diall, then,by.the Sunne,fhewing the diftinction of tunc.. By Sunne, Mone,and Sterres,this Dialling may be performed,and the precife Time of day or nightknowen. But the demonfiratiue delineation of thete Dialls,of all fortes, requireth good {kill both of Astrozomie,and Geometric Elementall,Spheericall,Pha- nomenall,and Conikall. . Then,to vie the groundes of the Arte, forany regular Superficies, in any place offred : and (in any poflible apt pofition therof ) theron, to deféribe (all maner of wayes ) how, viuallhowers, may, be ( bythe Susues.fha- dow: ) truely determined : will be found no fleight Painters worke. Soto Paint, and -prefcribe the Sunnes Motion,to the breadth ofa heare. In this Feate(Gin my youth )Idnuented away, How in any Horizontall,Murall,or Aiquino- Ctiall, Dialljécc. Atall howers(the Sunnefhining)the Signeand De- oréé afcendent,may be’ knowen. “Which is a thing very’ neceflary for the Rifing of thofefixed Stetres : whofe Operation in the Ayre, is of great might, evidently. I {peake no further,of the vie hereof. But forafmuch as,Mans affaires require knowledge of Times & Momentes,when,neither Sunne,Mone,or Sterre, can be fene: Therefore,by Induftrie Mechanicall, was inuented, firft,how,by Wa- ter, running orderly,the Time and howers mightbe knowen: whereof, the famous Ctefibius, was Inuentor : aman, of Vitruuius, to the Skie (iuftly) extolled. Then, after that, by Sand running, were howers meafured ? Then, by Trochilike with waight: And of late tine; by Trochilike with Spring : without waight. All ame d.t. y 4.Reg.20. Ai perpetual MMotson, Iohn Dee his Mathematicall Preface. by Sunne or Sterres direction ( in certaine time ) require ouerfight and reformati- on, according to the heauenly Aquinodiall Motion: befides the inxquality of their owne Operation . There remayneth (without parabolicall meaning herein ) among the Philofophers,a more excellent, more commodious,and more maruei- lous way, then all thefe : ofhauing the motion ofthe Primouant (or firft equino- Giall motion, by Nature and ’Aite, Imitated:which you fhall( by furder fearch in waightier ftudyes ) hereafter,vnderftand more of. And fo, itis tyme to finifh this Annotation,of Tymes diftinction,vfed in our common,and priuate affaires: The commoditie wherof,no man would want, that ean tell,how to beltow his tyme. Zographie,is an Arte Mathematicall which teacheth and de- monftrateth , how , the Interfection of all vifuall Pyramides , made by any playneafsigned, ( the Centre, diftance,and lightes ,beyng de- termined ) may be, by lynes,and due’ propre colours, reprefented. A notable Arte,is this:and would require a whole V olume,to declare the proper- ty thereof: andthe Commodities enfuyng. Great {kill of Geometric, -Arithme- tike,Per|pectine,and Anthropographie,with many other particular Artes,hath the Zo- grapher nede of, for his perfection. For, the moft excellent Painter,(who is but the propre Mechanicien, & Imitator fenfible,of the Zographer) hath atteined to fuch perfection,that Senfe of Man and beatt,haue iudged thinges painted, to be things naturalland not artificiall:aliue,and not dead. This Mechanical! Zographer(com- monly called rhe Painter)is meruailous in his{kill:and feemeth to haue acertaine diuine power: As,of frendes abfent,to make a frendly , prefent comfort; yea, and of frendes dead,to giuea continual, filent prefence : not onely with vs , but with our polteritie,for many Ages. And fo procedyng, Confider, How , in Winter,he can fhew you,theliuely vew of Sommers loy,and riches:and in Sommer,exhibite the countenance of Winters dolefull State,and nakednes. Cities, Townes, Fortes, Woodes,Armyes, yea whole Kingdomes (be they neuer fo farre, or greate ) can he, with eafe,bring with him, home(to any mans Iudgement ) as Paternes linely, of thethinges rehearfed. In one little houfe, can he,enciofe(with great pleafure ofthebeholders,)the portrayture lively, of all vifible Creatures, either on earth,or in the earth; lining:or in the waters lying,Creping,flyding,orfwimming:or ofany foule,or Ay,in the ayre Aying. Nay,in refpect of the Starres the Skie,the Cloudes: yea, in the fhew of the very light it felfe (that Diuine Creature ) can he match our eyes Iudgernent,moftnerely, Whata thing is this:thinges not yet being,he.can reprefent fo , as,at their being, the Picture fhall feame (in maner)to hane Created them. To what Artificer,is not Pidture,a great pleafure and Commoditie? Which ofthem all will refufe the Direciton and ayde of Picture? The Architect, the Gold- fmith,and thé Arras Weauer:of Pi@ure,make great account. Our liuely Herbals, our portraitures of birdes, beaftes,and fifhes : and our curious Anatomies,which way,are they moft perfectly made,or with moft pleafure,ofvs beholdenz Is it not, by PiGure onely? And if Picture , by theInduftry of the Painter, be thus commo- dious and meruailous: what fhall be thought of Zographie,the Scholematter of Pi- Gure,and chiefgouernor? Though I mencion not Scu/ptwre,in my Table.of Artes Mathematical ; yetmay allmen perceiue,How,that Picture and Sculpture, are Si- fters germaine:and both,right profitable , ina Comm6 wealth.and of Sculpture,af- wellas of Pi&ture,excellent Artificers haue written great bokes in commendation. Witnefle I take, of Georgio Vafari,Puttore Aretino:of Pomponinus Ganricus: and other. To thefe two Artes, (with other, )is a certaine od Arte, called Althalmafat, much beholdyng: more, then the common Sculptor,Entayler,K eruer, Cutter,Grauer, Foun- der, Sa ° lohn Dee hisMathematicall Pr#face. der, OF Paynter (oe) know their Arte,to be commodious, ba | wArchitectur €,to many may'feme not worthy, or not mete, to be reckned Ax obiection. among the Artes ~2 * - . —s a Sy an Se Sr on bes - ae . ee ; 4 : a nf Ta See . * ay lohn Dee his Mathematicall Preface. Hed,the Prouoft , the Directer,and Iudge of all Artificial workes ; andall Artifi- cers.For,the true Architect,is hable to teach, Demonttrate,diftribute,defcribe fand Tudge all workes wrought. And he,onely,fearcheth out the caufes and reafons of all Artificial thynges. Thus excellent,is Architecture:though few (in our dayes)at- teyne thercto : yet may notthe Arte, be otherwife thought on, then in very dede itis worthy. Nor we may-not,ofauncient Artes,make new and imperfect Definiti- ons in our dayes:for {earfitie of Artificers : No more,than we may pynche in,the Definitions of Wifedome,or Honeftie , or of Frendefhyp ox. of luftice.. No more will I confent,to Diminith any whit,of the perfection and dignitie , (by: iuft-caufe ) al- lowed to abfolute Architecture. Vader the Dire@ion. of this Arte y are thre prin- cipall,neceflary Mechanicall Artes. Namely , Howfing.y Fortification and Nanpegie. How/fing, | vnderftand,both for Diuine Seruice,and Mans.common viage: publike, anehpauane Ot Fortification and Nampegie, ftraunge matter might be told you: But perchaunce,fome will be tyred,with this Bedcroll, all ready rehearfed: and other fome, will nyccly nip my groffe and homely difcourfing with you :, made in poft haft for feare you fhould wante this true and frendly warnyng, and taft giuyneg, Of the Power’ Mathematical. Lyfe is {hort, and vncertaine : Tymes are periloufe: &e Jo And fill the:Printer awayting, formy pen ftaying : All thefe thinges,with farder mattcr of Ingratefulnes, gine me occafion to pafle away , to the other Artes remainyng, with all {pede pofsible. A He Atte of Nauigation » demonftrateth how ,by the fhorteft _ good way, by the apteft Diretid ,& in the fhorteft time, afufficient .. Ship, betwene any two places (in‘paflage Nauigable,)afsigned: may hod : Setcigy Anno. 559% becoducted: and in all ftormes;écnaturall difturbances chauncyng; how ,to vfe the beft pofsible meanes , whereby to recouer the place firft alsignied - What nede, the Maffer Pilote, hath ofother Artes ; heré¢ before recited, it ts.eafie to know:as, of Hydrographie, Astronomies Astrologie , and H OY OME « trie. Peeluppoting continually the common Bafe,and foundacion ofall: namely “Arithmetike and Ceometrie. So that,he be hable to vnderftand,and Iudge his own neceflaty Inftramentes,and furniture Neceffary: Whether they be perfedly made orno:and alfo can, (ifnede be)make them, hym felfe. As Quadrantes, The Aftro- nomers Ryng, The Aftronomers ftaffe, The Aftrolabe vninerfall. An Hydrogra- phicall‘Globe.Charts Hydrographicall,tru¢, (not with parallell Meridians); ‘The GommonSea Compas: The Compas of variacion: The Proportionall;and Para- doxall Compafles (of me Inuented,for our two Mofcouy Matter Pilotes, at the re- ueft of the Company). Clockes with {pryng: houre,halfe houre,and three houre Sandglaffes:& fundry other Inftrumétes: And alfo, be hable,on Globe, or Playne _todefcribe the Paradoxall ‘Cothpafle? and duely to vfe the fame,to all maner of purpofes, wheretoitwas inuented. And*alfo, be hable to Calculate the Planetes places forall tymes, «: ey aah Moreouer,with Sonne Mone or Sterré(or without)be hable to define the Lon- _ girude& Latitude of the place,which heisin:. So thatthe Longitude & Latitude of: theiplace,from which hefayled,be giuen:ar by him,beknowne.wh creto,apper- tayherh expert meanes,to be certified ener,of the Ships way.. &c. And by forefe, inthe Riling,Settyng , Noneftedyng , or Midnightyng of certaine tempeftuous fixed Stertes : or their Coniunttionssand-Anglynges with the Planetes,, &.he oughtto haue- expert coniecture of Stormes. Tempeftes,and Spoutes:and {uch lyke Meteorological effectes,daungerous on Sea, For(as Plato fayth,) Mutationes, | i. opper- lohn Deéé his Mathematical Prieface. opportunitate/ 4, temporume prefentire, non minus rei militari, quam Agriculture, Nam@n= tionig, conuenit. Lo forefeethe alterations and opportunities of tymes 1s connes nient , no lefSe to the Art of Warre , then to Huf bandry and Nanuigation. And befides fuch:cunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to be knowen: fuch as(the Philofophicall Poéte)#giliws teacheth, in hys Georgikes. Where he fayth, | | | Sol quog, cy exoriens cr quum fe condet in undas, Signa dabit,Solem certiamea figna fequuntur Ge. Nam fepe videmus, Ipfiusin vultuvarios errare colores. Georgic,%. Caruleus, pluuram denunciat igneus Euros. Sin ye J incipient riitilo immilcerter ign, Omnia tum pariter vento,nimbi{ 4g, videbts Feruere: non illa qui{quame me notte per altum Ire, neg, a terra moneat conuellerefunem. Oe Sol tibi figna dabit Solem quis dicere fal{um _Audeat? ———_— Cri: | And {oof Mone, Sterres,Water,Ayre, Fire; Wood, Stones, Birdes,and Beaftes; and of many, thynges els,a certaine Sympathicall forewarnyng sar be had: fome> tymes to great pleafure and profit, both, on Sea,and. Land... Sufficiently, for my prefent purpofe, it doth appeare, by the premiffes , how Mathemsaticall, the Arte of Nanivation, is:and how it nedeth and alfo vieth other. Mathematicall Artes: And now; if would’ go about to fpeake of the manifold Commodities, commyng to this Land, andothers, by Shypps'anid Nasigation , you might thinke , that I catch at occafions’, to vfeimany wordes where no nedc is. | i. Yet, this.onethyng may], (inftly) fay. In) Nawigation,none ought to haue grea» ter care,to be fkillfull,then our Englifh Pylotes.. And perchaunce,Some, would moré attempt: And other Some,more willingly would be aydyng, if they wiftcer- tainely, What Priniledge,God had endued this Iand with, by reafon of Situation, moft commodious for Nasigation, to Placés moft Famous & Riche. And though, nes, with good hope, and great caufes of perfuafion,to haue ventured, for a Dil- couerye, (cither Weflerly, by Cape de Paramantia : or Ejferly, aboue Nowa Zemla; and the Cyremiffes and wasjat the.very nere tyme of Attemptyng,, called andem- ployed otherwife(both then,and fince, in great good feruice to his, Countrey), as the Irifh Rebels haue.* tafted: Yer, I fay, ( Loe the fame Gentleman , doo not hereafter,dealetherewith)Some one,or other, fhould liftento the Matter: and bi good aduife,and difcrete Circum{pettion , by little, andilittle; wynneto,the fa cient knowlédgerof that Pradeand Voyage: Which ; now; I would be fory, (through Carelefhefle, want of Skillsand Courage, ) fhould remayne Vnknowne andvnheard of. Seyng, alfo,weare herein, halfe Challenged, by the learned, by halfe requeft,publifhed.. Therof,verely, might grow. Commoditye ; to this Land. (of* Late) ayoung Gentleman,a Cotirragious. Capitaine , was in a great ready- ‘ * Anne.15 69 chiefly , ‘and to the reft of the Chriften Common wealth, farre- paffing all riches - cand worldly Threafure. | | = *Thaumatti teike sis that Art Mathematicall; which giueth cer- taine order tomake ftraunge workes, ofthe fenfeto be perceiued, and of men greatly to be *wondred at. By fundry meanes, this Wonder- workeis wrought. Some,by Pueumatithmie . As the workes of Ctefibius and Hero, Aj. Some lohni:Deéd his Mathematicall Preface! Ta = SS =—_ : —_ ery ——— + ~~ De his qua Mundomi- ye rabiliter eue- ~ ninnt. cap.3 : Tufe. i, -oosht Bute at large-which Fomit.. “The Doue of wood »Avhich the Mathematiciem.dr- a cbytas didmaketo fiyets by Cagellius{poken of .Of Daedalus ftraunge Images, Plato reporteth: Hemere Of) wicans Sélfinontrs, (by fecret wheles) leaucth in writyng . Arz- in hys P efirakes, of both maketh mentions *\Mefuaylous was'the workeman- Say, of late dayesspexformed by good fkill of 2 rachitike! oc. For in Norembersé, A fyeotlerh jdynglerout efthe Artificershand did (as it were) Ay about by thie ets" sgeftesat rhe cablejand aclengthsas'though it wéfe' weary , retourrie tohis matters - shabdagayne sMoreouet}dnAtuificiall Egle was ordred 5 to ly out of the faite Towne,omightyway;aid:tharaloftin the Ayrestoward the Emperourcomifihg thethes:and followed hy, beyug-come to thegate-ofithe towne.* fhus;yourfee, cP * Wwaat, Arte Mathematicallican-pestorme when Skilhy will , Induftry; and Habili- dyjare duclyapplyed.to Prafe.susd isied siwow.olls cn yo Aebipotiery DMB bas ANd forthefe,and fuch like marueilous Aes and Feates, Naturally, Mathie- €., s - / - f { i ; : J 1; ( i \ ( 7 ’ Dit 4 j al at hd ‘ 4 Ki net | ‘ t bald Ale mt sah, 4 The 54 4 Aue kh ¥ 1g, ist } iP +e ‘ ALTA eae ; ; ‘i pad 4 ; , a i | " 4 a { oI 4 iy {pa} wa FEB De 4 tea Pe as A) 4 Pee tal) 4 i eee \ ii Aer A.) | » Ree it fone « ‘ ¥ £i | HA ¥ \§ge t bY . 2 4 4 ne Na On 7g Pe af » Peet ' i 4: a, x 4 FAG wr Me ete ; fh. oa) ay ues abel F at ve He ia 4 28 $3 : 7 ry 5 the y beat ty ue were 5 as te ee Pe | ae “peters By et 1 pal nu {+ 4 ; ett © a; rey Ke eals f: Ble ~ cet . PaREIsE: ar & terefiion. 4 + > ee Me on reel’ } Oe 40 et ot .0) ait 4 fh pole reat HERE AA AE Mechanically, wrought and con tried : ought any honeft: Student, and Modeit Chriftian Philofopher,be counted,& called a: Coniur t ? Shall the folly GfLdio 28,4 the Mallice oF the Score fomuch pretiaile, that He. who fegkéth no worldly gainé orglory at theirhandes . But onely,6f Godsthe threafer _otheaucnly wiledome;& knowledge of pune versties Shall he f fay):in the nieane ges big warded 10 2o210W Mt 2A. . sesatinaend Y a 2G CARI Apades: . % . pe ; ¢ i 54.0% AlAs &- a ‘ ~~ John Dee his Matheniaticall, Preface; fimce, be robbed and fpoiled of his honeft name and fame ¢ He that feketh ( by $3 Paules aduertif{ement ) in the Creatures Properties,.and: wonderfull vertues, to finde iufte caufe, to glorifie the Aternall,and Almightie Creator by: Shall that man, be (in huggermugger) condemned, as aGompanion of the Helhoundes, anda Caller, and Coniurer of wicked and damned Spirites¢ He that bewaileth his great wantof tine,fufhicient(to his contentation)for learning of Godly wifdome, and Godly Verities in : and onely therin fetteth all his delight : Will that maleefe | and abufe his tume,in dealing with the Chiefe enemie of Chriftour Redemer; the deadly foe of all mankinde : the fubtileand impudent peruerter of Godly Veritie: the Hypocriti¢all Crocodile: the Enuious Bafilifke, continually defirous, in the twinke ofan eye, to deftroy all Mankinde, bothin Body and Soule, xternally ¢ Surely (for my part,fomewhat to fay herein) L:hauenot learned to make fo brutish, and fo wicked a Bargaine. Should-I, for my xx.or xxv.yeares Studie : for two or three thouiand Markes {pending : feuen or eight thoufand Miles going and .trauai- ling,onely for good learninges fake : And that, in all maner of wethers : in all ma- ner of waies and paflages : both early.and late: in daunger of violence by man : in daunger of deftruction by wilde beaftes:: in hunger ‘in thirft : inperilous heates by day, with toyle‘on foote : indaungerous dampes of colde,by night, almoft be- reving life (as God knoweth): with lodginges; oft times,to {mall eafe::and fom- tume.to lefle{eeuritie. And for much more (thenaall this) done & {uffred; for Lear- ning and attaining-of Wifedome: Should I ( I pray you) forall this;no otherwife, nor more warily :.or (by Gods mercifulnes ) no more luckily, haue fifhed, with fo large,and coftly;a Nette, folongtimein drawing (and that with the helpeandad-, uife of Lady Philofophie,& Queene Theologic):: butat length,to have carched; and drawen vp,*a Frog ¢ Nay,a Deuill 7 For,fo,dotlnthe Common peuifh:Pratler Imagine and langle: And,fo,dorth the Malicious fkorner,{ecretly wifhe,& brauely and boldly face down,behinde my backe:. Ah,wHatamiferable thing, is thiskinde of Men? How greatisthe blindnes &:boldnessof the Multitude, in thinges aboue their G@apacite-< What a Land: whata People : what Maners : what Times are thefe: Are they become Deuils,them felues:.and;by falfewitneffe bearing againft their Neighboiir, would they alfo, become Murderers. ¢: Doth God, fo long geue them refpite,to reclaime them felues:in, from this horrible {laundering of the gilt lefie:. contrary to.théir owne Confciences ::and yetwillehey norceafe¢ Doth the Innocent, forbeare the-calling of them, Luridically to aunfwere him,according to digrigour of the Lawes : and willthey defpife his Ohatitable pacience? As they, againit him; byname,do forge, fablejrage;and raiféflainder, by Worde & Ptint: Will they provoke him; by worde and Print, likewafe;to N ote their Names tothe World : withtheir particular devifes;tables,beaftly Imaginations, and vnchriften- like flaunders ?) Welk : Well! Frendes, leaneof, fo to commend your vnacquainted frend,vpon blinde affection: As, becaufe he knoweth more,then the common Student: that, therfore, he muft needes-be fkilfull; anda doet;in fach matterand maner, as you terme Coniaring? Weening,thereby, you aduauncehis fame : and that youmake other men, great marueilers of your hap, tohauefucha learned-frend . Ceafeto afcribe Impietie, where you pretend Amitie. ‘For, if your tounges were truc, then: were that your frend, Vntrue; both to God,and his Soueraigne. Such Frendes and Fondlinges, I fhake of, andrenounce you: Shake you of,your Folly. Imperfec#ly zelous,to you, 3+ do I fay: that (perhaps) well, do youMeane: Batfarre you mifle the Marke: Ifa Lambe you-will kill, tofeede' the flocke with his bloud. Sheepe, with Lambes blonds hawe no:naturall fiftenaunce : No more,is:Chriftes flocke, with horrible flanndets:; duely edified » Noryour faire pretenfe, by fuch rafhe ragged Rheto- rike,any whit,well graced. But fuch,as fovie me, will finde a fowle Cracke in their Credite. Speake that yowknow : And know; 2s:you ought : Know not,by Heare fay, when lite lieth in daunger: Seatch to the quicke,& let Charitiebe your guide. 4. (Malicious Ignorant, what thalll fay to thee? Prohibe linguamtuamamalo. A ae- srattionie parcitelingne . Canfe thy toung to refraine fro euill. Refraine your toung from flannder . Though your tounges be fhatpned, Serpent like, & Adders poy- fon lye in your lippes : yet take heede,and thinke,betmes, with your felfe, ir lin. guofus non fabilietur in terra. Virum violentum venabitur malum , donec pracipitetur. For,fure lam, Quia faciet Dominus Iudicinm affutht cp vindsttam pauperum, Thus, I require you, my affured frendes, and Countrey men (you Mathemati-_ ciens, Mechaniciens,and Philofophers, Charitable and diftrete) to deale in Soy : Iohn Dee his Mathematicall Preface. bebalf,with the light & vntrue tounged, my enuious Aduerfaries,or Fond frends. And farther, I would wifhe, that at leyfor, you would confider,how Bafilius Mag- nus, \ayeth «Mofes and Daniel, before the eyes of thofe, which count all fuch Stu- dies Philofophicall.( as mine hath bene ) tobe vngodly , or vnprofitable. Waye well S.Stepher his witnefle of .Wofes.. Eruditus est mM ofes omni Sapientia LEgyptiorit: cy erat potens inverbss Gy operibus fuis, Mofes was inStruééed in all maner of wifes dome of the Azgyptians : and be was of power both in his ~wordes, and “workes. You fee this Philofophicall Power & Wifedome,which %ofes had,to be nothing imifliked of the Holy Ghoft. Yet Plinius hath recorded, Mofesto bea wicked Magi cien .And that (of force) mutt be, either for this Philofophicall wifedome,learned, before his calling to the leading ofthe Children of J/rae/: or for thofe his won- ders,wrought before King Pharao, after he had the conducting of the //raelites. As concerning the firft,you perceaue, how S.Stephen, at his Martyrdome (being full ofthe Holy Ghoft) in his Recapitulation of the olde Teftament, hath made men- tion of Mofes Philofophie : with good liking ofit: And Bafilius Magnus alfo, auou- cheth it, tohaue beneto Mofes profitable (and therefore, I fay, to the Church of God, neceflary ). But as cocerning Mofes wonders,done before King Pharao:God, him felte, fayd : ide vt omnia oStenta, qua pofui in manutua , facias coram Pharaone. See that thou do all thofe wonders before P harao, which I haue put in thy hand. Thus, you evidently perceaue, how rafhly,P ius hath flaundered Mofes, of vayne fraudulent Azagike, laying : Est ¢ alia Macices Factio, atm ofe, Lamne,e I otape, 1u- dais pendens : fed multis millibus annorum poft Zoroastrem.¢rc. Let all fuch, there- fore, who, in ludgement and Skill of Philofophie, are farre Inferior to P/nie, take good heede, leaft they ouerflioote them felues rafhly , in ludging of PAilofophers Straunge Adtes and the Meanes,how they aredone. But, much more,ought they to beware of forging, deuifing, and imagining monftrous feates, and wonderfull wotkes, when and where, no fuch were done : no, not any fparke or likelihode,of fiich,as they, withoutall fhame,doreport. And (to conclude) moft of all, let them be afhamed of Man , and afraide of the dreadfull and Iufte ludge: both Fo- lifhly or Malicioufly to deuife : and then,deuilithly to father their new fond Mon- fters om me : Ifinocent, in hand and hart : fortrefpacing either againft the lawe of God, or Man, in any my Studies or Exercifes, Philofophicall, or Mathematicall: As in due time, I hope, will be more manifeft. . Nowend Lwith Archemattrie. which name, is not fo new,as this Arteis rare.For an other Arte,vnder this,a degree(for {kill and power) hath bene indued with this Englifh name before. And.yet,this,may ferue for our purpofe, fufiicientlysat this prefent. “This Arte,teacheth to bryng to. actuall ex- perience fenfible,all worthy conclufions by all che Artes Mathema- ticall. popes ed, & by true Naturall Philofophie concluded: & both addeth to thern a farder (cope, in the termes of the fame Artes , & al- fo. by hys propre Method, and in peculier termes, procedeth., with helpe of the ce ayd Artes , to i performance of complet Expe- neces, which of.no particular Art, are hable(Formally)to be challen- ged. If youremember,how we confidered e4chitedure, in refpect of all com- mon handworkes: fome light may you haue,therby, to vnderftand the Souerain- ty and propettie of this Science. > ‘Science I may callit,rather, then ati Arte: for the excellency and Mafterfhyp it hath ., ouer fo'many \,oand fo mighty Artes and A.lij. Sciences. MEE VE. John Dee his Mathematicall Preface. Sciences. And bycaufe it procedeth by Experiences,and fearcheth forth the cayles of Conclufions,by Experiences: and alfo putteth the Conclufions them felues, in Experience,itis named of {ome,Scientia Experimentalis. The Experimen tall Sci ence. Nicolaus Cufanus termeth itfo,in hys Experimentes Statikall, Andan other Philofopher , of this land Natiue ( the foure of whofe worthy fame, can neuer dye nor wither ) did write theroflargely, atthe requeft of Clement the fixt. The Arte carrieth with it, a wonderfull Credit: By reafon, itcertefteth , fenfibly,fully,and completely to the vtmoft power of Nature,and Arte. This Arte,certifieth by Ex- perience complete and abfolute : and other Artes,with their Argumentes,and De- monftrations , perfuade:and in wordes,proue very well their Conclufions. * But wordes,and Argumentes,are no fenfible certifying: nor the full and finall frute of Sciences practifable. And though fome Artes,hauein them, Experiences,yet they are notcomplete, and brought to the vttermolt,they may be ftretched vnto,and applyed fenfibly. As for example:the Naturall Philofopher difputeth and maketh goodly fhew of reafon: Andthe Aftronomer,and the Opticall Mechanicien,put {ome thynges in Experience:but neither, all,that they may:nor yet {uficiently, and to the vtmott,thofe,which they do, There,then,the Archema/fler fteppeth in,and leadeth forth on, the Expersences by order of his doctrine Experimental , tothe chief and finall power of Naturall and Mathematicall Artes,Of two or three men, in whom,this Defcription of Archemastry was Experimentally verified, I hauc read and hard:and good record, is of their fuch perfection. So that,this Art, is no fan- taftical! Imagination: as fome Sophifter, might, Cum fuis Infolubilibus, make a flo- sith: and daffell your Imagination:and dath your honeft defire and Courage,trom beleuing thefe thinges, fo vnheard of,fo meruaylous,& of {uch Importance. Wells as you will. Lhane forewarned you.I haue done the part ofa frende:1 haue difchar- ged my Duety toward God:for my finall Talent, at hys moft mercyfull handes re+ ceiued. To this Science,doth the Science Alairangiat great Seruice. Mufe nothyng ofthisname. I chaungenot the name, fo vfed, and in Print publifhed by others beynganame, propre to the Science. Vnderthis,commeth —4rs Sintrillia, by Artephius, briefly written. . But the chief Science , of the Archemafter, (in this world )as yet knowen , isan other (asit were), OP TIC AL Science: wherof, the name fhall be told(God willyng)when I fhall haue fome, (more iuft) occafion; therof, to Difcourfe. " Here, I muft end , thus abruptly ( Gentle frende, and vnfayned louer of honeft and neceflary verities.) For,they,who haue(for your fake, and vertues caufe)re- quefted.me,(an old forworne Mathematicien) to take pen in hand : ( through the confidence they repofed in. my long experiefice:and tryed fincerity) for the decla- ryng and reportyng fomewhat, of the frure and commodity, by the Artes Ma- thematicall,to be attceyned vnto:euen they, Sore agaynt their willes , are forced, for findry caufes, to fatifhe the workemans requeft, in endyng forthwith: He, fo feareththis, fo new an attempt,& fo.coftly: And in matter foflenderly Chez therto)among the common Sorte of Studentes,confidered or.eftemed., | | Andaehere Twas willed, fomewhat to alledge,why,in our vulgare Speche, this part ofthe Principall Science of Geometric, called Euclides Geometrical Elementes; is publifhed;to; your handlyng ::being vnlatined people, and not Vniuerfitic Scholers : Verily, I thinke it nedeleffe. | | : For, the Honour, and. Eftimation of the Vniuerfities, and Graduates; is, hereby, nothing diminifhed’, Seing, from, and by their Nurfe-Children , you receaue all this Benefite show greatfoeuer it be. | fell br: ‘se Neither re ‘fe matical! Quadtiuie’? and yetnéither Paris OHéaice, or any of the other Vniuer- {ities of Fraunée, at any time, with the 'T ranilaterssor Publifhers offended : Orany mans Studre thereby hindred? soe | odie Db: before. his comming to the Hainer fitie, thall'( ot Inay) be, now(according to Plate there to belearned.And,fo,in leffe time,profite more,then (otherwife) he fhould; wittes ; where,els (perchaned ) otherwife they wouldil find exercifes,{pend ( or d} nor furdering the Wcale,cotimon And great Comfort, With-go6d ‘hape) may the Pare fiiehatie: by reafon of this Evgijhe Geometrie,and Mathematical] Preetace,that they (hereafter) fhall be the ‘more. regarded, efteemed ,and reforted ynto. For, when it fhall be knowenand reported, that ofthe Mathematicall Sciences on ely,fuch great Commo- their owne Skill so0d helar John Dee his Mathematicall Preface. | fightabaiiftmyne:owne fhadowe. For, noman (1am fure) will open his mouth | | againft this Enterprife.N oma (I fay) who either hath Charitie toward his brother (and would be glad of his furtherancein vertuous knowledge) : or that hath any care& zealc for the bettering of the Cémon ftate of this Realme.Neither any,that make accompt, what the wifer fort of men ( Sage and Stayed ) do thinke of them. Tonone (therefore ). will Imake any Apologie, fora vertuous acte doing : and for eémending;or ferting forth, Profitable Artes to Englifh men,in the Englifh toung. »» But, vnto God our Greator, letvsallbe thankefull ; for that, 49 he of bis Good= » nes by his Powre:, andin bis wifedome hath Created all thynges, in Number, (S >> Waight,and Meafure:So, to vs , ofhysgreat Mercy, he hath reuealed Meanes, ” whereby , to atteyne the fufficient and neceflary knowledge of the forefayd hys > three principall Inftrumentes : Which Meanes , I haue abundantly proued vato »> you,to be the Sciences and _Artes M athematicall. - ‘And though I haue ben pinched with ftraightnes of tyme:that,no way, could fo pen downe the matter(in my Mynde) as I determined : hopyng of conuenient layflire':, Yct,ifvertuous zeale,and honeft Intent prouoke and bryng you to the readyng and examinyng of this Compendious treatife,I do notdoute, but,as the veritie therof(accordyngto our purpofe ) will be evident ynto you : So the pith and force therof, will perfuade you : and the wonderfull frute th erof,highly plea- fure you. And that you may the eafier perceiue,and better remember., the prin- The Ground Cipall pointes , whereof my Preface treateth , I willgiue you the Gro un dplate latt of this otmy whole difcourfe,ina Table annexed:from the frit tothe laft,lomewhat Me- Prefaceina thodically contriued. Table. TE Halt, hath caufed my poore pen,any where, to {tumble : You will, lam . fure)in part ofrecompence, (for my carnelt and fincere good will to plea- ~~. fare you) , Confider the rockith huge mountaines, and the perilous oS yabeaten wayes,which (both night and day , for the while ) it hath toyled and labored through,to bryng you this good | Newes;and Comfortable profe,of Vertues frute. So,] Commit you ynto Gods Mercyfull direction , for the reft : hartely » ~~ befechyng hym, to profper your Studyes,and honeft Intentes: to his Glory,& the Commodity of our Countrey, Amen, 7 \ | ; Written at my poore Houfe At CMortlake. Anno.t $7 0.February. 9 7¥ t ah (i we , aaa. ; vi? a { Plrern a { \ || a 5 > iy e+ 1 a? a lee t i? We | ul boii } ¥ a . ghreth : ( Ef ' vry , ‘ raat 4 . t Li ha U Ws , Ky / t i: - ; : ‘ : M : - 7 re) 4 ts my MATHEMA 3 [ Simple, Here haue you according to publifhed in our Englifhe tollnge, F. D&E. my TICALL Preface: annexedto Euchde (now firft) Which dealeth with DN umbets onely = ay An. 1570. Febr. 3. d demonftrateth all their properties and apper- 7} oo Arithmetike. Principall, which are two, onely, Sciences, and Arrtes Mathe- aticall_ 4 Geometrie. tenances : Where, an Unit, is Indiuthible, | Mix L, Which with aide of Geomentie principal, demenStrateth fome eAtrithmeticall Con- clufion, or Purpofe. [ Si mpl C, Which dealeth with Magnitndss, onely + and demonStrateth all their properties, paffi- f ens, aad appertenances : whife Posns, is ladiurfible. Mux i 9 Which with aide of eAvithmttike principal, demon frat eth fome Geometrical purpefe:as BVCLIDES ELEM ENTES. Arithmeti ke, fArithmetike of mof yf]! whole Numbers: And of Fractions to them appertaining, vulgar : which Arithmetike of Proportions. confidereth< Arithmetike Circy]ar. The vfe whereof, is either, Re BR ay ham, om promifle) the Groundplat of “In thinges Supernat e rall,etcrnall,c® Diuine: By Application, Afcens ding, In thinges Mathema- tical: withext farther Application. % } degree lower ) inthe A | Mathema In thinges Natural: both Subftatiall,c® Ac- cidentall Vifible, e& In- uifible.cxc.By Applica: ton: Defcending. } The like Ves and et pplie % Cations are, ( theugh in a ticall Deri. uatine. The names of Niibers:Simple,Compound, Mixt: And of their Fra@ions. be thmetike of Radical] are, either the Principalls: Arithmetike of Cosfike Nibers: with their Fractions: And the great Arte of Algiebar, a, [All Lengthes. - Mecometrie. { Athand — i All Plaines: As, Land, Borde, GlaiTe,&c. Embadometrie, | All Solids -As, Pimber,Stone,Veflels,&c. | | Stereometrie, Geometrie, vulgar: which teas cheth Mi eafuringd | { How farre, from the Meafurer , any) thing is: of him fene,on Land or Water: called | _| Apomecometrie. } G eodefie ° more cunningly to Meafure and Suruey Landes, Woods Waters.cy¢. 2 ? With diffe Of which Geographie, How Ligh or deepe, from the leuell | 47€ growen ‘ | Muf ike Which denonftrateth by rea ou,and teacheth by fenfe,perfeLly to indoe and order the dinerfitie of Soundes , bie or low. 3 y ySenfe.perfettly to indg ; Cofm Ograp hie 9 — Which, Wwholy and perfectly maketh defcription of the fLeauenly and alfo Elemental part of the World : and of thefe partes, make homologalapplication, and mutual! cohation necefary. o¢ Aftrolo g1 S Which redorably demonftrateth the operations and effetles of the naturall beames of light and fecrete Infinence of the Planets and fixced Starts , in entry Element and Elementall body : at all times, in any E1oriz.on affigued. o Stati ke ; Aes ditonftrateth the cafes of heauines and lightnes 7 allthinges : and of the motions and properties to beanines and lightnes elonging. Anthrop O Ma raphie, Which derebeth the Nuber, Meafure, Waight,F. igure,Situation,and colour of euery diners thing contaisted inthe perfecte body of ef Ai :and geneth certaine kuowledge of the Figure,S symmetric, Warght ,Charatterization,c due Locall motion of any percell iT ; | of the faythody alfigned : and of numbers tothe Said percell appertaining. | ro chi ike, Which qéionfirateth the properties of all Circular motions: Simple and Compound. . Prop ¢ names Helicofophie 9 ———— Which dionstrateth the dcfigning of all Spiral lines : in Plaine,on Cylinder,Cone, Sphere, Conoid, and Spharoid : and their prea aS, perties. | 2 AARNE Er Tegan, Ress esa cl | Imprinted by John Day. An.1570. Feb.zs. | | S from the | of the (Meafurers tanding ; any thing iss the Feates Meelereaes Scene of hym jon Land ar Water : called PCa Artes of Chorographie. Derinatiue 2 Hyplometrie, y fro the Prince y ca oO ne | Hydrographie, OMe Dave 6 | Flow broad 94 thing is ,which is in the Meafurers vew : foit be fituated on Land or | Water: called Platometrie, | Stratarithmetrie, $ | P erf] p ectiue 9 Which demonftrateth the wanersand properties of ak Radiatious:DireEe, Broken, and R efictted. Aftro homie ? Which denonStrateth the DiStances, Magnitudes,andall LN aturall motions, Apparences,and Paffions , proper to the Planets aud "fixed Startes:for any time, paft, prefent, and to come : in re/pecte of acertaine Horizon,or Without refpecte of any Elorizon, P neumatithmie 9 — Which dienfiraterh by cloft hollow Geometrical! figures (Regular and Irregular ) the firannee properties ( in metion or Stay ) of the Watn-T} re,Smoke,and Fire,sn their C outinuttie,and as they areiayned tothe Elementes next them. . Menadrie, Which deonftrateth how, aboue N atures Vertuepand power imple: Vertue and force may be multiplied : and fo to dirette, to Lift, to pula, and to put or cast fro,any multiplied, or. fimplédetertined Vertue, Waight, or Force: naturally, not, fe, directible, or moueable, wha # | Which déeftrateth how ,onder the Spherical Superficies of the Earth ,at any depth, to any perpendicular line affigned ( whofe die france frdthe Perpendicular of the entrance: and the Azimuth lkewife, in refpette of the fayd entrance,is knowen |) certaine Way, may be pirebed and gone, ec. ; Which damftrateth the poffible leading of Water by Natures law,and by artificial belpe, from any head( being S pring standing or running Wer ) to any other place affigned. 7 Hypogeiodie, — Hydragogi (= H Orometrie > Which daaftrateth how,at all times appotated,the precife,vfual denomination oftime, may be knowen,for any place affigned, * Z ographie, ~ = Which demftrateth and teacheth,how, the Interfettion of all vifnall Pyramids made by any plaine affigned( the. Center, diftance, and lightsting determined ) may be,by lines and proper colours reprefented. A r chitecture > ——_ Which isgcience garnifhed With many doctrines and diners Inftructions : by whofe indgement all Workes by other Wor kenen firti- fhed, are ae ed. Which amnstrateth, how, by the Shorteft good Way by the aptef direttion and in the fhortest timeia fufficient Shippe, betwene a- zy two pact( in paffacenantgable) a Signed may be condutled:and in all formes andnaturall disturbances chauncing , how to vufé - the best ro) ble meanes,torecouer the place first affigned, | SC,- Which geuth certaine order to make firaunge Workes,of the fenfe to be perceiued:and of men ereatly to be Wondred at. Nauigation,—— Thaumaturgi 7 { Ar chemaftr ie, ~~ Which tegheth to bring to aftuall expertence fenfible,all worthy conclufions by all the Artes Mathematicall purpofed : and by true | Naturall\bilifophie concluded: And both addeth tothema farder Scope , in the termes of the fume Artes: and alfo , by his proper Qn Maethod,ad in peculiar termes, precedeth with helpe of the forfaya Artes,tothe performance of cx mplete Experiences: which, of no : particulerdrte,are hable( Formally ta be challenged, a lhe Grit booke of Eu- clides Elementes. Zs N rurs FIRST soo x Fis intreated of the molt 7 fimple, eafie, and firlt matters and groundes of Geo- M\ metry, as, namely, of Lynes, Angles, Triangles, Pa- MW rallels, Squares, and Parallelogrammes. Firft of theyr 2 hf definitions,thewyng what they are. After that it. tca- 5 Zeke cheth how to draw Parallellynes,and how toforme EAN EN diuerfly figures of three fides& foure fides according IRASNN{ to the varietie of their fides, and Angles : & cépareth eee them all with Triangles & alfo together the one with $)59) the other. In italfo. ts taught how a figure of any ST ANG—# forme may be chaunged into a Figure of an other Soest Zc") forme. And for thatit entreateth of thefe moft com- mon and g?nerall thynges:, thys booke ismore vniuerfall then is the feconde, third,or any other, and therefore iuftly occupieth the firft place in order : as that without which, the other bookes of Euclide which follow, and alfo the workes of others which haue written in Geometry, cannot be perceaued nor vnderftan- ded. Andforafmuch asall the demonttrations and proofes of all the propofiti- onsin thiswhole booke, depende of thefe eroundes and principles following, which by reafon of their playnnes neede no greate declaration, yet to remouc all (beit neue fo litle) obfCuritic, there are here fet certayne fhorte and manitelt expofitions of them. | Se Definitions. 1. A figne or point isthat swhich hath no part. The betiee tovnderftand what maner of thing a figne or pointis,yetauft mote that the natureand propertie of quantitie(wherof Geometry en treateth )isto be deuided, fo that whitfoeuer may be deuided into fundry partes;is called quantitie.but a point, although ilpertayne to quantitie, and hath his beyng in. quantitie, yet isit no quanti- tie, for tha: it cannot be denided. Becaufe(as the definition faith). it hath no partes in- to which itfhould be deuided.So thata pointe isthe leaft thing that by minde and yn- derftandirg can be imagined and conceyued : then which,there.can be nothing letie, 3s the poirt 4.in the margent. A figre or point is of Pithageras Scholers after this manner.defined: +4 poynt is an unitie whic) hath pofition. Nibers are conceatiedin mynde without any forme & figure, and therfcre without matter wheron to receaue figure, & confequently without place and pofition. Wherfore ynitie beyng a part of number, hath ho pofition, or determi- nate place Wherby it is manifeft,thatumber.ismorefimple and pure then is magni- tude,and :Ifo immateriall: and fo vnity whichis.the beginning of number, is leffe ma- teriall thes a figne or poynt, which is the beginnyng of magnitude,For apoynt is ma- teriall, and requireth pofition and place,and therby differeth from vnitic. 2, Aline is length without breadth. There sertaine to quantitie three dimenfions; length, bredth,& thicknes,ordepth: and by thefe thre are all quatities cieafured & made known, Thereare alfo, according Bile to The argument of the firft Booke. Definition of a pont. Definition of a poynt after Pithagoras. Definition of a line. 4 : An other defi- ition of a line. An other. T he endes of « ; Line. Difference of a posnt fro Gasty. Viaitse is a part of number. A poynt #3 #8 part of quan- £sf ie. Definition of B right line. Definition of a wight line after Campanus. Definitsa therof after Archi- giedes. Definitio thereof after Plato. Jn osher Ach OSES ON hw osher. T he firit Booke to thefe three dimenfions, threc kyndes of continuall quantities : alyne, a fuperficies, or plaine,and a body.The firft kynde,namely,aline is here defined in thefe wordes, e4 tyne is length Without breadth.A point, for that itis no quantitie nor hath any partes into which it may be deuided, but remaineth indiuifible,hath not,nor can haue any of thefe three dimenfions.It neither hath length,breadth,northickenes.Butto aline,which is the firft kynde of quantitie,is attributed the firft dimenfion; namely, length, and onely that, for it hath neither breadth nor thicknes, but is conceauied to be drawnein Jen eth onely,and by it,it may be deuided into partes as many as ye lift,equall,or vnequall.But as touching breadth it remaineth indiuifible. As the lyne AB, whichis onely drawen in length, may be deuidedin the pointe C equally, or in the point D ynequally,and fo into as many partes as yelift, There oh gigi are alfo of diuers other geuen other definitions of a lyne: as A cD B thefe which follow. od lyness the mouyug of a poynte,as the motion or draught ofa pinne or a penne to your fence maketh a lyne, Agayne,e lyne ts a magnitude hauing one onely /pace or dimenfion, namely, length Wantyng breadth and thickves. 3 The endes or limites of a lyne are pointes. Fora line hath his beginning from 4 point,and likewife endeth in a point: fo that by this alfout is maniteit,that pointes, for their fimplicitie'and lacke of compofition, are neither quantitie,nor partes of quantitie.but only thetérmes and endes of quantitie, Asthe pointes «4, B, ar¢ onely theendesoftheline 4B, and no partes thereof ,. And herein differetha poynte in quantitie, from ynitic in number: for that although ynitie be the beginning of nombers, and no ee number(‘as apointis the beginning of quantitie,and moquan- A B titie )yet is ynitie a part of number.For number is nothyng els but a collection of vnities,and therfore may be deuided into them, as into his partes. Buta pointis no part of quantitie,or of alyne: neither isalynecom pofed of pointes,as number is of vnities.For things indiuifible being neuer fo tnany added together, can neuer make a thing diuifible,as an inftant in time,is neither tyme,nor part of tyme,but only the beginning and end of time,andcoupleth &ioyneth partes of tyme together. 4 Aright Lyne is that which lieth equally betwene his pointes. As the whole line «4 B lyeth ftraight and equally betwene the poyntes 4B without any going vp or comming downe on eyther fide. . Campanus and’certain others,define aright linethus:) A B ° } i : i er eee ie eee eae ee Al right line ts the fhortest extenfion or draught that is or'ma be from one poynt to un other. Archimedes defineth it this. A right line is the fhortest of all ines, which baue one and the Self fame limites or endes: which is in maner alone with the definitié of Campanus.As of all thele lines 4B C,ADC,AEC AEC which aré alt drawen from the point 4, tothe ; : poynte J, as Campanus fpeaketh, ‘or whichhave the felffame limites or'endes,as Archimedes fpeaketh,the jyne 4A BC, beyng arighriine,is thethortett, “Plato defineth a rightline after thismaner: Arighe line ts that Whofe middle part fhadowerk theextremes. Asif you put any thytig in the middle ofarightlyne,you fhall not fee from. the oneende to the other, which thynghappeneth notin a crookeddyne: The Ecclipfe of the Sunne/(fay Aftronomers) then happeneth,when the Sunne,the Moone, &our cye arein one right ltne-For the Moone then being in the midfbetwenevs-and the Sunne, caufeth it to be darkened. Divers other define aright line diuerfly,as followeth, ed rseht lyne is that which [tandeth firme betwene his eactremes Agayne, A right liae is that which With.an ether lite of lyke formee cannot C WAR 4 fiGMTEs.. >>... Agayne, F 7 q ee ee ee of Euclides Elementes . Fol.2. Agayne,ed right lyne is that which hath not one part in a plaine Juperficies, and an other ereited Another. * an high. Agaynci right lyness that ,all whofe partes agree together With all bis other partes. Agayne,/ right lyne ts that,whofe-extremes abiding cannot be altered. Euclide doth not here define a ctooked lyne, for it neded not, It may eafely be vnder- ftand by the definition of a right lyne, for euery contrary is well manifefted & fet forth by hys contrary-One crooked lyne may be more crooked then an other,and from ‘one poynt to an other may be drawen infinite crooked lynes: but one right lyne cannot be righter then an other, and therfore from one point to an other, there may be drawen but oncright lyne.As by the figure aboue fet,you may fee. 5 A ‘fuperficiesss that, which hath onely length and breadth. A fuperficies is the fecond kinde of quantitie, and to it are attributed two. dimenfi- ons, namely length, and breadth. Asin the fuperficies -4 BCD, whofe length is taken by thelyne 48, or€ D, and breadth bythe 3-7 lyne 4 C.or BD: and by reafon of thofe two dimenfions a fuper- ficies may be deuided two wayes, namely by his length, and by hys breadth, but not by thickneffe, for it hath none.For,that is attribu- ted onely to a body,which is the third kynde of quantitie,and hath all three dimenfions,length,breadth, and thicknes,and may be de- “nided according to.any of them. Others definea fuperficies thus: 4 /uperficses 1s the terme or ende of a bedy.Asalineis the ende and. terme ofa fuperficies, 6 Extremes of afuperficies,are lynes. As theendes,limites,or borders of alyne,are pointes, inclofing theline:fo are lines the limites,borders,and endes inclofing a fuperficies. As in the figure aforefayde you maye fee the fuperficies inclofed with foure lynes. Theextremes or limites of a bodye; -are fiperficiefles And therfore'a fa perficies is Of fome thus defined: 4 juperfictes # that, which endétb or incloferh a body: as is to befene inthe fides ofa die, or of any other body. 7... A plaime [uperfictes that which lieth equally betwene bis lines. As the fuperficies 4 B C DJyeth equally and {moothe betwene::’ " thetwolines..4&.-and C D:, or betwenethe twolines 4C, and A— BD: fo thatno part therofeyther fivelleth vpward,or is depref- | | fed downward.And this definitio much agreeth with the defini- tion of aright line.A right line lieth equally betwene his points, € D anda plaine fuperficies lyeth equally betwenehislynes, Others define a plaine fuper- ficies atter this Maner: | OSs | et plaine fuperficies,is the {hortest extenfion or draught from one lyne toan other-likeas aright lyne is the fhorteft extenfion or draught from one point to an others Enclide alfo leaneth out here to fpeake of a crooked and hollow fuperficies,becanfe it may cafely be vnderftand by the diffinition of a plaine fuperficies,being hys contrary, And cuen as from one point touan other may bedrawen infinite ¢rooked lines, & but one rightlinc, whichis the thortett+fo from one lyne to'an other'may: be drawen infi- nite croked fuperficiefles,& but one/plain fuiperficies, which isthe thorteft. Here muft you confider when there is in Geometry mention made of pointes;lines,circles,trian- gles,or of any othef figures,ye maynot conceyue of them as they-be in matter, as in woode, in mettall/ in paper; orin any fuch iyke forfoisthere no lyne;but hathfome breadth atid may be deuided:nor points, but that fhalthaue fomepartes, and may alfo be deuided,and {fo of others.But you muft conceive them in mynde,plucking them by imagination fromall matter,fo.fhall ye enderftande them truely and perfealy,in their owne patureas they are defined.As alyne to be long,and not broade:anda — to Es ij. e sin other. An other. VPhYE uclide here defineth not a crookeh ie. Definition of a | fuperficses. A fuperficses may be deuided twe WAYes. An other defins- tion of a fuper= freses. . Theextremes of 4 fuperficses. Another defini- trom of a [uper- preses. Definition of F plasne fi uper fin ees, Another defins-- tion of A playne [aperfic {C56 NOTE. ~ ae ———— So . Le Eee is i. SY Re es - 4 ——. re ree Lewes 2 ane ~ des ere — 5 _ se +e an ™ - "Au ~ Dacia Sa = J Se Tv “ae : Rrra oes me 3 ol : See ee " a tates 7 ~~~ ———— ee ae - 5 op ittaet see: - = a, “ , ——— : 2 Se ee ee ein nade dee See Be RS ‘ Be eres eee : “ 4 sere aes = tin eS i a eee — aren i i ee " i} ne : | j at 1 ahi 1 t i] ‘ ipl vale ‘ ) 4 i if u { i . j i { ‘ [ ‘fs : TAR! ) Pi | (4 Pere | 1% EI eT! it i? ine Ti “ ‘da re ed Le ig f WAL ee BA dda’ } H ; " ny he gh 4 i 4 } ey dha 4 ay | ef i rie) Hy 5) re ie - 4 ' ib ' " oa ae ee : 1h 7 n \e RL a : di ad The aL | ij 1y) Le ae! ) i a 4 hy Me ray 1 ane , feat ii “bias i] @ ; Ha Wier th ft, ] ‘7 H i Aa yo ‘ : 1} See : 9 iu As 1 m ay aL eet oa P\\dia) + 4 hi ’ a ‘ SE ee sf _ ' hood q Pr B.] alt, yn fe y Shots ify) aad tpi 4 7 ot . 59} ‘ way’ an ie . w) | eae ae j 4 wh o] figs f 1 f aa , ithe 3) , 4, ry Meh id is } be Ay ela ae j Py i) ‘enert L oe i weR Aa By ie? | ae | : bs = : 7 6 rf , ja i] 4 ef J] eat 4: Me aan SS Another defins- tion of a pla cyne fuperficies ; An other defi- esitior. Az other defr- ss5tton. Az other deft sition. Definition of « pla cyne angle. 7 Definitrn of « redilined an- gt. Three kindes of angles. VV hat aright angle, €5 GGhat alfo « perpends- ewlar lyne a. VV hat az ob- tufe angle m. Sap T he first Booke befolittle,thatit fhall hane no part at all, r Others otherwyfe define a playne fuperficies:eA plaine Superficses is that , Wwhsch ts firmly Jet betwene his extremes ,as before was faydof aright lyne. | Agayne,eAd plaine fuperficies is thatyunto all whofe partes a right line may Well be applied. Again, A plaine (uperficies s that, which is the fhortest of al fuperficies, which haue one & the felf extremes: As a right line was the fhorteft line that can be drawen-betwene two pointes. Againce.A playne fisperficies is that, whofe middle darkeneth the extremes, as was alfo {ayd o€ aright lyne. ) 8 Aplaine angle ws an inclination or bowing of to lines the one to the otber and the one touching the other,and not beyng directly ioyned together. As the: B ; two lines $ | | ; ABL&B 1 pes C.incline ° D Fala : the one . | to the o- oe | OA, V A 2 | ther,and touch the ae ce ; one the other in the point B, in which point by reafon of the inclination of the fayd lines,is made theangle .4 BC. Butif the two lines which touch the onethe other,be without all inclination of the one to the other,and be drawne dire@ly the oneto the other,then make they not any angle at all,as the lines C D, and D E,touch the one the other in the point D, and yet as ye fee they make no angle. | 9 And ifthe lines whichcontaime the angle be right lynes,then is it called a rightlyned angle. | bon 7 | : As the angle 4 B C,in the former figures,isa rightlined angle, becanfe itis contai- ned of right lines : where note,that an angle is for the moft part defcribed by thre let- ters,of which the fecond or middle letter reprefenteth the very angle, and therfore is fer at the angle. | | | 4 By'the contrary,a crooked lyned angle,is that which is contained of crooked lines, which may be diuerfly figured, Alfo amixt angle is that which is caufed of them both, namely, ofa right line and a crooked, which may alfo be diuerfly figured, asin the fi- gures before fet ye may fee. There are of angles-thre-kindes, aricht angle,an acute an- gle,and an obtufe angle, the definitions of which now follow. i ; 10. VV ben aright line ftanding ypona right line maketh the angles on either fede equall:chen either of thofe angles is a right angle. And the richt lyne which ftandeth erefled, is called a perpendtculer line to that vpon which at ftandeth, eaters si Bee As ypon the right line CD, fuppofe there do ftand anotherline AB ,An {ach fort chat it maketh oe angles.on either fide at | “ quall: namely,the angle ABC onthe one fide eq uall- to! the angle AB Don the other fide: then isecheofthe two angles ABC, and. A 8D aright angle,and the line 4.2,which ftandeth ereQed vpon thelineC D, without inclinationto either partis a perpendicular | dine,commonly calledamong artificers aplumbelyne., |. Sa 3 u An obtnfe angle ts that which is greater then dvight angle. ag As - of Euclides Elementes Fol.3. _ As the angle CBE intheexample ts anobtufe angle, foritis °° A _ greater then the angle 4 8 C, whichis aright angle,becaufe itcon tayneth it,and containeth moreouertheangle ABE. w.. Anacute angle is that, which w le/sethena right angle. Asthéangle EB D inthe figure’before putis an acute angle,for = : Se that itis lefle then the angle 4B D,which is a right angle, forthe right angle contai- neth it,and moreouer the angle ABE, 13, A limite or termeyes the ende of entry thing. Foras muchas of thinges infinite (as Pistefaith) thereis.no {cience, therefore muft magnitude or quantitie(wherof Geometry entreateth) be finite,and haue borders and limites to inclofe it, which are here defined tb. be theendes therof. -Asapointis the li- mite or terme of aline ,becaufe itis thend therof : Aline likewifeis the limite & terme ofafu Perhiies -and.likewife.a {uperficies isthe limite and terme of a body,as is before declared, 14 Afigure that, which ws contayned vuder one limiteor terme,or many. As the figuré 4is contained vnder one limit, | which is the round line, Alfo the figure B iscon : tayned vnder three right lines, rhe the figareC- ro vnder foure,and fo of others; which are their li-’ mites or termes, YH IE 15 Acircle is a plaine ficure conteyned vnder one line, which called a cir cumference;ynto whichall lynes drawen from one poynt within the figure and falling-vpon the circumference therof are equal the one to the other. As the figure here fetis a plaine figure, thatis a figure without groffenes or thick- nes,and is alfo contayned ynder one line namely ,the crooked lyne BC D,which is the circumference therof, it hath moreouer in the middle therofapoint, namely, the pointe, from which, all the. lynes drawen to the fuperficies ate equal; as the lines 4B, AC, A D, and other how many foeuer.. °°) a: sere Ofall figures acircle is the moft perfea&, and therfore is it here firftdefined, arts eg N | a 15 And that point iecalled the centre ofthecircle; as is the point A, which ss fet in the'middes of the former diréle. For the more eafy declaration,that all the lines drawen from the centre of the circle to the circumference, are equallye muft note, that although a line B be not made of pointes: yet a point, by his motion or draught,de- fcribetha line. Likewife a line drawen,ormoued,,defcribeth a fu- perhities: alf@ a fuperficies being moued maketh a folide or bodie, Now théimaginethe line 4 B;(the point 4 being fixed) to be mos _ ved aboutinaplaine fuperficies,drawing the point B continually about the point e/, till itreturne tothe place where itbegan frdt to moue: fo fhall’the point B, by this motion, defcribe ee. ee ference of the cirelé and the point ¢ being fixédjisthe — the circle, whi in = »lij. a VV hat an acute angle #6. The Lmste of any thing. Na fcsence of thinges infinite Definstion of P" figure. Definitions of a cirele. A crcle the weft perfed of all figures. The centre of “a circle. ’ 1) Boia |! ba) , ad — _ . Ac ihlen - _—_ ae = ——— - * . — v4 — — <3. eee “i " : —— = 2a ~ - Sane » ? - > ae oy - - rt a - — Ng ty fo GS Sa es eee ene ot iic=d — ? Rr ae La aS. . a a ea Bee TRNE ns nae on — woe 2s - = WL 2. ee ae of ik SS Be Sinn. Se “ = Shree = nates lt — Lata, CS meeps! eib = ane —— ~ _ SS = a = =D et TSS = + —— ¥ 5 halted AAA A “ ~ 8 ee Sg ey ee ES sD + yee oe mor ~— duthlhns todas a a aid © etme > een — we a co ~ . —~"t = oe « 7 = Sneed — = mal SSS < - ay mia ess $9 s t. ——_ —™ ~ry ee _ eo ey ot ee Den oy — ee bat aor > ey Cae shi ~ a a ear Definstion of a diameter, Definition of a femscircle, Definition ofa fection of a cir- ele, Definition of redslined fi~ L4res. Definition of three fided f- LY Gs. Sa The first Booke all the time of the motion of thé line, had like diftance; from the circumference ,ngne- ly the length of the line 4 B.And for that al the lines drawn from the centre to the cirs cumference are defcribed of that line,they are alfo equal ynto it, & betwene théfelues. 17 A diameter ofa circle,is.a right line, which drawen by the centre thereof, andending at the circumference on either fide, deutdeth. the circle into to equal partes, As the line B 4 Cin this circle prefent is the diameter,becaufe it paffeth from the point B, of the one fide of the circumferéce, to the point C,on the other fide of the circumference, & paffeth alfo by the point e# being the centre of the circle. And moreo- uerit deisideth the circle into two equall partes: the one,name- ly BD C,being‘on the one fide ofthe line, & the other namely, BE C,on the other fide which thing did Thales Milettus (which brought Geometry out of Egiptinto Grece ) firft obferue and proue, For ifa line drawen by thecentre,do not denide a circle into two equal partes: all the lines drawen from the centre to the circumference fhould not be equall, 13 A femicircle,és a figure which ws contayned vnder the diameter, and yne der that part of the circumference which is cut of by the diametre, Asinthe circle ABCD the figure B_ACisafemicircle,becaufe . it is contained of the right line B GC,which is the diametre,and of the crooked line B AC, being that part of the circumference, 5 whichis cut of by the diametre# G C. So likewife the other part i ofthe circle namely. B D C, is a femicircle as the other was, D 19. A fection or portion of a circle, is.a figure whiche is contayned vnder a right lyne, anda parte of the circumference, greater or leffe then the femicircle. Asthe figure e4 BC, in the exampie, is a fe&tion of a Circle, & is greater then halfe a circle,and the figure AD C,is alfo a (eati- on of acircle,and is leffe then a femicircle.A feion,portion, or partofacircleis all one, and fignifieth fucha part which is ei- ther more orléfle, then afemicircle: fo that a femicircleis not here called a feGtion or portion ofacircle, A right lyne drawen from one fide of the circumference ofa circle to the other, not paflyng by the centre, deuideth the circle into twoi ynequall partes,which are two fections,of which thatwhich contayneth the centre is.called the ereater fection,and the other is called the leffe feGtion. Asin the exam ple,the part of the circlee-4 B C,which containeth init the centre E,is the greater fection, being grea ter then the halfe circle: the other part;namely ef DC, which hath not the centre in it,is the lefle {eGtion of the circle, being leffe then a femidircle? "°° 20 ‘Rightlined figures are [uch which are contayned vader right lynes. As are fuch as followeth of which omer sone d ee right li ¢ : yned-vnder threerightlines, fom vnder foure;fome vader fiue,and fomevnder mo, 9.50" OQ - a . 2\\Lhre fided figures,or figures of thre/ydes.are fuch which are contays * ned vnder three right lines, a of Euchdes Elementes. Fol.4. AX the figure in the example 4. BC,is.a figure of three fides, becaufe itis cOtained vnder thre right lines, namely,vnder the A. linese 4 B,BC,C et. | /\ A figure of three fides, or atriangle, is the firft figure in order of all right lined figures , and therfore of all others it 1s \ Grft defined. For vnder lefle then three lines, can no figure be \ comprehended. \ B C 22, Foure fided figures or figures of foure fides are fuch, hich are contained Definition of ded ynder foure right lines. ats ' Asthe figure here fetsis. a figureof foure fides,for thatitis cO- 5 prehended vader foure.rightlines,namely,A B,BD,DC,CA. rae Triangles,and foure fided figures ferue commonly to, many v- | {esin demonftrations of Geometry. Wherfore the nature and = —-—--—-—-_ properties of them,are much to be obferued, the vic ofotherfi- ~ D gures ismore obfcure, | 22, Many (ided fieures are uch-whichhaue.mo fides then foure. Definition of f fi fg 2 fi f muny fided Right lined figutes hauing mo fides thenfower,by continual adding of fides may be figures Oe ay rhe te i algun | my inogye \ : | | ' AOR —— we ad ae ‘—- -nfinite, Wherfore to define them all feverally,according tothe number of their fides, fhould be very tedious,orratherimpoflible, Therfore hath Excide comprehended the yvnder one name,and vnder onediffinition : calling them many fided figures, as many aéhaue mo fides then foure nasifthey-haue flue fidessfixe, feuen,or mo, Here note yes that cuery rightlined figure hathas many angles,asit hath fides,& taketh his denomi+ nation afwell of the number of hisanglesas ofthe number ofhis fides. Asa figure cO- tained vnder three right lines,of the number of his three fides, is called a thre fided fi. gure vegenforof the number of his three angles, itis called a triangle: Likewife a figure contained vnder foure riglit lines, by reafon of the number of his fides,is called a foure fided figure : and by reafon of the number of his angles, itis*called a quadrangled fi- cure,and fo of others, i“ 9 vop {howe pee voy Gyneme ita , ts th Definition 24. Of three fided figures or triangles, an equilatre triangle is that, which sed , eae ha th three equa Nfides.f ) a> triangle, _Triangles hauetheir differefices partly of their.fides, and partly oftheirarigles . Ag touching the differences of theit fides,there are three kindes. For either all thre fides of the tr1- angle age equall,or twa onely.areequall, & the third ynequal: / or elsall thréeeré vineq uallthe oné.ro the other. Fhe firft kind / A of triangles, namely,that which hath three equall fidessismolt 90% fimp!e,and cafieft to be knowen:andis here firftdefined, and = B.ily. 1S Definition of an Lfofceles. Definition of a Scalenum. An Orthigent~ 5 9B Sr $ee73 rg be P An Ambligons- wens triangle. AX. Oxigonte rt tran gle, The first Booke is called an equilater triangle,as the triangle e-# in the example, all whofe fides af2 of one length. : | 25. I/ofceles is a triangle which bath onely two fides equal. The fecond kinde of triangles 4 hath two fides of onelength,but the third fide is either longer or fhorter thé theothertwo, asare | the triangles herefignred,B,E,D 23)» In the triangle B,the two fides A Eand E Fareequal the one to 2 * Mpsexcaa the other,and the fide A F,is16- - 3 “3 iba - gerthen any of them both:Likewifein the triangle C the two fides G Hand H K, are e- quall,and the fide G Kis greater.Alfoin the triangle D,the fides L 4 and 44 N, are €4 quall,and the fide Z Nis thorter. | 26. Scalenumis a triangle -whofe three fides are all ynequall. As are the triangles E,F, in which there is no > one fide equall toany of the other.Forin the tri- A angle E,the fide 4 Cis greater then the fide B C, and the fide B Cis greater thé the fide 4B. Like-°- wife in the triangle F,the fide D His greater thé the fide D G,and the fide D G,is greater then the fide GAZ, B ¢.6U6UcGlwlwé“(i‘dC 27. Againe of triangles an Orthigonium or a rightangled trian le, is & trie angle which hath aright angle. As there are three kindes of triangles; by reafon of the diuerfitie of the fides,fo are there alfo three kindes of triangles, by reafon of the varietie of the angles. For euery triangle either containeth one right angle, & two acute angles: or oneobtule angle,& two acute? orthreeacute angles: for it isimpoffible that one triangle thould containe two obtufeangles,or two right angles,-or one obtufe an. gl¢,and the other aright angle. Alb which kindes arehere defined. Firft a rightangled triangle whiche hath init aright angle, As the triangle B C_D,of which the angle B C Djis a right angle, WwW re ) 28. Anambligonium or an obtufe angled triangle, is a triangle hich hath an obtufe angle, : . ae eS Asis the triangle B, whofe angle 4C A D,isan obtufe angle, andisalfoaScale. |. - non,hauing his three fides vnequall : the triangle E, is likewife an Ambligonion, whofeangle FG H,is an obtufe angle, & is an Ifofceles, hauing two of his fides — equall,namely FGand GH, an D G Fi 29. Anoxigoninmor an acutean oled triangle, isa triangle which hatha his three angles acute. ) | ao a Pan ae Cnet at Ok A mee, of Euchdes Elementes . Fol.s. \s the triangles .4,2;C,inthe example,al » whofe angles are acute:of which Ais an e- quilatertriahele, B, an Iofceles, and C 2 Scalenon.An equilater triangle fs moft fim ple, and hath one vniforme conftruction, and therfore all the angles of it are equall, and neuer hath init either a right angle,or an obtufe:but the angles of an Ifofceles or a Scalenon, may.di- uerfly vary. Itis alfoto be noted thatin comparifon of any two A. fides ofa triangle, the thirdis called bafe.. As of the triangle ABC inrefpeé of the two lines AB and 4 C,the line B C,is the -bafe: and in refpea of the two fides 4 Cand C B,the line 4 B,is the bafe, and likewyfein refped of the two fides C B & B -d, the line 4 C,is the bafe, B C Definition of 30 Of foure fyded figures, a quadrate or [quare is that, Wwhofefydes are e4 farcre quall,and bis angles right, As trianglés haue their difference and varietic by reafon of their 4 3 fides and angles: fo likewife do figures of foure fides,take their varie. * tie and difference partly by reafon of their fides, & partly by reafon of their angles,as appeareth by their definitions: The four fided figure D Cc ABCD isafquare ora quadrate, becaufeitis aright angled figure, al hys angles are rightangles,and alfo all his four fides are equall. 3 Afigureon the one fyde longer, or [quarelske, or as fome callita long — ‘fqnare,ts that which bath right angles,but bath not equall fydes, : This figure agreeth with afquare touching his angles,in 4 _ thateither of them hath right angles, and differeth from it. [- onely by reafon of his fides,in that the fides therof be note- quall,as are the fides of afquare. Asin the example,the an- gles of the figure ABCD, are right angles, but the two fides thereof .4B ,and CD, arelonger then the othertwofides ¢ D ef C,& BD, 32 Rhombus(or a diamonde )is a figure baning foure equall fydes,but its sfuirions > notre 4 he ang led. Diamond figure This figureagreeth with a fquare, as touching the equallitie of st lines, but differeth fromitin thatit hath not right angles, as hath the {quare.As.of this figure,the fonre lines 42,2 C,CD, D A, bee- uall,but the angles therofare not right angles. For the twoangles > ABCand ADC, are obtufe angles,greater then right angles, & the. other two angles B_4D and BC D, aretwoacute angles lefle then two right angles. And thefe foure angles are yet equall to foure right \ angles: for,as muchas the acute angle wanteth. of a right angle, fo \ A much the obtufe angle excedeth aright angle. = Cs, Rhombaides Definition of @ diamondlike fi- gure. Z rap ezsa or fables. Definition of Parallel lines, What Petics- T he fir t Booke 33 Rhombaides(or a diamond like )is a figure, whofe oppofite fides atc ee quall,and whofe oppofite angles are alfo equall, but it bath neither es quall fides nor right angles. Asinthe figure 4B C D,all the fonre fides are not equall,but the two fides 4 2B and CD,being oppofite i a Me a the oneto the other,alfo the other two fides _4C and se B D,beng alfo oppofite, are equall the oneto the o- / ther.Lilewife the anglesarenotrightangles,butthe / angles (4B, andC D&B, are obtufe angles,and op- / / pofite aid equall the oneto the other. Likewyfe fe j angles 4B D,and 4C D,are acute angles,and oppo- | ° p/ fite,andalfo equall the one to the other. eT ae 34 All other figures of foure fides befides thefe are called trapexia,or tables. Suchtre all figures,in which is obferued no equallitie of fides nor anges: as the figures 4 and £,in the margét,which hane nei- © —>——., ther eqtall fides nor equal angles, but are de{cribed at all aduen- ee / ture without obfernation of order, andtherefore they are called Ce J | irregular figures. | 35 Parallel or equidiftant right lines are fuch, which bring in one and the felfe. fame fuperfictes, and pros diced infinitely on both fydes, doneuer in any part CUUOUT Ba ca ee tae Asare the lines 4 B,andC D,in the example. Sao Teticions or requeftes. 1 __Fromany point to any point,to draw a right line. After the definitions, which are the firft kind of principles,now follow petitions, which we the fecond kynd of principles: which are certain general fentences,{o plain, & fo pefpicuous, thatrthey are perceived. to be true as foone as they are vttered,8no ian thit hath but common fence,can,nor will deny them. Of which, the firt is that, which is here fet. As from the point e4,to the point B, who wil de- ny,but:afily graunt that a right line may be drawn?For two points howfogier they be fet,are imagined to be in one and the felfe fame “A Be plaine fiperficies,wherfore from the‘one to the other there is fome : shortefidraught,whiche is a right line.Likewife any two right lines howfoenet they be fet,/arcimagined to bein oné fuperficiés, and therefore from any one ling to any one line,may bedrawetnafuperficies, | pnd : 2 ‘Toproduceareght line finite firaight forth continually, As te draw in length continually theright line 4B, who will not graint’ Forthereisno magnitudefo great, but that there 4 B rc Hay bea greter, Ror any fo litle,but that there may bealeffe,And ————7———= a line of Euchdes Elementes . Fol. aline is.adraught from one point toan other,therfore from the point B, whichis thie end@of the line ef Bymay.be.drawn a line to fomeother point,as tothe point ¢, and from that to. au other,and fo infinitely. 3 .Vponany centreand at any diftance,to defcribea circle. A playne fuperficies may in compaffe be extended infi- a nitely: as from any pointe to any pointe may be drawen a a right line, by reafon wherof it commeth to paflethatacir- "FE = cle may be defcribed vpon any centre and at any {pace or / c~ diftance.As vpon the centre_4,and vpon thefpace 4B, ye _ ns / may defcribe the circle BC, &vponthefamecentreyyvpon |) 4 ~ - the diftance -4‘D,ye may defcribe the circle DE,orvppon..\ \. \ the fame centre-4,according tothe diftaunce.4 F,ye may. \ / defcribe the circle F G, and foinfinitelyextendyng your, \. ~~" / {pace. 4 All right angles areequall the one to the other. This peticion is moft plaine, and offreth it felfeenen to the fence. For asmuchasarightangleis canfed'ofone rightlyne | | falling perpendicularly vppon an other,andnoonelinecan fall | | | more perpendicularly vpo a line then an other: therforenoone |. (4. right angle can be greater thé an other:neither do the length or | fhortenes ofthe lines alter the greatnes of the angle.For in the E example, the right angle 4 B C,though it be made of much lon- ger lines thenthe right angle D E F,whofe lines are much thorter, yetis that aigle no greater then the other.For if yefet the point E init ypon the point B, then fhal the line E D,cuenly and inttly fall vpon theline 4 Band theline E F,thall alfo fall equaly vpon the line BC,and fo thal the anele D E F,beequallto the angle 4 8 C,forthat thelines which caufethem,are of like inclination, “_ It may euidently alfo be feneat the centre ofa.circle, For if yedraw in.acircle two diameters, the one cutting the other in the centre by rightangles, ye hall denide the circle into fowre equal! partes;of which eche contayneth one right angle, fo are all the foure right angles about the centre of the circle equall. \ d 5 VVben aright line falling vpon two right lines, doth make on one cm the Jfelfe fame fyde, the two inwarde angles lefSe then two right angle:, then hal thefetworight lines beyne produced at length concurreon tha part, in which are the two angles le/se then tworight angles. Asifthe right line 4 B,fall vpon two right lines, . namely,C D andE F,{o thatit make the two inward angles on the one fide,as the angles DAIJ& EIH,. © lefle thentworightangles (asinthe example they. .. do) the faidtwo lines € D, and E F, being drawen forthin légth on that patt,wheron thetwoangles_ _, | Being léeffe the two-tight angles confift, hal atleeth~ F_- concurre and meete together: asinthe point D,as {.o% it is eafie to fee, For the partes of the lines towardes DF = more enclined th: one te ij. the the other,thenthe partes of the lines towardesC E are. Wherfore the more thefe parts are produced ;the more they fhall approch neare and neare,till at length they thal Inete 4n one point. Contrariwife the fame lines drawnin légth on the other fide,for that the angles on that fide namely,the angle C 4 B,and the angle E J _4,are greater then two ‘right angles, fo muchas the other two angles are lefle then two right angles, shall ne- uer mete,but the further they are drawen,the further they fhalbe diftant the one from ‘the other. 6 That tworight lines include not a Juperfictes, If thelines AB and AC,beingtightlines,fhould inclofe a fuperficies,they mufte of neceflitie bee ioyned together at “D ‘both the endes,and the fuperficies muft be betwene thé.Toyne 4 them onthe one fide together in the pointe A; and imagine the point 2 to be drawen tothe pointc, fofhall theline AB, : ie fall on the line .4 C,and couerit,and fo be all one with ir, and neuer inclofe a {pace or fuperficies, Cc Sa» Common fentences. tothe other. 1 Thinges equallto one and the felfe fame thyng: are equall alfo the one shied common After definitions-and petitions,now are fet common fentences which are the third fentencesare. and laft kynd of principles-Which are certaine general propofitiés, commonly known ofalimen,of themfelues moft manifeft & cleare,& therfore are called al dignities not Differencebe. able tobe denied ofany-Peticions alfo are-very manifeft, but not fo fully as are the cO= twene peticions mon fentences,and therfore are required or defired to be gtaunted. Peticions alfo are €F common fen~ more peculiar to the arte whereof they are: as thofe before put are proper to Geome- seuces, try: but common fentences are genéral! to’all things wherunto they can be applied, - | Agayne, peticions confiftin ations or doing of fomewhat moft eafy to be done: but common fentences confift in confideration of mynde, but yet of fach thinges which are moiteafy to be vnderftanded, asis' that before fet. git oes | As if the line Abe equallto the line B,And ifthe lineC be alfo. equall to the line B, then of neceffitie the lines A and C,fhalbe equal the one to the other,So isit in all fuperficiefles angles,& numbers,& in all other things (of one kynde) that may be compared together. 42... dad if ye.adde equall thinges to equall thingessthe whole Toalbe equall, Asif theline 4B be equal to the line C D,& tothe line 4 B,be added the line B E,& to the line CD,be added alfo an other line ‘D F,being equal to the line & E.fo that two equal lines,namely,B E.and D F,be— added to two equalllynes -4 B,& CD; then thal the” wholelyne eZ E, beequall to the whole lyne CF, and fo of all quantities generally. a 3 Andif from equall binges take away equall thinges:. the thinges res mayning {halt be equall. 28.5 >> or 3: a0 of Euchides E:lementes . Fol.7. Asif from the two lines 4B.and C Dybeing equ@l,ye take away two equall lines, namely, EB oipAei 2 | ob B o3ouP and FD, then maye youconclude by this com- mon fentence,that the partes remayning,name-. » © Ls B die: ly,.4 E,andC F are equall the one to the other: and fo of all other quantities. 4, And if from vnequall thinges ye take away equall thinges: the thynges which remayne fhall be vnequall, | - — SS ——— ; — =; —— -- === a SS — = ———= — — - — = ae . a ~ > y | | { 1 fi Hi) "I Hi i | W i As ifthe lines 4 B,andC D, be wnequall chalinrer< B,beyng 4 —e i | longer then the line C D, & ifye take fro them two Panaules,. ST See i | —w as E B,and F D:the partes remayning,which are the lines e4 E and C F,fhall be vnequall the one to the other,namely, the lyne af E, {hall be greater then theline’C F,which iseuer true in all quantities whatfoeuer, ~~ eee — ¢ And ifto mnequall thinges ye addeequall thinges; the whole hall be vne equall, | : ae = = : ———— tt, 5 Asif ye haue two ynequal lines,namely,4 E the greater,and C F the leffe,& if ye adde ynto thé two equalllines, EB & F D, —-——_— then maye ye concludethat the whole lines compofed are vn- ail aero equall: namely; that the whole lynee4 B, is greater then the wholelineC D,and fo of all other quantities. 6 Thinges which are double to one and the felfe fame thing: are equall the one to the other; . Asif thélinee 7B be doubletotheline E F,and ifalfo the A B line CD,be double to the fameline EF: thé may you by this <————-—+—————* common fentence conclude,that the two linese4 B,& CD, E FE are equall the one tothe other.Andthisis trueinall quanti- ¢ 2 ties,and that not only,when they are double, but alfo if they. be triple or quadriple; or inwhatproportion foeuer it be of What proporti- the greater inequallitie, Whichiswhen the greater quantitic is compared tothelefle. 07 of the grea- rer snequality wn , Thinges which are the balfe of one and the felfe fame thing:are equal the one.to the other. Asif the line AB sbe the halfeofthe line EF; andifthelyne |, 5 6D, bethehalfealfoof the faméline EF: then mayyecon- + clude by this comrhonfentence that the twolines -4B and “F | Bs CD, areequallthe one tothe other. Thisis alfo‘true in all kyndes of quantitie,and thatnotonely when itisa halfe, but alfo if it be athird,a quarter, or in What proportion foeuer it be of the leffein equallitie. Whichis when theléfle quantitie is cOpared tothe greater, what proporti- on of the leffe inequalitic H 8 Thinges which agree together: are equall the one tothe other, Such thinges are fayd toagree together,whiche when they are applied the oneto the other: or {et the one vpon.theother;the oneexcedeti not the other in any thyng. C. ii}. As What a Prepa- ftton fhe Propofirsons of twe fortes. What «a Pro- Sleme #. Wheat «a Ties- TOME he T he first Booke Asifthe two triangles 4 BC, and DEF, weéte 6 1 the one to the other,and the triangle 4 B C, were fety. pon the triangle D E F,if then the anglew4, doiuftly a- gree with the angle D,and the angle B ,withtheangle £, and alfo the angle C, with the angle F: and moreouerif the line 4 B,do iuftly fall vpon the line D E,and the line 4C,vpon the line D F, and alfo the line BC, vppon the line E , fo that on euery part of thefe two triangles, there is inftagreement, then may ye conclude that the two triangles are equall. 9 Exery whole is greater then'bis part.’ As the whole is equal to all his partes taken together, fo isit grea- ter then any one-part therof,Asif the line C B bea partoftheline 4. 4....' ¢ B &,then by this cominon fentence ye may conclude that the whole “7 line 4 B, is greater then the part, namely,thé the line C.B.And this is senerall in alt thinges:; 7 “hosp » He principles thus placed & ended,now follow the propofitions, which ye are fentences fetforth to be proued by reafoning and demonitrations; yy and thetfore they are agayne repeated.in theend. of the demonftrations a} Leandiy For the propofition is euer the conclufion, and that which ought tobe proued. Propolitions are of two fortes, the one is called a Probleme, the other a Theoreme: A Probleme,is a propofition which requireth fome action,or doing: as the makyng of fome-figure,or to deuide a figure or line,to apply figure to figure,to adde figures to- gether,or to fubtrah one from another,to defcribe,to in{cribe,to circumfcribe one fs gure within or without another,, andfuche like, -As-ofthe Arf propofition of the firft bookeis a probleme, whichis thus: Upon aright line genen not bein snfinite;to defcribe anes qutlater treangle,or a triangle of three equall fides. For init, befides the deinonftration and contemplanon- ofthe mynde,is required fomewhat to be done: namely, tomakean equilater triangle vpon a line geulen. And in the ende of euery probleme, after the de- monitration, is concluded after thismanner, Which is the thing, Which was required to be done. 4 i age a “= * & a A Theoreme,is a propofition,which requireth the fearching out and demonftration of {ome properticor paffion of fome figure: Wherin is onely {peculation and contem- lation of minde,without doing or working of anything }.-As the fifth. soiiakientn 4 bie firft booke,whichis thus,e4n Laieulsee trhteete Semnte nalifidesshat bas : — “a = bafe,equall the one tothe other,eesis a.Theoreme.Forin itis required only to be pra»: > ued and made plaine by reafon and demonttratid, that thefetwo angles be. equall,without farther working er doing. And.in-the.endeof euery Theoreme,after.the demonttration.is concluded after this ma- her, Whichthyng was required to be demonstrated or proneds - of Euchides Elementes . Fol.8. Sap The firft Probleme. Lhe firft Propofition. Upon.a right line geuen not beyng infinite, to defcribe an e- guilater triangle,or a triangle of three equall fides. oie C)) —,~ G) t< 3 Ae ae 2S > SS eK A a 44 Af a ’ le wee ee ey NOI gS 2 fides.Now therfore making the centre the pomt\d ae. wie and the [pace A Bdefcribe(by thetlird peticion) ‘ ANC AS KGS Ve | . NID SEAS the centre the point Band the {pace B A,defcribe EAMES an other circle ACE. And( by the fir/t peticion) GS POR A from the point C,wherin the circles-cut the one the other,dra%y one right line — point B us the centre of thecircle CAE, thers fore (bythe fame definition) the line B Cts es guall tothe line B AAndit isproued, that the line A Br but thiages which areequall to.one and the fame. thingsare alfo.equall the one tothe ather( by the firft-common fentence )wherfore thelineC A, alfo isequalltotheline C B;VVherfore thefe three ricbt'linesC ALA B, and BC are equal the one'tothe other.V/V, berforethe tridoleA B Cis equilater VV ber fore vppon'the line A'B, is defcribed an equilater triangle ABC.VV berfore Yppon.a line geen not being infinite, theres defcribed an equilater triangle, VV hichis the thing which was required tobe done. A triangleorany otherreRilitied figure ts then (aid to be'fét or defcribed vponalinc,when the lines one’of the fides of the figure, This firft propofition is a Probleme,becaufe it requireth acte or doyng, names lyjtodeferibeatriangle And this is to benoted;thateucry Propefitten, whether it beaPrablemesor aT heoreme,comtmonly containeth in it 4 ching ¢enen,and a thing requt- redtabefexrchedoat: although ic be noralway es fo And the thing geven,.is cuet ferbe- fore: thething required.\n forme propofirtons-thereate more things geue then one, and mo thinges required then one, in fome there is nothing geuen at all, Moreotier ctiery Probleme OT heereme bey ng perfectand abfolute, ought to hane all chefepartes wamély’, Birt the Prepofirien.to beproued. Then the expofition Teg Gilt, which Construction. SY circle BC D:and agayne ( by the fame ) makyng Demonstration Thing geuen. Thing required Prope Cti0n. Expofitiom, : 4 ’ ’ Peter minvatrcon. - t Ceafsructsert. Desmsonflration. Cc gucls fier. Cafe. The thing geuen sn this Pree bleme. The thing reguired. The propofition. The expofition. The det crm - gation. Theconftracts The dewsonft te “$807. Ihe perticalar , Pl COBCLUROH The ssiverfall 4 . CORCiBISGH. = Thenuete where by st £3 bnowne te bea Pre- biexec, Nocafles 12 thys — 7 proposition. Three bendes of demonfiratton, T be firSt Booke which is the explication ofthe thing geuen. After that followeth the determjnatio, which isthe declaration ofthe thing required. Then is fer the construétion of fuche things whichare neceflary ether forthe doing of the propofitid,or for thedemod {tration, Afterward followeth the demonstration, which is the réafonand proofe of the propofition, And laft ofall is put the conclufion, which is inferred & proucd by the demon(tration,and is ener the propofition, But all thofe partes are notof ne- ceffitie required in euery Problemeand Theoreme,But the Propofition, demonstra- tion,and conclufion,arenecellary partés, & can nevérbe abfent: the other partes may fometymes beaway, Further in diuers propofitions there happen diuers cafes: whichare nothing els but varierie of delineatiou andconftru@ion or chaunge ofpofition, as when pointes, lines fuperficietles, or bodies are chaunged. V Vhich thinges happen in diutrs propofitrons, N Ov then inchis Probleme,the thing geven,isthe line geue: the thing required,tobe ferched out is,how.vpo that line to defcribe an equilater triangle, The Pre- pefition ofthis Probleme js ,Cpon a right line Leuen not beyne infinite,to defcribe an equilater tri- angle. T heexpofition is ,Suppoje that the right line geuen be ABs and this declareth onel the thing genen.T he determination is jit isrequivedvpoa the liue -4 B, to deftribe an equilater triangle: for therb y as youfee,is declared onely the thing required, The conStruétien hee ginnech atthefe wor ds, ew therfore making the cetrethe post A,crrhe [pace A B, defcribe (bythe third peticéon a circle Gt,and eontinucth vntil you come to thefle wordes,.dnd forafmuch as the point A cc,Fortheshetto are de(cribedcircles and lines, neceflarye | both forthe doy ng of che propofirion, andalfo for the demonftration therof, V Vhich demonftration beginneth arthefe wordes: e4nd Sorafmuche asthe point A is the centre of tise circle C B D é&c: And fo procedeth till youcome to thefe wordes //her- fore upon the line A Bis deferibed au equilater triangle A BC.For vneill y oucometherher . s . § ‘ | , is by groundes before ferand conftrutions had ;proued,and made culdent, that thetniangle madeyis equilarer And thenin thefe wordes sWwherfore vpon the inte 1B, ts defcribed an.egutlater weangleeA B.Cis put the firltcouclufion.For there are Common: ly in euery propo fyi0n two.conclufions: the one perticuler,the ochervaiverfal: and irem the firft you so cothelaft. and this isthe firftand perciculer‘conclug- olor chat icconchudeth,charvpon thelyne AB ts defcribed an equilater tria. ste,whtch fs according to the expoficion,afrer ir, followeth thelaftand vniuerfal coneliifton,W lex fore wpon aright line veuenthor Being infinite is deftribed an equilater trian gle. For a hg fet rt m2 | 1. 3c 3 7, : ‘ whether the line geuen be greaterortefle thenthyslyne, the fame conftru@ics and cemonitrations proucthefameconclufion. Lait ofall 1sadded this claufe Which 1 therhing which was required.to be done: wherby as-we haue before noted,is de clared,that this propolition isaProblemeand not.aT heoveme. p s for varietie of ca- es inthi iti 1ere is §: . | . {es in this propolition there is none, for that the line geuen,can haue no diuerfi. ticof pofition. so | As you haue inthis Problemefene plainelye {er foorthe. the thing gene, and the ng required, moreouer the Propofition,expofition dererminationsconstruttion. demonftration 4 itlelsns te ee C - ana conclifios( which aregenerallalfo to many, other both ProblemesandT-beoremes ) {0 may you by che example therot difting; | ifkin& them,and fearche them oy Problemes,and alfo Theoremes. es ines tin other This alfo,is to be noted,that thereare three-kyndes of demonftr aa : tl . ation T oneis called Demenéfratio.a priori,or compofition, The other “oe is called Denoustratie a pofteriori. of Euclidés Elementes. Fol.9. apederioriorrefolution. Andchechird isademonftration leadying to animipof- A-demonfttation prieri,or compolitionis ;whenin reafoning, from the priti- ciplesand firft groundes,we paffe difcending continually ,rill attermany reatons: madc,we comeat the lengih to conclude that, which we firft chiefly entend.And this kinde efdemonttrationv fet Euchdein his bookes forthemoft party A demontftration 4 posteriort,or refolutionis,when contrariwife inreafoning, we pafle from the laft conclulion made by.che prémiffesjand by the premiffes of the premiffes,continually afcending til we come to the firft prs nciples and grounds, which are indemonftrable,and tor they r fimplicity can fuffer no farther rclolus tion, : BES) | | A demonftration leadyng toan impoffibilitieis thatargament, whofe co- clufionis tmpoffible: that is,when irconcludeth direS&ly again{tany principle, ot againft any propofition before proued by. principles, or propofitious. beiore proued, | ; | Premifles in an argument,are propofitions goyng before the conclufion by which the conclufion is proued. Compofirion paffeth from the caufe to the effe&t,or from thinges {imple to thinges morecompounded.Refolution contrariwife paffeth trom thinges com: pounded to thinges more limple,or from the effe& to thecaufe, . » €ompofition or the firftkynde of demonftration, whichpaffeth from the principles,may eafely be fene in this firft propolicion of Euelide,: Thedemon> ration wherof beginneth thus, And fora{mech‘as the poiiit’ A is the centre’ of thecircle G.B D, therfore theline AC, is equat rothe line AB. This reaton(y-ou fee)takethshis beginny ng ofaprinciple namely ,ofthe definition ofa circle.And thisisthefitft reafon, agayneforafmuchas B isthe ceorreof the circle CAE, therfore the linéB C is equallro thelyne B.A:which ts the fecond reafon. and it was before proucdthat thelyne A C is equalbto theline A.B, wherforeeither ofthefelines C A & CB ts equal to the lyneABJand chisis the chirdrealo, But things Whichard equall to one sethefelfe'fame thy ng, are alfo equal! theone to theorber: VVherfore'the line CA is equal'to the line CB, and this isthe fourth argument: VV herfore thefe threc tines CA,AB sand B C ate equall the onc to the other which is the conclufion,and thething to be proued,:, ... SS hoe “You may, alfo in the fame firft Propofitid,calely. takean.exaple.ofRefolurios y fing a contrary, order pafly.ng backward fro the latt couchifio of che former dex monttration,tilyou come.to.the frit principle ot groundiwheron it begant F or shelaft argument or teafon incompolitionyis the frit in Refolution: & the Are icompofitidn;isthelaftinrefolution,T hus therfore mut yeprocede, T he tri angle AB Cis'contained ofelirte equal SE ei inamely,A BA Cand BC, 13 édefnition.of an,cquilarer triangle; - a - Gndtherfore itis aticqut facet triangle byt W Yad this is tHE AH rcalon, That the three lines be equall, 1s thus proucd.,.1 he lines A.C and CB are equall to theline A.B, wherfore.chey ase-cquali che.one to "4 i i other;.and thisisthe lecond cealon.T hat chelines AB and B:G,are equaliis thus proued; dibedings A Band iA:C;aredraweinftom the cedtreroftheciicle*e Eto thecircumferent@ofthefame: wherforerhey ate equalrby the definite ofa citckeriandthis1s the thitd reafon brkewifé thar the lines A-C and A B, are Sw» oct She mons 1g eA INI ISNT HIVOFS YS -* Yee equalt™ rt oF ~ ve rei ’ | ,- 3 ) . ZJhbe Jaws tiie whee eeeer = = J, 7 ° <9 . > ie » . a =F Damonfitatéom a4 priors,or come pofitiens Demonstration a poftersor$ or ref olution, Demonftratiow leading 164% sa poffibslit ye « +’ Premiffes what they Are. An example of com pofition Pt 3 the firft propofs® tion, Firft rea[on. Seeond reafow Third reafom Fourth rea[ot. Comelufion. Example of re [olution te thE Sir prapaphsria Firft reafon. Second realom, Third reafcn. Fourth reafon whichis the end of the whoierea J ] bss leg. a. = Ss = ae ———=— == — — : a ——— - — j iy ' | hai 7 ii mn - ms » i if } 0 ae ——o—s- ee eo . TSS SS ’ — ee equall,is proued by the famereafon: For the lines A Cand A Baredrawn fom the centre of the ciccle BC D: wherforethey areequallby the fame definition’ ofacircle: this 1s the fourth reafon or fillogi{me, and thus is ended the whole celolution: for thatyouare cometoa principle, which is indemoftrable, & can pot berefolued, Ota demonttrationleadmg to.an impoffibilitiejorto anabfurditie, you may: hauean example in the fourth propofition of this booke, eAn addition of Campanus. Vtnowe if vponthe fame line geuen, namely, ef B, ye wil defcribe the other two. kinds of triangles, namely,an //o/éeles ora tridgle of two equal fides, & a Scalenon,or a pig. atriangle of three vnequall fides.Firft forthe defcribin g of an J/o/celes triangle produce or dcferibe’ eH eline A Bon ether fide, vntill it concur with the circumferences of both the circles ee | intes Dand F,and makine the he point-4,defcribea circle AF G ac™ sriangle. inthe pointes D and Fan bmaking the.centre the pont _4,delcribe a circle Gg ac | cording to the quatity of the line 4 F, 9 Likewite making the centre the poynte B,defcribeacircle AD G,according to the quantitie of the line BD . Now thé thefe.circles fhall cut the one thesother in two poyntes , which let be H,and G: And let the endes of the line geuen be ioyned with one of thefayd fections by x ae tworightlines,whichlet bee¢GandB f Ray Pei | {a ake G. And forafimuche as: thefe twolines::: at ” AB and 4D aredrawen fro the centre - ofthe circleCD E vnto the circumfe- _ £&ce therof,therforear they equal. Like... wifethe lines B dand B F, forthat they are‘drawen from thecentre of the cir- cle Ev CF to thecircumference ther-* of,are qual, And forafinuch as ether of > thelinese# Dand B Fis equalbto the line, 4B, therfore they are equal théohe | Ve 2 . tothe other,Wherfore putting the line.4 B cOmoéto thé both,thewholcline BD fhials Pe i | | be equali to the whole line e4 F, But B.Dis equaltoB G,forithey are both drawen4ro ae, | the cétre of the circle 2D G ta the eiroumferéce therof.And likewile bythe fame rae P| " fon the line 4 Fis equal to thelinee4G, Wherfore by the comé {cntéce thelines 4. pean | and BG are equal the one to the othet,and eitherofthem is greater then the line 48, ea | for that citherof the'two lines PD and U4 Fis greater then the line 4B; Wherforéy- ne ih : rete ery 1 ; | | | Het lk | ome = pon theline geweiis defcribed an Yoftdles or triangle oftivo equal fidesiioo" 85 pee Few todsferibe. . Yemay alfodefcribe vpon the felfe fame linea Stxlency | or ttianele oPeaedevnen ae #Sealnum. ". quall fides,ifby. two tight lives; ye toyne boththeendes of the line sonentofome one Pia | | gic oa ae paint thatis in the circumference of ong of the two Preater circles; be notin one of the,two fections,and.that the line D F.do not concunwithit, awvhen i¢ ison either fide prodiced continuallyeand direG] yc.Forlet the. oynte Kbe taken ia | the citcumiference of thecircle #7 D Gand tit not beinany ofthe fections neyther | let theline D Fconcat withit,whepitis produced continually and dire@ly vnro o : | | _ Greumference therof, And draw thefelines A Kand B Kand the tine <4 & thal out fie en * circumference of the: dirdeHFGrs Let iccur itin the poynte L's ‘now then hy thé commonfentence the line BX thalbe equal to thélinee 7 L ,tor(by the definite s ofa circle)the line BK isequall totheline B G,and the-line 4.1 is equall tothe: ined4@ K which is equal tothe ne BG,Wherfore the linee4 Kis greater thenthe line B K and ssreoies’s\s by the fame reafon maye it be proued that the line B Kis greater then the line ef B, Wherfore fothatthat poynt of Euchides Elementes ‘ Fol.10. Whtrfore thetriangle 4 BK confifteth of three yhequal fides, And'fo.haue ye ypon the line geuen, defcribed all the kindcs of triangles, This isto bencted that ifa man will mechanically andré2 dely not regarding demonftration vpona fine peuen deferibe atriangle of three¢quath{ides.be needech aot todetcribe® the whole forefayd circle, butonely alittle parry ofeche: nameby; wherethey cur the one the other,and{o from the point of the {e&tion fo draw the linesto the endes of theline geuen, \as in this figure here pur, rN “oS and likewife,ifv pon the faid line he willdefcribea trian, G How to defcribe gle oftwo equall fydes, let him exrende! the'compatfe accor- i. rng ie is ding tothe quancitie that he will have che fyde to bey wherher \ = oe longer them the line geuen or fhorter: and lodraw onely ais \ tle part ofechecircle,where they cutthbe oneche orher,.8¢ {6 the poinr ofthe fection draw the lines to the ende of theline \ geuen. asinthe figureshereput, Note that.inthis thetwo — fy des mutt be fuch, that bey ng ioyned together, they belon- .~ ger then the line geuen, and fo alfo if ypon the favdricht line hewill defcribea oe, triangle of three vnequally.des,lec him exctad the compafle. NS - pe ao Firft, according to the quantitie that he will haue one ofthe No angle redsly, vnequallfydes robe, and fo draw alictle part oi the circle, se thenextend it'according to che quautitie that he wil haue the other vaequabfyde ro be\and draw likewyfealicele part ofthe 3 ‘circle, and that done, fromthe point ofthefedion draw the “ lines to the endes of the line geuen,as inthe figare here put, Note that inthis the two fidésmuft be fuch,thatthecircles deicribed according co their quatitic may. cutthe one the other, ; SmT he fecond Probleme... Lhe fecond Propofition. Fro.a point geuen,to draw aright lne equal to arightline genen. ‘9 How to deferibé a7 CGE ALEK ivs a angle redily &9 Re See Me CARI AL}. — See 8 ene a SF > = ra = — ee, = — “ A oe ee . ~ a = ee ee ee J SS ————— = : 22-33 ™ o Aveo j f a i i J i ae V ppofe thatthe point cent be A eo let the right line gee AkxZ |'uen be BC, It WAS isreguired[ro \ the pomt A,to MSG drawe arioht SN Lyne equall to Zo Al the line BC, o_-- palin rena Draw (by the firft peticio) fromthe |. point Ato the poynte Barightlned _. B: and vpon the line AB defcribe(by - the firft propofitio) anequilater tris angleand let the fame be DA B,and exted by the fecond peticio the right 3.x,siomsa. | E a. Confruitian. lines D AQ DB; to the poyntes E 087 D4, ana *~ os $ : : Nw . e T he firt Booke and F,ex( by the third peticio making thecentre Band the [pace BC defcribe f circleCG H: ex againe(by the fame )making the centre D and the Jpace DG defcribea circle G KL, And forafs . Dewenfiration: much asthe pointe Bis the centre of the civcle G H,therfore( by theas.~ definitio ) the line BCs equal to the” line BG: and forafmach asthe poynt” Dis che centre of the circleG.K.L:. therefore( by the fame)the line DL. . is equall to the line D.G:of which the line D Ais equall toa line DBCby the propofttio coing before ywherfore the refidue namely,the line AL is ec gual to the reftdue,namel) tothe line BG( by the third common fentence ) And it is proued that the line BCises fh quall tothe line BG.VVberforeeyther of thefe lines.A L c~ BCis equal to the line BGBut things which are equall toone and the fame thing are alfo equal the one tothe other( by the firft commo fentence VV herfore the line AL is ea qual tothe line BCV V herfore from the poynt geue,namely, Ais drawna right dine AL equallto theright line geuen BC; which was required to be done. eA | Ne ‘Of Problemes and Theoremes,as we haue before noted,fome haue no cafes atall; ‘which are thofe which haue onely one pofition and conftruction:and other fomehaue many and diuers cafes:which are fuch propofitions which haue diners defcriptions & Tworbingesse- conitructions,and chaunge their pofitions. Of which forte is this fecond propofition, wen sn this pro- hich ig alfo a Probleme, This propofition hath two thin ges genen:N atnely,a pointe, oe j2 andalinesthe thing re.juired is,that from che pointe geuen wherefoeuerit be put, be shis propofition. drawena ine equall tortie line geuch’ Now this poynt geuen may haue diuers pofitios For it may be placed eyther without the right linc geuen,or in fome point init. [fit be without it; either itis on. the fide-ofir,fo that the right line dtawen fom itto the ende of the right line geuen maketh an angte:or els itis put directly vnto it.fo that the right line geuen being produced fhall fall vypon the point geuen which is without,But if it be in the line geuen,then either it isin one of the endes orextreames thereof : or in fome place betwene the extremes.Soare there foure divers potitions of the poynrin refpea of the line. Wherupon follow diuers delineations and conitructions, and confequent- ly varietie of cafes. | The frit cafe. For the firft cafe the figure before put,ferueth, Thefecond cafe, To the fecond cafe the figure here onthe fide fet belongeth . And as touching the or- der both of conftruGion and of demonftra- tion itis all one with the Arft, > The third cafz The third cafe is eafieft of all,namely,whé the poynt geuenisin one of the extreames. As for exaple,ifit were in the pointC, which of Euchides Elementes . Fol. isote oftheextreames of theline BC : ‘Then making the,centre the poynt Cyahd the {pace CB defcribea circle BL G:and fromthe.cen- tre C drawe 2 line ynto the circumference, whichlét the € L,which by the definition of acircle, fhalbe eg tall to the line geuen BC. ¥°"Phe-fourth café 2s touching conftruction hérein differethftom the two firlte', for that whereas in th@you are willed to drawsa right linefrom the poyntgeuen,nainely; «4st0 the poynt B whichis one of theendes ofthe line geué,hcre you fhal not nede to draw that line, for that it is already drawen.As touching the reft,both in conftruaion and demonitration you may proceedeas in thetwo firlte, As itis manifefte to fee in thys figure here on,the fade pu The fourth cafe. This propofitiom thourh it be Gem ry eafie to be done mechania cally, yer is n@ prénceple, t. at This propofition for the playnes & eafi- nes thereof,feemeth to be as it were a_princi- ple,and may eafly mechanically be done, For opening the compagffe to the quantitye of the linegeuen, and fetting on foote of it fixedin the poynt geuenand marking with the other another poynt wherfocuerit fall, & fo by the firft peticion drawinga rightline frothe one of thofe poyntes to the other, the fayd righte line fhall beequall to the right line geuen: yet in deedeisitno principle, for that it may by demonftrationbe proued: but principles can not be proued,as we haue before declared. a epee = a —— a +, a —— o » - x | ; " fc ¢ a i } ‘ i] SV V ppofe that the two vuequal right lines geuen be ABS Gin, =< C, of which let the lyne A B be the greater. It is requic x LLDpy)\""4 from theline AB being the greater, to cut of aright GueN line equal co the right line C “which is the le/Se linedrawt ee oe Ate ; 4 HS | . { iy : ‘ : SSE x an( by the fecond propofition )fro the post Aarightline aé py 2 Dy) « . Wr By: lequall to the line Cand let the fame be A D:and making SEY : be (by thethird : FA\the centre A andthe [pace AD defers peticion) acircle DEF, And forafmuche as the : point Ais the centre of y circle DEF, therfore A Eis equalto AD, but the lineC wsequaltotheline AD.VV ber fore either of thefe lines AE andl us equallto AD, wherfore the line A Ew equall to \ the line C,wherforetwo vnequall right lines being \ geuen,namely,A Band C there w-cut of from AB | being 9 greater, aright line A Eequall to the le [Se o/ —— lyne, namely, to C: which was required tobe done, Daiij. This F Two thingesgee ynequallrigh ies, ee fe ines yeu Divers cafes in MOU theorher:or are 1oyned rogerherat one ofthert get or they cuttheone sr. the orker,orche one cuttech the orher in one of the extreames, V Vhich may be two wayes.For echer the greater cuttech theleffe,orthe leflethe greater. [fthey curthe one the drher,ertherech cutteth th’ other.into equall partes:.or into vne- quall parces : or the one into equall partes sandtheorher iacowvnequall partes. V Vhich may happ Thefeurth cafe By his extreameC. Now theneytherit cuttethitinthe ~ Cc = py middeft ortiot in the middeft. Firbletit cutitin the mid LIS | deit:then C Dis ether the halfeof 4B,& fo is_A Cecual | ' ii ee D dike third cafes toc D, | | af Lhe fifth cafe, Oritisleffe then thehalfey and then making the: centre C & the fpace C D defcribeacircle;which'fhall cut The fxtcafe. Of fromthe lineedBaline equalto.theline CD. 5. < Or itis greater then the half. And thé ynto the point Flas A put the line A F equallto the line’€ D, by theifecond, A 5 cl the line 4F,thatis,ynto the line € D, “a And making the centre 4 & thefpace F deferibéa dir. e, which thall cut offrom the line. 6. a line,eyuallto \ + +? ae a. But of Enchdes Glenschttes. Fol.i2. Batif the line C D do not cut the line 4B in the midft: C D*thal either be the halfeof the Tine A Btor greater then'the halfe, or lefle . If CD be the halfe of 4 B,or leffe then rhe half of 4 B thé making the centreC,andthefpaceC D defcribe A ‘a circle whi¢he fhall cut of from the line.4 Ba line equal tothe line CD, The feuenth €$ sight cafes. But ifit be greater then the halfe,then againe ynto the pointe put the line 4 F equal tothe line C'‘D(by the fecond propofitio: ) & making thecentre 4, and the {pace 4 F defertbeacirele which fhall cut of from the line 4 B a line equall to the linee4 F,thatis,to the lineC D. The tt b tafe. But ifehéy cut the one the other as the linés C D& AB do,Thé making the cétre B & rhe fpace B Adefcribeacircle 4 F,& draw aline from the point B to the pointC,& produce it to thepoint , F,And forafmuch as the tworight lines BFand . CD are ynequall,andthe lineC D cutteththe \, line B F by on¢of his extrgames, therefore itis 4 ‘poffibleto cit of fromCD aline,equall tothe line BF.For how to doit we haue beforedecla- sed, whérefofeit is poflible from the lineC Dto cut ofaline equall totheline 4 B:or.4 B and B F are equal the onc to the other. a The tenth café. . a. | P z . —* _ a - PTT 10 SIS i Lf 193: PD “err 4% , ‘no This isto bepoted, that in all chefecafes,a man may bothascouchyng com- 2» es : * ln . . d J . p Pc . : - . , ° . >». : @ : r= it ‘ {44 aU Jade SLE Mtl a hhawdi We erakhadstas ti 2 47 SS Sisk Gil | OF a LP) as ~~ ‘ gba J " 3 : ~ . e 0UQ SOU 29170 90310 Suignicod oni ai visisa ba ‘Fthere be two triangles,of which two fides of th one be equat Aotiofides of the other eche fide to his corre/pondentfide,and ( bauing alfoon angle of the one eqiral to one angle of the other, namely, that anglewhich ts contayned vnder the equall righ lines >the bafe alfoofthe one hall be equal{ to the bafe of the \_ ather,and the one triangle fhall be equal to the othet triangle, “and theother angles remayning fal be equall tothe other an- gles remayning tie one to the other, under which are Jubten- dedequall ides. ! tii. Suppofe of Eclides Elementes. Fol.13. QIN ZA ppofe thatthere bet wotriansles A BC Aad in che A criagle DE Fylecthe fide DE be 2.and the fide D F bes. whiche added tage - ther make al {6 7.8clo'rhe B fydes of the one triangle added together are cquall tothe fides of the other trite gleadded together, Yetare both the triangles vnequall,and alfo thetr bales. For the area ofthe triangle A B Cis 6 and his bafeis’5. andthe area ofthe triangle D E Fis: and his bafe9'29.So that to hatiethe areas oftwo triangles to. be ¢ quall,it is requifite thatall the fydes ofthe two triangles beequall, eche to hys correfpondent fy de, It happenethalfo fometymes in triangles, that the areas of them beyng tquall ,tcheirlydesadded together fhall be vnequall. and contrari- ae wile En, Hb 2 a of Euclides Elemente s. Fol.t4.. wifeStheit. fides beynaequall,theirarcas be ynequallias inche(efigures here put {= 56. it is plaihe to fee, In the firft example thefareas ot the two, triangles are equal, for they are’: 13 eche12,andthe'fides in ech ad" NS | ded together are vnequall, for "\ >. ¢ 5 in the onetrianglé the fides adm y\5 aS ded together make 18, and ing? py, the other they. makej6.Butin «4 |. ms the fecondexapletheareasof °° | thetw6 triigles are vnequal, *°? ** for the one is 12,and th’other 8914 9) is 1, But the fides added toz hea : - getherinecheace equall, for. © ineghethey make38.). . 0 oS, \ daze <“Tharangle staid robs tend a {ide ofatriagle, which A> ye How Pte) angle . oe hoe R ey #3 [ayd to fubted is’ placed dive@ly oppofite;ss °: : a fiderand a fide & againtt that fi de,T hat fide. 2a oy Ar angle. alfo.is faydto fubtendan any, .\ gle,whichs oppofiteto the angle. Foreuery angle ofa triangle is contayned of twoly des of the triangle,and is fubtended to.the third fide, ° | ow ThISGs thefirtt Propofition in which is vfeda demonftration leading toan Théspropofition abfurditi¢}oran impoffibil miesViV hich isa demonftration thar proueth pot di ame ‘Beinn : i monfration lea rectly the thing.entended, by principles; onby.thinges before proued by thefe ding to anabfur principles:but proucth theconttary therofto be impoffible, & fo doth indirests diy. ly prouethethingentended, =.) | See sv lovas,, we 2 4e2-T heoreme,” Thes.Propojition. CAN. Lfofceles, or triangle of twoequal fides, hath his angles at “the bafe equal the one ta theotber.eAnd thofeequal fides be: ing produced, tbe angles which are ‘vnder the bafe are alfo ¢- quall the onetothe other. oe 5 V ppofe that ABC bea triangle of tio ONS equall fydes called Hofceles , bauing the POW de 4Bequall tothefide AC. And I Sa = eae R (by the fecond peticio)produce the lines ABS AC direétly toy points Der E/The Lay, that angle ABC is equal tothe angle ACB:and y » angle BD is equal to th angle BC E.Takein the line B Da point at all aduentures, and let.the Ey, Jame Confirutiex, Dewoufiration. = Theyirst Booke te fame beF cand( by the third propofition)from eT te lite, wamely, 2 Ey : cnt of a line :quall to A F being the lefse line,an let toe fame le AG: and draw a right line fro the point F co tht point C,and an other from the point G to the poin: B.Now thenfor as muche as AF ws». | equallto AGand A Bis equall to AC, therefore’ thefetwo lins F Aand AC areequall to thefe two linesG A anc A B,the one tothe other, and they containe a conmonangle,namely,that which ts coo tained ynderF AG: wherfore ( by the fourth pros pofition)the lafe F Cis equal to the bale G B: and <0) bo) hse9b" s , the triangle 4 F Cis equall to the triangle AG Band the otber'angles remate ning,are equllto the other angles remaining the one to the other ynder which are fubtendel equall ides: that is,tbe angle AC Fis equal totbe.angle 4 BG, and the anglA F Cis equall to theangle AG.B, And; forafmuch.as the whole line AF is equall tothe whole line A G,of which the-line A Bis equal toy lyne AC, therforethe refidue of the line A F namely the line B Fis equal tothe res fidue of the Ine A G,namely,to the lineC G (by thé third common fentence) And itis proved that C Fes equal to BG. Now therfore thefe two.B Eee FC are equall to :hefetwoC G and G Bthe one to the other, and theangle BE Cis _ equall to theangle G B,and they bane one bafe,namely,B:C:common tatbem both: w herfare ( by. the 4. propofition) the triangle BFC is equalt-to the tris angle CGB, and the other angles remaynyng are equall to the: other ane glesremaming eche to other, vader which are fabtended equall fides VV here fore the angk F BCisequall to the angleGCB, andthe angle BC Fis equall totheangle( BG, Now forafmuch as the whole angle ABG is equall to the whole angle AC Fas it bath bene proued )of which the angle CBG is equal to the angle BLE: therfore the angle remayning: namely,A B Cis equall to the angle remaining namely to AC BC by the third common fentence)And they ar the angles atthe bafe of the triangle ABC, rnd itis proued thattheangle FB Cis equall tothe angle GC B,and they are angles vader the bafe.. VV herforea triangle of tvo equall (ides hath his angles at thebafeequall the one to} other, And thofe equall fides being produced, the angles which are ynder the bafe are alfo equall tle one to the other: which Was required to be proued, For thar this pro A pofitid is fome what hard to erceue © y rca~’p fon the B Cc of Exuclides Elementes . Fol.15. fides of the triangles A F C:and:A-G Bae alferthé fydes'ofthe triagles B FC & C GB run fo one withinan other,therfore I haue here puethé diftnély names ly ,the triangles F A C and BF C ononefy de of the figure of the propofitid & the triangies\A G@ Band ©: G:B on the othexfyde:fo tharyou may with leffe la- bor fee the demonftration playnely. ~* That'tranTfofceles rriangle,the two angles aboue the bale arzequall,may orherwifobe demonftrared withour drawing lines benearl: the bafefomwhar alz tering the conftru@ion, Namely ,drawing the lines within the tniagle , whiche before were without itafter this manner. Suppofe that 4B Cbe an Ifofceles triangle:and inthe line A Btakea point atall aduentures,and fet the fame be D, And from the line\4’C cut ofa line eqnalltothe tine 4D. Which let be 4 E,And draw thefe right lines B E,D C,and D.E, Now forafmuch asin the triangles,4.B E,and 4C D, the fide 4 Bis equall to the fide 4 C,by fuppofition , and the fides 4D and AE are alfo equall by conftruction,and the angleat the poynt Aiscommon tothem both: therfore, by the fourth propofiti- on,the bafe B Fis equall to the bafe D C,And,by the fame, the angles remayning ofthe one triangle are equall to the angles remayning of the other triangle. Wherefore the angle 4 & E1s equal to the angle 4 CD.Againe forafmuch as in the triangles aa C BD E,and C E'D the fide D Bis equall to the fide E C, and the fide B E to the fide D Cand the angle D BE is equal to the an- . | le EC D,and the bafe D E being common to both triangles is equall toit felfe:there- ae the angles remayning of the one triangle, are equall tothe angles -emayning of the other triangle. Wherfore the angle E DB is equall to the angle D EC: & the angle DEB isequalto the angle E D C.And forafmuch as the angle E DB is‘ecual to the an gle D E C, fré which are taken away equall angles DEB,& EDC, therfore by the co- mon fentence the angles remayning,namely.B D C and C EB are equall: Andasit was before manifett the fides B D and D C are equall to the fides C Eand EB the one tothe other,thatis,ech to his correfpondent fide : and the bafe B Cis comma to both the triangles: wherfore the angles remayning areequall to the angles remayning the one to the other,vnder which are fubréded equall fides. Wherfore the angle [L B Cis equall tothe angle EC B.For the line D.C fubtendeth theangle D BC,and the Ine 8 fubté- deth the angle EF CB: which two lines are as we hane before proued equal. Wherfore in an Ifofccles triangle,the angles at the bafe areequall, though the righ: lines be not produced, Soe To proue this alfo,there is an other demonftration of Pappus much fhor- ter which needeth no kind ofaddition of any thing at all:as followeth. Suppofe that A B C be an Iofceles triangle ,& let the fide AB be equall to the fide e4 C.Now then vnderftand this one A triangle to be as it were two triangles,And thus reafon - For- afmuch as in the two triangles 4B Cand 4C Be Bis equal to AC & ACto A B,therfore two fides of the one are equall to two fides of the other, ech to his: correfpondentfide, & the angle B ACis equalltotheangleC 4B, oritis one and the fel fe fame angle,Wherfore by the 4.propofition the bafe C8 jsequall tothe bafe BC, andthe triangle e# BC is equal to the triangle 4C B:and the angle 4 B Cis equall to the angle - ACB, andtheangle 4C Btothe angle 42 C:for ynder them are fubtended equall fides namely,the lines 4B & 4 C.Wher forein an Iofceles triangle,the angels at the bafe are equall. E, iit, aS An ether demo- frration inuen- ted by Proclus. Jin other deme STY ATION INNO ted by Pappis. PS ge: 2 rn SSI ———— a ng = — . 7 " a Se ne - ea _ ———— nas : —- — ee ee meme ge yi He as | ie i 4 ' i OO AED, BS ——— a SS SE me ——- 5 Sy “ * =e. — =, ee —s. > pe Tae ee a ee re bales Milefises fhe inuentor of Chess propofition. leading toan — muse’ 4 > | Demenfiration actly agreing with the triangle D E ~~ Te = ~ aa: — ~ ~ = - < te — — . —— eae wae ~— oe a © 3 aS ene ~ » . ss _ ~- ———— 2 en : aS - “FERS “Soe . = —— = — * ~< —— = Oo a es — = Ss — - “ i = wnat Ps = atin, conte —- : nr ee ee a oe — —~— ied — - : — - ——=— ——_ = >: =: — a eS SS od . a - ae -_ ar = ae a 8 Sa re A eRe A 4 — . ee - are : RT mn tg. lhe oa ‘<= ~ cs 5% - ated <0 2 se = —— ” a = am = a we = mh ame ve —, ee — ee - = = = - - es ~ —— = -——— : ~ = = 7 = 2 or Fe a SS = << ae = re - — = —~ : se ast a es Er = - TO ts Sait. SRE : ae . - nas Se ~ , ret ee 1 ——— ; whdien cost - —— ie = es <————o : SS SSS = a = : : — <= = or = Se - ‘mpofttiliey F,and the point B being put vpon the D oint E,and the right line BC vpon I the rizht line E F: the point-C fhall \ exatHlyagree with the point F (for | the line 'B Cis equallto the line EF) | : And B.C exatily agreeing with EF | the lines alfoBA and AC fhall ex actly agree with the lines E Dee D EF, Fonifthe bafe B.C doexattly as gree with the bale FE: but the fides a | BA ce AC doo uot exactly agree w eas hs 9 dilo Vy the fedes:ls Diep DE but “per ash oto) bit ows ‘ial G or G Fido, the fromy eudesofone ' ° ‘as Se [aati Lyue fhalbe drawn twontght lines toa poynt,cx from the Jelf fame endes.on the. eat | Jame fide [balbe drawn two otberlines,equat tothe two firft lines,5 one tothe os : ie | ther,and ynto an other poynt:but that is impo/Sible, (by the [euenth propofitiay Weill VV berfore the bafe BC exactly agreeme with the bafe EF the fides alfo BA | and AC ao exactly agre with the fiderED and D FV her fore alfo theanole BAC fhall exatily agreD the angle EOF and therfore fhall alfo be equal to at. Lf therfore two triangleshaue tho fides of the one equallto. two fides of the other ech to ins corréfpondent fide and bane alfo the bafe of} one equallto the bafe of the other:they fhalthaue alfa the angle contayned vader the equallright - lines of che one,equall to: the angle coutayned vader the equall right linesiof.-the other: which was requited to be prowed.®-— 28 603 oli sunooni This propofition This Fheorethe is shegonuerle of ch@fourth , but itis nor thechiefefttand ericcnuerl’ prineipall kind of conuerfion, For it curneth not the whole {uppofitioninto the batik Me chee cOntclufionsand the whole conclufion ince che ty ppoficionnforthefourth pros fof yodof con polis whofe conuerfeshisisyis acépound rheoreme, hawing two things geug ser fait. ot fuppofed which are chefe:the one, thactwo fidesofthe one teiiglebe equal to . ewo fidesofche other triigle:ch'orher,tharche angle coraaned of the two {des of _ th’onets equal ro he anglecontained ofthe two fides of th other: bur hath amo geftorker one thing required, whiche ts, that the bafe of the onc,is equal to the bafc ofthe ather, Now inthis 8, propofitié being. che conuerletherof:that the bale of che oneis equal to thebafe of th’other, is the {uppofition orrhe thing ges we; which in the former propofttid was the conclufié. and this,that rwofides of the U of Euclides Elementes . Fol.18. the one areeduall torwo fides of the other, isinthis propofirion alfo afiips pofition like as 1t'was imche former propofitrontoihar it 1s. a thing gewen In ci ther propofition, The conclufion ofthis propofttion 1s that the angle enclofed of the two equall fides ofthe one triangle is equall to the angle enclofed of the two equall fidesef the other triangle: which in the former propofition was one of thethings geuen, Philo and hh4s (-hottdemonftrate this propofition without the helpe ofthe former propofition,in this maner, | . equallto two fydesof the Gther namely, A Band 4€ equalltoD E and DF, the one to the other, & the bale BC equal te Plie bafe'B Fy’ And forthat the bafe' B Cis equall to the bafe E F,therfore the one being applied to the other they agree , Place the two tridgles 4B C& DE Finone & the felf fame plaine fuperficies,& apply the bafe of the one to the bafe ofthe other:But yet fothat the triagle ABChbe fet one the other fide of the right line EF; that the top ofthéone may be oppofite to thetop of the other. And in ftead. ofthe triangle.4 BC put the root a D trianvle E F Gas inthe figure, And let - DE be equalkto Gand. DF to, FiG. « Nowe then by this meanes fhall hap- pendiuers cafes, For the line F G may fall diteGly vpo'the line DF ,orit may fo fall that itmay make with the line DF an angle within the figure,or with Out. gto | | ~- « aA Firfttet it fall direftlye . And foraf- mucheas the lineD Eis equall'ta the line EG,and D FG, is onerighte line: therfore D E Gisan Iofceles: tridgle: and fo, by the fifth propofition the an- gle at the point Dis equal to the angle at the poynt G:which was required to be proued, But ifit fall not dire@ly , but make with the line D Fan angle within the figure, drawea line fronrD to G, Now forafmuch as ED and E G areequall,and théline D Gis the bafetherfore by the fifth propofitio, the an- gle E D Gis equalltothe angleEGD . Agayne foraf- much as D Fis equall to F Gsand D G is the bafe = ther- fore by the fame, the angle F D G is equalto the F GD: and it was proucd that theangle ED G isequall cothe angle E G D:wherfore the whole angle ED Fis equal to 7 wholeangleF GE: whiche was required to: bepro-~ ued, Butiftheline FG make with the line-D F an-angle without the figure:draw aright line without the figure F,ij. | from Suppofethatthere be two triangles 4 B Cand D E F, haning two fydes of the one: ian other demia fration sens tedby Philo, - After thes de- mnonfiratte thre ; cafes 47 this pi §= Pefition a The firft cafe. The fecond cafe. \ The third cafe, T he firft Booke from the poynt D to the poynt G.And forafmuch as D E and EG areequall,andDG as the bafe, therfore by the fifth propofition,theanglesEDG and DG Eare equall.A- D gaine forafmuchas D Fis equall to F Gand D Gis the bafe, therfore, by the fame,the angle F D Gis equall to the angle FG,D = Anditwas pronedthat the whole angles E DG&DGEare equalithe one tothe other: wherfore the angles remayning EDF &- EG Fare equall the one to the other, which was required to be proued, . | SeThe 4.Probleme. The 9.Propojition. Todeuide a rettiline angle geuen,into twoequall partes. if ppofe that the rettilineangle.cenen be BAC, It is ree: Mae ||quired to deuide the augle BAC into two equal partes,. lee ~ = rs the line AB takea point at all aduentures, ¢¢ let the. “GHES| [ame be D. And( by the third propofition )from the lyne 'AC cutte of theline AE equallto AD, And (by the Conttrudtion. ey ES firSt peticion) draw aright line from the point D tothe 7 iS SF} point E. And( by the first propofition )vpon the line DB: defcribe an equilater triangle and let the fame be DF E, and by the first petis cion) drawe a right linefrom the poynte 4 to. 7 the point F. Then I fay that the angle B A Cis by | Demonstration, 5 line AF deutded into two equal partes.For,fors afmuchas AD isequall to A E, and AFis comon to them both:therfore thefe two DA and AF are equall to thefe two E Aand A F,the one to the o« ther.But( by the first propofitson) the bale D Fis equall to the bafe E F:wberfore( by the 8, propos _ fition)theangle D A F sequal to the angle FAE, VV berfore the reétiline angle geuen, namely, B A Cis denided into two equal partes by the right line AF VV hich was requie ved to be done, | In this propofition isnot aught to deuide aright lined angle into. mo partes then two: albeit to deuide anangle,{oit bea right angleinto three partes, it is. gcd ape not of Euchdes Elementes. Fol.t9. Its smnpof- not hard, Anditis taught of Vitelio in his firktboke of Perfpeaiue,the 28, Propo- (lero dewide fition,For to deuide an acuteangle into three equal partes ,is(as faith Proclxs)1m- poffible:vnles it be by the helpe‘ofother lines which are ofa mixt nature, Which thing A scomedes did by {uch lines which are called Concoides linea,who firft {erched out the inuention,nature, & properties offuch lines, And others didit by other meanes,as by the helpe of quadrant lines inuented by Hippias ee Nicomedes.Ochers by Helices oc Siical lines inuented ot Archimedes.Bur thefle are things of much dif- ficulty and hardnes,and nothere to be intreated of, Here againtt this propofition may ofthe aduerfary be broughtan xinftance. For he may caui}l that the hed of the equilater triangle fhall not fall berwene the tworight lines ,but in one ofthem,or withoutthem both. As for example, Suppofe that the angle to be deuided into two equal] partes be B AC, andinthe line Be take the poynt D.,afid vite the line DA put the line A Eequal(by the third propofition. JAnd drawa line fro D to E,And vpon the line DE deferibe(by the firft) an equilater triangle, whichletbe DFE, Now then if it be poffible that the point F do not fal betwene the lines 4B & AC,then it thal fal e- ther in the line 4B or AC, or without them both. Suppofe that the point F be fall ypon line 4B, fo that let D FE be an equilater triangle.‘ Wherfore the line D Fis equal to the line F E: & the angles atthe bafe are equall, namely, the angles EDF and DEF. Wherefore the whole angle D EC ts B ; greater then theangleEDF. Agaiheforafinich ase4 Dis equallto AE, therefore AD Eisantfofcelestriangle. Wherefore (by the fifth propofition.) the angles yn- der the bafe are equall,Wherfore the angle D E Cis equall tothe angle ED#, Butit was alfo greater: which is impoffible. Wherfore the-top of the equilater triangle canot bein the right line-4 B.And in like fortalfo may we proue that it canot bein the right line 4 C,Wherforéfuppofe that it be without them both,ific be pofflible. And foraf- much as D Fis equal to F E,the angles at the bafe are equal, namely the angles DEF. & ED F.Wherfore the angle D E Fis greater then the angle EDF.Wherfore theangle D ECis much greater then theangle E J) F,But itis alfo equal vntoit.For they.are angles vnder the bafe D E of the Ifofceles triangle 4 D E.Which is impofiib'e , Wherfore the poynt F fhall not fall without the two right lines on that fide . And in like forte may we proue thatit fhall not fall without theni on the other fide , Wherfore it fhall of necefli- ty fail betwenc them:which was required to be proued. “> There may alfo in this propofition be divers cafes.ifit fo happen that there be no {pace vnderthe bafe D E to defcribe amequilater triangle, but that ofnecef- fitie you muft defcribe it on the {ame fide that the lines AB and A C are. For chen the fides ofthe equilater triangle either exa@ly agree with the lines AD and AB, ifthe faidlines A Dand AE beequall with the bafe DE, Or they fall without them,1fthe lines A D.and A Babe leffe' then the bale DE. Or they fall withinthem,if the faidlines be greater thea the bafe D E, Firft let them exactly agree.And let D 4 E be an equilater triangle.And in the fide A D take the poynt G.And from the fide 4 E cut of aline equal to the line 4 G( by the third propofition )which let be 4 H,And draw thefe right lines G E,H D and G #,and A F,Now forafmuch as 4D is equal to 4E er AGynto 4 #,therfore er! om lil, an an achte re- Biline angle sna to three equall partes without the helpe of lines which are of a mixt naa tures An instance ¢$ an obietion or a doubt wherby letted or trou- bled the con- firudtion, or de monitration,€S conta ryneth Aan Guruth, andan sympoffibsliry: and therfore sat mult of nece[fity be anfivered Gn- to,and the falfe- bode thereof made manif er, Diners cafes im this propofitsen. The fri cafe, ant “fore the angle GA F isequall tothe angleH A But the line D G is equal tothe line EH: wher ‘gleE GH is equalltotheangle DH G. Wher. | Thefn 2A and A Hare equallto thefe two lines EA and AG:and they contayne one and the felfe tame angle, Wherfore by the fourth. propofi- tion,the angle GD His equaltothe angleH E G.And the bafe DH is equall tothe bafe EG, fore againe by the fourth propofition , the an- ore by the fixt propofition , the bafe GF is e- qualto the bafe HF.And forafmuchas A His equail to AG,and'A Fis commonito thé both; and the bafe G Fis equall to the bafe HF, ther -F.Wherefore the angleG A His deuided into two equall partes: which was required to be . .done, Whe fecond cafe. the lines DF & E F thé draw a line from F to.A & p - Now forafmuch as the lines D F and FEare equal, -& the line FA'is common to them both; & the ba- fes D AandA Eareequall: therfore(by the eight) the aagle D F Ais equalto the angle EFA, Againe fora{miuchas‘D F and FE areequall) and FG. is common to them both, and they containe equall” “angles (as it hath bene proved ) therefore (by the. fourth) the bafe-D G is equall to the bafeG E. And - forafmuchasA DisequalltoAE,and A Giscom- ~ mon to them both, Therfore(by the eight)thean- gle DAGisequaill to the angle EA G.wherefore ‘the angle DA Eis denided into two equall partes: Tbe third cafe. ‘Te deuidea eA © red slime ang: $820 L196 eqisalh parts M ecxans- £4 By. "Which wasrequired to be done. But if the fides of the equilater triangle fal with- in the rightlines B Aand A C, as dothelines DE and FE,then againe draw aline from Ato F, And forafmuchas D A isequall toA E,and A Fis com- man to then both,and the bafe D Fis equal to the baie F E-therfore the angle D A F is( by the eight) equall tothe angle BAF, Wherefore the angle at the point Ais denided into two equall partes,how focuer the equilater triangle be placed: which was required to be done. This isto be noted, that ifa man will me- ‘ chanteally or readily, not regardyng demon- ftration, deuide the forefaid retiline angle B AG, and fo any other rectilineangle geuen wha into two ¢quall partes, he fhall neede onely with one oz pening ofthe compaffe raken at all aduentures to marke thetwe pointes D and E,which cut of equal partes‘ofthe lines A Band A C,howfocuer they happen, and fo ma- ¢ point F.to draw a rightline: which fhall deuide gle BA C iato two equall partes, And here no you thall not nede to draw the circles all whole,but ones king the centres the two points Dand E, to defcribetwo circles according to the openyng ofthe compafle:'and from tae point A to their interfeAion, which let be the frit Booke B | CG Butifthe fides of the equilater triangte falf without the right lines BAR AC,as da_ toduce the line FA to the pointG, tfoeuer, the an- :, te, that — of Fiuclides Elementes . Fol.r0. ly aportion where they cut theonethe other: Asiathe figurehere intheend of the other fide put. {aT hes.Probleme. Lhe to. Propofition. To devide aright ine geuen being finite,into two equall partes. ATK V ppofethat theright:line cenen be A B It isrequired todenide the CASK line A Binto two equal partes. Defcribe( by the firf? propofition ype conftruttion Q) CS) on theline A B an equilater triangle and let thefame be ABC. And = (by the former pr opofition )deuide the angle AC Binto twoequall partes by the right line CD. Then L fay that the | | right line AB is denided into two equall partes in the poynt-D:For forafmuchas ( by thefirft propos fition AC is equalltoC B, andC D is commonto the both:therfore theferwo lines AC CDare.ee gual to, thefe two lines BC eo CD, y. o7@ to y other, and theangle AC Dus eguallto the,augle BED. VVherforelhythe s.propapsion )ebe bale A Dis equal tothe-bafe BOY Vhexefore the righte line geuen A B,is deuided into twacquall parses ja.tbe poynt D: which was required tobe. done, Demonfiratio 9, Apolloniusteacheth to deuidea.right line being finite inte two equall partes after this manner, emit ee | An other tay te : a , denide aright line being finite, Suppofe that the right line being finite inuented by de be AB:whiehit is required to demide ta: ere TIE S polonsus. to two equal parts. now thé making the... of LAN centre the point 4 & the {pace 42 de- “ff re \ {cribe'a circle. Again making the centre: ~ ~* P | ) the poynt B & the fpace$ A defcribean hater? 2A E B other.circlesand fromthe comon feci-.... ons draw. the right line C‘D, which let — . chtthe line AB inthe poyatE +. Then fay;that rhe rightline C Rdepideth the, suhN line -4 8 mto two eqnuallpartes in the... 495 7 TH point E.For draw thefe right lings D 4, _ | DBC Mand eB. Which thal be equalthe one to the other, for that they aredrawne fro the centres to the circumferences of equail circles.And forafmuchas the lines C4 & ADareequalltothelines CB and B D,and the line C D is common to either ofthem: therefore, by the eight, the angie AC Disequall.to the angle BC D. Again forafmuch as the line CE is common to thelihes e4C andC’B, which are equall the one ta the 2- ther, therfore thefetwe lines 4 Cand CE areequalto thefe two lines B Cand CE, the © one tothe ther,and they'contayné equall dngies,as it hath behe proued. Wherefore by the 4-propofition the bafe #£isequall tothe bafeB E. Wherefore theline 441s deuided into two eqnall partes in the paynt £ :which was required to be done. Fill, By ‘Conftrudtion, Demonftration. T he firft Booke By this way ofdeuiding aright line, into two ; ‘equall parts inuented by Apollonius, itis manifeft, that if a man wil mechanically or redely not confi- dering the demonftratid,;deuide the faid rightline, and fo any rightline geuen whatfoeuer,into two e- quall: partes he nede onely to marke the poynts of the interfeions ofthe circles, 8 to draw a line frd the fayd interfeions,which fhall deuide the righe line geuen into two cquall partes : asin the figure here puc. | ; Se Tie 6.Probleme. — Thewt.Propofition. V pon a right line geuen, to rayfe yp from 4 poynt genen in the Jame line a perpendicular line. | TN cd SEN ORF 4 wm in it geuen be CIt ts required from the poynte C toray/e % : 9 FAS RAY | a, BS _ » Although the po ynte geuen fhould be fet in one ofthe endes ofthe + ghte line geuen,it is eafy {o do itas it was before, For producing theline in leagth from the poy nt by thefecond peticion,you may workeas you didbefore, But if one require to erea right line perpendicularly fromthe poyntatthe end ofthe line V ppofe that the right line geucn be A B,eo let the point of of Enclidés Elementes. Fol.21. lyne, without producing therightlyne;. chatalfoumay wellbeedone after thy s Manner, Suppofe that the right line geuen be 4 B,& let the point in it geuen be in one of the endes therof, E namely,in 4.And take inthe line 4 2apointatall aduentures,andlet the fame be C. And from the {aid point raife vp(by the forefaid prépofition)vn- to 4 Ba perpendiculer line,which let be C E, And (by the 3 .propofition)from the line C£ ent of the line CD equall to the line C_4.And (bythe9.Pros pofition )deuide the angle AC Dinto,two equall partes by the line C F, And from the point D raife vp vnto thelineC Ea perpédiculerline,D F,which . let concurre with the lineC F inthe point F, And C drawe aright line from F to A, Then I fay that the ‘¢ angle atthe pointe is aright angle. For, foraf- muchas D Cisequall to @e4, and C Fis commonto them both,and they containe e- quall angles (for the angle at the point Cis deuided into two equall partes) therefore (bythe 4. Propofition) theline D Fis equallto the hine¥4; and fo the angle atthe point A1s equal to the angle atthe pointD, But theangle at the point D is aright an- ee eee gle. Wherfore alfo theangle atthe point e-7is a right angle,Wherefore from the point 4 A ynto the line AB,is raifed vp a perpendiculer line 4F,without producing the line 4 B, Which was required to be done, Te: Appolloniusteachethto rayfe vp-vnto aline geuen;from a point in it getten, a perpendiculer line, after this maset, | } Suppofe thatthe right lirie geuen be 4 B. And let the point init geué, be €,Andin- the line-e-f Cytake a point atall ad- uétures,&let the fame be D. And fré the lyne CB,outef~aline equall to the line CD, whiche™*} let be CESanid'makyn¢e. % the \centre, D,and. the: {pace D E, defcribe a circle. “And againe ma- king the centre C,& the {pace £.D., defcribe,an,., other circle,and let the point of their interfeQién be F,and draw a right linefrom F to C.Then I fay that theliné FCis cré&ted perpendiculerly vnto the line’ 4 #. Fordrawe thefelines F D and-FE«yhich thal bythedcfintion of acircle be either of them equal to.theline.D.E; and therfore (by the firftcommon fentence) areequall the one to the other.But the lines D Cand CE are by conftrnG@ion equall,and theline F Ciscommon to thenrboth. Wherfore'the ansles alfo atthe point C are equal (by the 8. propofitio:) whetforetheyare rightangles. Wherfore:theline CF iserected perpendiculerly yato theline 42 from the point.C: which was.required to be done; ) By this way ofere&ing a perpendiculer line mnuented by Appollonius, it ts alfo manifelt that ifa man will caret Shh ag dem tae vato Ct SEHR. Ga a Si 8 SERPS UGA | Ay a line cs An other tafe ii this propofitsict, C onttr ach Ion, Demonftratien. An other way oe “ = SS < -—- ae > ——~ ~ a _——. Paneer = =a oe a — . a — ro 7 — — . ee — = : - : as se Se EEE 5 - —— = : : = —_——— 2 : oy onftradtioms’ | Demontration wwieteve gles CB Eand EB Dare equal tothe felfefame . An other de- mous cation af ger Lelitarstms. » A BC are equall to two right angles. VV berfore whena right line fanding va The firtBooke 7 ppofe that the right line A B ftanding vppon the right \\line CD do make thefe anglesC'B Aand ABD. Then [fay, that theanglesC BaAtand ‘AB Dare eyther two XGHEN\ riot angles,or els equall totworight angles.If the angle Area \OASS ASIA ha the 11. propofition) ynto.the right line C D,and from the ponte geneninst; namely, Byaperpendiculer line B EVV berfore (by the x definition the angle C B Eand EB Dare right angles. Now foralmuch as the angleC B Byisequall tothefet wo anglesC B Aand AB E, put the angle EB Dcommon tothem both: wherfore the anglesC BE and EB D,are equal to thefe three anglesCBA,ABE,and EBD, Agayne forafmuch as the angle. DB A is equall vnto thefe two angles D BE and EB A put the angle A BC common to them both: wherfore the angles DBA\ *o9 | and A BC,are equabto thefe three angles, DBE, E: EBA,and ABC. Andit is proted that the an-.. | : A ny three angles: but thinges equall toone co the Jelf att fame thingyare alfo( by the first comma fententey: > 220< equall the one totheothe VVberforethe anotes E>" B EandEB Dare equall to theangles DBA gm, | ABC, But the angles CB EandE BDiaretwe oD: 8 right angles: wherfore alfotheangles DB Aand’”’ a! pon aright line maketh any angles: thofe angles {halbe either two right angles; or equal to two right angles: which was required to-be demon/trated, - An othet demonftration after Pelitarins..- Suppofe thatthe rightline A B do ftand vpon the right line CD. Then fay,that the tvoangles 4 BCaud.d BD, areéeither two right anglesior equaltotworight angles, Fordf 28 be perpediciilerynto CD: thes it manifeft,that they.are rightangles(by the conuerfion of the definition )But ifit incline towardés the end Cthen(by the 1 1,pro¢ potition jfrom the point B,erect vnto the line C D a.perpendiculeriine'B E By whiche conitruction the propofitio is very manifeft,For forasmueh as-the angle 14 B Dis greas terthentheinight angle 2.2 Lby sheanglewtB E,-and .-- »- bs amie 31:3, yl to Beiorstin: the other. angle 4B Gis lefle. thenthe right angleiGy, ..' BE by the felfe fame angle ABE, if-ftom the greater») >- bee taken away the exceffe; andthe fame bee added to the lefie,they {hall be madetwo rightangles. Thari s,if 5 ey ther - ; - ey SS SOx iene ewan Bose h Seem OT Sh : gle Certs chereshalttace made thenight-ang W herefore it is mantfe of Euclides Eleméites Fol.23. Se The7. Theoreme: Phet4.. Propofition. Ffunto aright line, and to a point in the fame line,be drawn two right lines, not both on one and the fame fide,making the fide angles.equall to two right angles : thofe two. right ynes [ball make direttly one right line, aes i. , : a Se $NA EP a <= fy one right line: which was required to be proved. 2 coat edeie guy 1a other demonftration after Peltarts. oe . = Suppofethar there bea right line 4 B, vnto Whole pointe B; let there oe ye Another den. two tightiines CB'and BD,vnto contrary fides ‘and tet the twoanglesC'B 4 jane! ey omg A A,be either two right angles, or equall to two right angles »Then I fay, thatthe tWO sey reistersoe,® lines CB and BD, do make dire&ly one right line, : = namely,C D. Foriftheydo not.the lend Ebefo drawn LUSAOD S941 vntoC B, that they both make dire@ly oneright line aS eS ——s > — Pay oy a * eee SSS a, . guar oi i aking nn tal CBE: which hall patie cither aboug the, line 8 50k... i teddy = He vnder it ,.FintJer.at pafle aboueit,And for asmuch as... vis Ube E | thetwoangles CB Aand ABE;are(by theformerpro- © : : position) equall-totworightangles, andareapart.of Sow nik aH and 42 0ibuttheangkesCBA oh tg and. BD are by Cuppofitionjequall alfo tbo aes angles; therefore the parte is equialltg the whole which u fs impollible. And thelikeablurditie wilfollowifC2 ¢ BD Gili. ‘The firft Booke E pafle vnderthe line B D:namely,that the whole fhalbe equall to the part:which is alfo impoflible. Wherefore CD is one right line: which was required to be proucd- The&.Theoreme. The 15.D’ropojition. Ff two right lines cut theone the other: the hed angles fhal be equal theone tothe other. Z55 V ppofe that thefe too right lines AB and CD, docut the one the ov \ | ther in the point E, ThenT fay, that the angle A E C,is equall to the Demmnitration WKS) angle D EB, For fora/much as the right line AE, /tandeth vpon the == right line DC, making thefe anglesCE A,and A ED: therefore( by the 1 3, propofitio)the angles CE A,and A E D,are equall to two right angles. Agayne forafmuchas the right line'D E,ftandeth _ vpon the right line AB, making thefe angles A ED, and DE B:therfore( by the fame propo/itis on the angles AE Dand DEB,are equall to ttre right angles: andtt is proued, that the angles CEA,and AED, are alfo equal to two rigbt ane gles. VV berfore the angles CEA,and AE D are equall to the angles AED, andD E B, Take ae way the angle AE D which és common to them both. VV herefore the angle remayning C EA, is # Fhe to the oe remaynine D EB, Andin like fort may it be proned , that’ theanolesC EB andDE Aare equall the one to the otber : If therefore two. right lines cut the one the other, the hedangles (halbe equall the one to theos ther: which was required to be demonstrated. | oy ae a Milefiusthe Philofophet was the firft inuenter of this Propofition, as see ob this vii witnelleth Exdemivesbut yerit was firlt demonftrated by Euclide. Andinit there is pofition. no conftruction at all, For the expofition of the thing geué,is {ufficientinough ee forthedemonftration, sn thes propofi= Hed Angles are appofite angles,caufed ofthe interfetionjof two right lines: f40n, Whathedan- and are {o called,becaufe the heddés of the two angles are ioyned together in ghsare, . one pointe, : fe The conuerfe of this propofition after Pelitarins, 7 Tf fower right lines being dra\ven from one point do make fower angles, of which the tho Oppe- b é od : > "o3 * = cous yA fiteangles are equall: the two oppofite lines fhalbe drawen por! Pisin. one right line. Bon ‘or Pelstarius. . Suppofe that there be fower right lines AB, AC,A D,and AE, drawen from the poynt A,making fower angles at the point A: of which let the angle BAC be equall to the angle D A-E, and the angle BA Dto the angle C A E. Then} fay, thatBE and Cp are onely tworight lines: thatis,the two right lines BA and AE are drawen diredtly, | : | a ve of. Enchides Elementes ; Fol. 24. and dow make one rightline, andJikewife the two right lines C dand A D are drawen directly. ,anddo make one fe right line. For otherwifeifit be poflible, lec EF beone 6 Y right line, and likewsfe let C G be one right line. And for- aimuch as theright line £4 fandeth vponthe right line CG, therefore the two angles EAC and E AG, are ( by \ the 13 propofition ) equall to two right angles. And for- \ afmuch as the right line G4 Randeth ypon the right line \/ E F: therefore (by the felfe fame) the two angles E AG _ and £4 G,are alfo.equali totwo night angles. Wherefore 4, taking away the angle E 4G, whichis common to them \ both, the angle E AC, thall ( by the thirde common fen- \ tence) be equalltotheangleF AG:buttheangleE AC , 8 ‘\ is fuppofed to be equall to theangle B 4D. Wherefore ~ > theangle B 4 Dis equallto the angle FAG, namely a G 5 artto the whole: which is impoflible. And the felfe ame abfurditie will follow, on what fide foeuer the lines be drawen. Wherefore B £ is one line,and CD alfo is one line: which was required to be proued. The fame conuerfe after Proclus. If untoa right line, and to a point thereof be drawen two right lines, not on one and the fame fide, in fach fort that they make the angles at the roppeequall: thofe right lines fhalbe drawen direttly one to the other, and {hal make one right line. Suppofe that there bea rightline A-B,and take.a pointin in C.And ynto the point initC,draw thefetwo.rightlinesC Dand CE ynto contrary fides, making the angles at the hed equal, namely, the angles 4.C D.and BC E.Then I fay, that the lines (D andC E are drawen dire@ly, and do make oneright line . For forafmuch as theright line C D ftanding vpo the right line 4 #,doth make the angles OC 4and DC & equall to two right angles (by the 13 propofition:)andtheangle DC A. is equall tothe angle #8 C E:therefore the angles D CB and B (E B are equal totwo rightangles. And forafmuch as vnto’a certayne right line B C,and toa point thereof Cyaredrawen two right lines | not both on one and the fame fide, making the fide angles equall o to two right angles, therefore (by the14‘propafition ).the lines C DandC Eare drawen dire@lly, & do make one rightline which was required to beproted. Sigs A E The fame may alfo be demonftrated by an argumentilea-.. .., ding to an abfurditie, For if C Ebe not drawen diredlly to N\ B CD, fo thatthey both make one right line, then (ifitbee v poffible) let C F bee drawne dire@ly vnto it. Sothat let D / C F be one rightline.. And forafmuch as the two.rightlines A Band D Fdocuttethe one the other, theymake the‘hed 3 f angles equall (by the 15. propofition) Wherefore thean- ‘NG gles dC DandB C F areequall ; but (by fuppofition) ithe (RO angles ACD and BCE are alfo equall. Wherefore (by. a? thefirft comnion fentence) the angle BC € is equal! to. the SOF angle B.C F: namely,the greater to the lefle ? which is im- \ poflible. Wherefore no other right line befides.C Eis dra- : wen directly to C D. Wherefore the lines CM and€ €are fe \ | drawenxtireGly and. make onerightine:whichwas requi- -A red to bé proneds: ilo mor | 261 § Demon? ration leading to an ablurdstse. The fame con werfe after Pelj= tarsus, which ss demonftrated — | divedly. The fame com uerfe after Pren clus demonftran ted sndirestlge \ 1 WW ae 4 , qi t i h t! ! i i j 4) tlh } 14 > ithe) ae Ha ile ey) «ll at ay ihe th ¥ UW bat aa He “i be i i ) iol @ af a’ ta hy - Hea { bat aCorroi- bity ay. ZAC orrollary folowing of ths prop ofitsan. wf wonderfull propofstiez tn- sented by Ps- thavoras 5 Exery angle of an equilater trs of 20 le ss egsttl 423 two tbird partes nis ; ofaright anzle * arectangle quadrilater figure isaright angle: wher Euery angle of a fixe angled fr- Lure 18equa il to a right angle, andtoathira part o f a right angle, _ equall tofower right angles,Forthey ©. T hefrft Booke Of this fiuetenth Propofition followeth a Corrolary,’ Where norethat a Corohary is 2 Propofition, whofe demonftration dependeth ofthe demonftration ofan other Propofition, and it appearerhfodenly , as it were by chance offering it {elfe ynto vs: and therefore is reckoned as lucre or gayne. The Corollary which followerh ofthis propofition, is thus, If fower right lines cut the one the other: they make fower angles equal to fower right angles. This Corollary gaue great occafion to finde out that wonderful propofitionin- uented of Pithagoras, whichis thus, 3 Gi ly three kindes of figures of many angles, namely,an equilater triangle, aright angled figure of fower fides, and a figure of fixe fides, hauing equallfides and equal angles, can fill the Whole /pace about a point their angles touching the fame post. Every angle of an equilater triangle contay- neth two third partes of a right angle: fixe tymes two thirdes ofa rightangle make fowér riche. ane gles. V Vhereiore fixe equilater triangles fill the hole {pace abqura.point.which is equal to fewer ¢ g - rightangles,as inthe s.fizure.alfo euery angle of fore fower ofthem fill the whole {paceas inthe 2. figure,Euery-angle ota fixe angled figure is equal to arightangle; and-moreover toa third pare ofa tight angle. Buta rightangle,anda third partéfa tight angle,také thre ttmes,make 4.rightangleés: wheretore thtee equilater fixeangled figures fill the whole fpace about.a point: which fpacecby this Covollary) is equallte-fower right.angles: as in the third figure, Any otherfigure ofmanyfids, howfoencty ou ioyneth€together at theangles, {hal either want of tower anelés orexcéede them. By.this Corrollary alforit tssmantfeft thar?" if av then twollines, that is, three, or fower, or how many fceuer do cut the one the other in one pointy all'the an= | glesby them made atthe'point fhalbe —*? fill che place offower righrangles,And .., itis alfo many felt, that the angles byrods thefe right lines made are:double in:i\’/ number tothe rights linesahich cutte : the one the other, So that iftherebe’ two lines which cut the ofie the other, Bee thé are there made fower angles equall.. - ome tofower rightanglestbutifthre, then ~~...) 1, es--.. Brick Nawthe ica are there made fixe angles.;iffower scight.aogles.and fo infiaitly ,Foreuer the muiticude,ornumber of of the anglesis dubled to the multitudeof rhe right liaes which cut the one the other. And as the angles increafe in multitude, ‘fo diminifh of Enclides Elementes Fol.25. ditmihith they inmagnicade, Por that that which is deuided is alway ésoneand thetelfc fake thing; namely, fower right angles,” $a The 9.T heoreme.. The16.Propofition. D angleECF, But the angle E CD js.greater then the angle BC EK V bevefore isitheancle AGC Dsiigreater then the angle A BE. VV benfoewer therfore in any triangle, the line of one fidets drawen forth in lenoth: the outward angle fhalbe greater, themauy,one of thet woin ward and oppofite an oles: Which was wads Tag aes 2 5. Aworber demon/tration after Pelitarius. : “Sup pole that the triangle geuen be ABC. Whole ide ABlct be produced ynto ’ Gonfirud ars An other De- monfiration afe ter Pelitarits, AC orrollary follwing of this Propofitson, an sther Cor- rotlsr ry folls- wsng alfo of the fer. wil be frft Booke the point D.Theén I fay, thatthe angle DB Cis greater then either of the angles B AC and 4CB,Forforafmuchas the twolines.AC and BCdo concurre inthe point C,and vpon them falleth the line 4 B:therefore(by the conuerfe of the firft peticion)the two inward angles on one andthe felfe.fame fide,are leffe then two right'angles,Wherefote the angles ABCand | C 4 Bare lefle then two right angles:but the angles 4 & 8 Cand DB C are (by the 13 propofition) equal totwo | right angles . Wheretore the two angles AB Card DB BT are greater then the twovanglesed BC and BAC. © Wherfore taking away the ahele e4 B C,whichistom. mon to them both, thereshall be left. the angle D.B.C greater then the angle B 4 C~And by the fame reafon, forafmuch as the twolines B_AandCA concurre in the point 4,and vppon them falleth the right line CBsthe Q MATS IST pee twoinwardangles ABCand ACB are lefle then two right angles.But theanglese 4B Cand DB Care equallto two right angles.Wherfore the two angles 4BCand DBC,are greater then the two angles ABC & AC B.Wher- fore taking away the angle e4 BC,whichis€éommon to them both, there thal remaine the angle D BC greater then the angle 4C B: which was required to be proued. Here is tobe noted, that when the fide ofa triangle is-drawen forth, the angle ofthe trianglé whichis nextthe outwatd angle, is called an angle inorder yuto” it: andthe other twoangles ofthe trianglearecalled Oppofiteangics ynto it, » Of this Propofition followeth this Corrollary, that itis not poffiblethat from one _& the felfe {ame point fhould be drawento one and the felfe fame right line, three equall’ right lines,/For from one point, namely, 4; ifit be RISA pollible,let there be drawen ynto the right line BD Ak thefe three equall right lines e4B,AC,& ADAnd , forafmuch as 4 Bis equall to 4C,the angles at the bafe are( by the fifth propofition )equall. Wherfore theangle 4B Cis equalhtotheahgle 4C B.Agayne forafmuch as.4 Bisequallto 4 D,the angle 4. BD is (by the fame) equall tothe angle ADB: but the angle 4 BC was equall to the dn ele AC B.. Whetes - \ fore the angle-4C Bis equalltothe angle ed DB; / « namely,the outward angle tothe inwarde & oppo- > fite angle: which is impoffible:Wherfore from one and the felfe fame point;can not be drawti toioneé & the felfe fame right line,three equal rightlynes; which was required to be proued, % By this Propofitionalfo may this be demonftrated,tharifa tight line falling vpon tworight lines do take theoutward aneleequallto the inward and Oppo fice angle,thofe right lines thallnort makeatriangle neither thal they concurte} For otherwife one & the felfefame.angle fiould be both greater, and alfo equal: which is impoffible,As for example. . | i: to Suppole that there be tworight lines AB and CD. andi on them let the right lite B E fall, making the angles dBDandCDE equall, Then Tfaysthat the right linesu4R and C D thall not concurre.For if they concurre,the forefaide angles abidyng equall, namely,the angles CD Eand 48 D: Then forafmuchas thea wwle CDE is the ont- ward angle it is of neceflitie greater then the inward and opattte angle, &it isalfo e- qual vato it:which is impoflible. Wherfore if the faid lines cocurre,thé fhal not the an- gies temayne equall but the angle at the point D ihall be encreafed. For whethere# B a ae abiding of Enchie®Elemeéntes. Fol.26. abidinp fixed yonfuppofe the line €D-to, be houed.s:-> yato it,fo thatthey,conturre,the{pace.and diffance: | - in the angle will be greater’: forhow much more C. . .. D approcheth toe4 B,fo much farther of goeth it from D E.Or whether C D abiding fixed,you ima- pine the Hicee4' to be nioued vito it; fo that they’ concutre,the angle e4 BD willbe lefle, for there- with all itscommmedione're Vntothedines CD 30B Dp! er Or whether youimiagine either of them toibe mosi> d«. hed the onéro:the other, you fhali finde that-the: |) Hne eB -comming neerdé:to: 6: 'D, maketh, thes: o« ingles B/E leffe and © D-going farther from D B25 by reafon of his motion to the ine B.D,maketh thes: 2° £» angle CD Etoincreafe. Wherdforeitfollowethof.': 1: 2219 fad: necefiitie thatifit bea trianglesand that the rightdines ef B and C.D.doconcnrre,the outward anglé alfo fhall be greater then the inwardand oppofite angle; For either the inward and oppofite angle abiding fixed,the outward isincreafeds or the outwarde a7 biding fixed ,the inward and oppofite is diminithed:dr els both of them being moued till they concurre,the inwarde isaliminithed, and the ontwarde is more inereafed.And the caufe hereofis the motion ofthe right linesthe one tending tothatipattewhere.it diminifheth theinwarde angle, the othertending tothat part where it increafeth the outward angle. Hit et iF ii © ST hero. Theoreme. TherzsPropoftion, 5 ; a 3 ie Ak > ps HIV et JT) SALI, eas T} et $i 7 3] \ Bude 5 i « ‘angles focner betaken,are \\ le/Se.the,two.right angles A Sy : Extend(by the 2.peticio) elgu te none. the lne.B Eto the poate. D, And forafmuch as (by the propofition going before) the outward angle of the triangle A BC, namely , the angle ACD is greater then the inward and gppoftte angle AB (put theangle ACB comma to them both: wherefore theaughs AC Dand ACB are greater then theans gles ABCand BC A.But( by the 13 propofition) the angles AC Dand AC B are equall to two right angles,. VV berefore the angles A BC and BCA are leSethen two right angles. Inlike fort alfo may weproue, that the anglesB A Cand AC Bare lefse then two right angles :andalfo that the anglesC 4 Be AB Care le[se then two right. angles. VU herefore inenery triangletwo ane gles, which two foeuer be taken,are le[Je then two.right angles: which was ree guired tobe proued, 2 oe ee H,ti, This ~-- ConfiraEion, Demonftration, 40 other demi- jiration inuen~ gen by Proclus, A Corrollary f lowing thss Propofitton. Conftradtion. Lhe firft Booke This may alfo be demonftrated without the helpe ofthe former propofition, by the conuerfe of the fifth petition, and by the 13. propoficton as ‘you aw was done in the former after Pelttarius. It may alfo be demonftrated without producing any of thefides of the tric angle,atter chis maner. Suppofe that there a be triangle 4BC.Andinthe fide BC take:a pointatall aduen- tures,and let the fame be D, and draw the line 4. D. And fora{much asin: the triangle AB D, the fide B Dis produced, therefore ( by the former propofition) the outward angle e 4 D C,is greater then the inward and oppofite angle 4 B D.Agayne forafmuch — angle 4 D B,is greater then the inwarde and oppofite anglee4C D: butthe angles atthepoint Dare equall to two tighranglées( by the tz. propofition: )wherfore the angles 42 Cand 4C Bare lefle then two right 'an- gles. And by the fame reafon may we proue that thean eles B.ACand BC A areleffe then two right angles,if we take a poyntin the lin¢e#C and draw aright line froin-ittothe point 2 : and fo alfo may it be:proued: that the:angles ‘Ce B and ¢4BC are leffe the: rwo ryght angies, if there be taken in the lyne 4 Ba point, and from it bea line drawen to the pointC, as in the triangle 4D C, the fyde CD is produced;therefore(by the fame) the outward A By thispropofition alfomay be prouedthis-Corrollary,that from oneand the felfe fame pointto one and the felie fame right line,can not be drawen two perpendicular lines, HS oy 3 For ifitbepoffible, from the point 4,let. there be drawen ynto the right line BC,two perpendicular lines 4B,and AC: wherefore the angles 4 BCande4 CB are right angles, But forafmuch as 4 8 Cisa triangle. therefore any two angles ther- ofare (by this propofition ) leflethen two right angles, Where- fore the angles 4B Cand AC Bareleffe then tworight angles: but they are alfo équallto two right angles, by reafon 4B and ef C are perpendicular lines vpon'B C: which is impoffible. Wherefore from one and the felfe fame point cannot be drawé to one and the felfe fame line two perpendicular lines ; which was required to be proued. : \Wppofe that ABC be a triangle, bauing the fide A C greater then the fide A s\\B. Then! Jay that the ane Cele ABC is oreater then Nitheangle BA, For fore yy afmuch as A( & greater “the AB put( by the 2 proe pofitton) vnto A Baneguall line AD. And (by B | Theu.Theoreme. The 18.'Propofition. Ineuery triangle, tothe greater fideis fubtended the crea ter angle. A \ GC of Euclides Elementes. Fol.27. ho fart peticion) draw-a line from the point Bro the point D. And forafmnch Pemefration as theout ward anole of che triangle DBC, namely, the angle 1D B isgreas ter then the inward and oppofite angle DC B (by the 16, propofition, )but (by the ;. propofition) theangle AD B 1s equal to the.angle A BD, for the /yde A Bis equal tothe fydé A D3 therefore che angle ABD is greater then the angle:AC.B. VV herefore the angle ABC is much greater then the angle AC B.VV berefore inenery triangle, to the greater /yde is fubtended the greater angle: which was required to be proued, You may alfo prone the angle at the point 2 greater then the angle at the point C An other de~ (the fide 4 C being greater thenthe fide 4 B if from the line 4C you cut of a linee- saphiiatndte & quall to the linee4 8,beginning atthe point C,as before you beganne at the point: Ss ett and that after this manner. Let the line D C be equall to the line e4 Banddraw theline BD: and producee# Bto the point £: A and put the line B £ equal to the line 4 D.Wherefore the whole line 4 E is equall to the whole line .4 C:draw aline from Eto C, And forafmuch as 4 E is equal to-4C, therfore the angle AEC is alfo equall to the angle 4 (E ( by thes. propofition-) but the angle 4 B Cis greater then the anglee-# E C.For one of the fides of the triangle CB E, namely, the fide B Eis produced , and fo the outward angle e4 BC is greater then the inward and oppo- fite B EC (by the i6 propofition:)wherefore the angle 4 BC 1s muchgreater then the angle «4 CB: which was required to be proued. “ats Note that that which is here fpoken in this propofici. —— tie . x F at which ds on, is to be vnderftanded in one and the felffame triangle, For itis poffible that poten in rhie one and the felfe fame angle may be {ubtended ofa greater line,and ofa leffe line: Propefitson is t0 and oneand the felfe fame right line may fubtend a greater angle , anda leffe an- ee pani See gie,Asforexample. ©: yoo Suppofe that there be an Ifofcelestriangle ABC, & angle. in the fide 4B take the point D at all aduentures: & fro A, the line .4 C cutof(by the 3,propofition) the lyne AE equall to the line _4 D.And draw a right line from D to E.Wherfore the right lines DE and BC do fubtend the angle at the point A, & of them the one is greater, and - the other leffe. And after the felfe fame manner a man Rf erommeb may putinfinite right lines greater & lefle, {ubtending ¥ the angle at the point -4, i, Agaynefuppofe that 4 BC be an Ifofceles triangle. | Andlet BC belefiethen either ofthelinesBdAand AC, 6 ee. And vpon B C defcribe (by the firft Jan equilater trian- gle BC D.And draw aline from Ato D,'and produceit tothe point E. And forafmuch asin the triangle 4 8 D,the ‘outward angle B D E,is greater rhen the inward & oppolite angle B_4D( by the 16,propofition )And by the fame in the tridgle 4CD,the outwardangle CDE,is ereater then the inward & oppofiteangle C_4D:ther- fore the whole angle B D Cis greater thé the whole an- gle B.AC.And oneand the felfe fame right line fubten- deth both thefe angles,namely,the greater angle & the leffe, And iris allo proued,that greater right lines && leffe fubtende one and the felfe fameangle,Butin | EX stile - one DemonfIr ation leadin ig 10 AD This prope fission ssthe conuerle the former. dn Affampt ov a Propofition ta ken of neceffitse ta the helpe of a dlemonftration, the certaimty spbereef 1s not fe platne, and therfore nedeth sr felfe firft to be demosftrated, 4: afiuimpt put by Procles for the demonftrae- seoie of thes Lro- pojstion. a a we ote So fore the fide ACits OF equall to the'fide ABAnd A ie | The fr Boke ‘9 oneandthe felfe fame triangle one righrliaefubtendeth@nea note ,and the #reat rightlineeuer fubtendeth the great angle,and the leflerche léde,as it was proued ta the propofition, | Oe ae he “app ae) ea , «Lhe 12z,Lheoreme.. .Lhes 9: Propofition. cacy Ae \ Fa enery triangle, under the greater angle ws fubtended the rn EC “pple that-d BC be a triangle, hauyne ‘nid the angle 4 BC oreater then the dngle B WE! CAT hen L fay that the fide A Cis ereater | theny fide A B.For if not, the the fide A Cis ether equal to 9 fide AB orels st isleffe theit/The fide A Cs not equalto y fide AB, for then( by thes, pros poftcion )y angle A BC fhould beequall to. the ane gle ACB: but (by fuppo/itia) it is not: VV beres - BR Y . » ww , -_ ; q : 7 » # ; == a ez the fide AC can not be le/Se then the fide -A'B, for then the angle ABC fhoulde beleffe then the anglé ACD (by the propo/ition next goyng: before), But (by Suppofuion itis not) VV herefore the fide AC is not leffe then the fid? AB, VV herefore the fide AC is greater thenthe fide. A B.VVherefore in enery tris angle, vader. the greater anole is {ubtended the greater fide: which Was requis red tobe demonftrateds: >. 0) asd 5 : ima: This propoficion is the conuerfe of the propofition next soing before. V Vher= fore as y ou fee,that which was the'conclufion inthe form er,isin this thefuppo. fition,or thing geuen:and that which wasthere the thing geuen,is here the thing required or conclufion. And ‘tt ts ptowed by an argument leading to an impoffiz bilitic,as commonly ail contérfes are, °** | : Pocus demonftrateth this propofitioa afteran other way: but firfthe putteth this KATumpt following, Fest : i ef TS OEE: Tf an angle of a triangle be deuided into two equall partes, and if thelinewhich denideth it being drawen tothe bafe, do deuidethe fameintotvpo uneguall partes :the fides which contayne that angle fhalbe unequal, and that fhalbe the greater fide, which falleth on the ‘grater fide of the bafésand thae the leffe which falleth on the leffe fide of the bafe. 2°. _5uppolec4 BC to bea triangle,and(by the. propofition )denidethe angleat the point 4,intotwo equall partes, by the right litie 4D. and let the line.4.D deuide the bafe B C,into two vnequall partes, and let'the part C’-D be greater then the parte BD, Then I fay,that the fide _A Cis gtéater then the fide 4B. Producétheline.4 Dto the point £,and (by the third )purtheline DF equalfto the line DA: And foratmuch ag D Cis by tuppofition greater then D'B, put (by the 3-propofition )D F equal to BD, 4 and draw aline fro Eto F,and produceitté the point G. Now foraftauch as 4 D is e- qualito £ Dand D Bis equall to D F, therfore in the two triangles 4B D,and EF D, tw fides of the one are-equallto two fides ofthe other, eche to his correfpondent fide; and ( by the 15. propofition) they contaype equall angles, namely, the hed an- gles ie “Oc. | ™ of Euclides Elementes. Fol.28. gles :;wherfore ( by the fourth propofi- tion ) the bafe B 4 is equall tothe bafe A EF :-and the angle D E F is equall tothe I angie D 4B, Buttheanple D AGis by con{truction equall to the fame angle D ef B: wherefore (bythe firft common fentence ) the angles E.4G and AEG are equall, Wherefore (by the 6. propo- | 2 \ fion ) the fide AG is equall to the fide E G.Wherfore the fide 4 Cis greater then the fide E G.Wherefore it is much grea- terthen the fide E F.Butthe fide EF is equall to the fide 4B, as it hath bene proued.Wherefore the fide 4 Cis grea- ter then the fide 4 2;:which was requi-+ red to be proued. . .. This aflumptbeing put,this Propofition is of Proclus thus demonftrated, *\ Suppofe.4B Cro be a triangle, hauing his angle at the point B eo thenthean 4, ortor >. gleat the point C, Then I fay that thefide 4 Cis greater then the fide 4B.Deuidethe monfration af line BC into two equall partes in the pene D,and draw aline from AtoD,And pro- ter Preciu. dice the line 4D to the point E: and put the line D € equall to the line 4 D,and draw dliné from Bto £. Now forafmuch as BD isequallto DC, and A Dis equall toDE therefore in the two triangles 4 DC and B DE;two , fides of the one are equall to two fides of the other, ech to his correfpondent fide, and they containe equall.an- gies (by the rs. propofition):wherefore (by the fourth propofition) the bafe BE ts equall to the bafe 4C, and the angle D B Eis equal to the angle at the point (-De- vide alfo th’angle ABEL into two equal parts by the line BF: wherforetheline EF is greater then the line FA. And forafmtch.a$in the triangle.4 BE, the angleatthe point B is deuided into two equall partes: by the right line B F, and the line E F is greater then the line 4 F: therefore by the former Afmmpe thefide BE is ereater then the fide B_A: but the line B Eisequall to the line.4 C, Wherforethe fyde AC is greater then the fidee- 4 B: which was required to be proued. The13.T heareme,. The 20. Propofition. / Inenery triangle twofides, which two fides foeuer be taken, are greater then the fide remayning. WG SA Sars | triangles T ben. I fay that two /tdes of the triangle A BC, which two fides foee . RC ner be taken, are greater: SS then the fide remayning EL) ithat is, the fides BA and AC are greater thenithe Bi, | fide Vuppofe that ABChe a PD g Ren Ser, ‘be NaIK ES = a ee =e st —— hee — oe — a AA ass a eaten a = —_ a a saa —_ = = - = —_ = —asmrv — - — oi — ews < = —s ™ - - —-& 4s ey em - a — — eS oe on om —— ~ Bie et DE fee in * i are - = atten _ ees = Si =n oe Veees hh ns — pipe eS ee ee ——— —— senate ~ = — ——- ns — — ees a —— = ~ SS OS ~~. = — : Se ~ rs bee Se> =o “~ Ae. 5 > a + a — in - ee . = > Ss ‘> 4, eet & - oom it ee. aye « eb ea ew = a a= ~ Se = == = = — ee. a ssl dined Ye. $53 ee ee ~ Wk “Sy < a SS De hye ah Re Sa Sera Sie. Sn gan ~ Coax tration. Dewsenflration, TL he fr Booke fide B Cand the fides A Band BC then the fide'A C: and the fides'A C and B C. then the fide B A,Produce (by the 2.peticion the line B A tothe point D,And ( 4y the third propofition)vato the line A Cputan equallline AD; and drawe a line from the point D tothe pointe C. And foraf- | much as the line D Ais equallto the line AC theres... D Jore(by the 5 propofition )the angle AD Cyisequall fo tie angle AC D.But the angle BCD is greater then the angle ACD, therefore the angle BCD is er eater then the angle A DC. And forafmuchas D Bis a triangle, bauing the angle BCD greater then the angle AD C,but( by the is,propo/ition)yne aer the greater angle is fubtended the greater fide: wherfore D Bis greater then BC. But the line D C - ; ’ B is equallto the lines AB and AC{ for the line A Dis equall tothe line A C) woerfoxe the fides B-Aand-A Cyare greater then-the fide BCAnd in like forte oom ape pronethat the fides A Band BC areorcater then the fide Ce that we : . ae a ¢ A other deme « pita £502 withe eur producin f4 axe of the fides. A cther De- mani rctiet, » thefides BCand € A arb preatenthen the fide AB VV berfore inewery tridits’ oe gletwayides, which two /rdes focuer be taken,are greater then the fide remaye ning which was required tobe demonftrated. »- 3.042 tored3 r> This Propofitionmay alfo, be demonttrated without producing any of the fides, after this mancr. GSR RE 1S sale gi 2 * le, Suppofe 4 2Cto be triangle, Then Lay; thatthe two fides\4 Band of Care gréde: ter then the fide BC: detide the angle at the point:4 (bythe 93 propofition)into twa. equall pargés by the right line ef B-And,forafinuch.asin the triancle4.B E,the outs, ward angle AE C is greater then the angle Be E (by fi bot ec the 16 propolition),and the anele BAB is put to be | “A equall tothe angle E 4 C,therefore the fide 4 Gis préae terthen the fide F, And bythe fame-reafon;the fide A Bis greater thé the fide BE For in the triangle 4 £0) the outward angle 4 E B, is greater then the angleC.A €,thatis then the angle € 4 By Whereforealfo the fide \ A Bis greater ten the fide BE: WHerfore the fidesu4B 2. & f! and 44 C aregreater then the whole fide BC, And “after taciame manér may you 'preiie ‘touching the ‘ooher fs re fides alfo, re aes a1 ASEM ETT MG The fame may yetalfo be demonftrated an other WAY cake Suppofe 4 B Ctobea triangte.\Now if 48 C bean es ay, equilater triangle, then without doubt any. two Sides. on Ae thereof are greater thenthe third. Forthe three fides NER. being equall any two fides of then? ate‘dotible tothe third. Butifit bean Ifofceles triangleyeither the-bafeis lefie then either of the equall fides or itis greaterIf the bafe belefle, then againe two of them are greater then thethirde, butif the bafe be greatér:let' BC being the baf¢.ofthe Ifofceles triangle 4BG.be, gréater.théeither. of she fides AB & AC and from it cut of (by the:3.pro- 3h % . o ‘pofition of Enchdes Elementes. Fol.29. pofition ) aline equall to any one of thé other fides, whiche let bee BE,and dtawe a line from to £.And forafmuch as in the triangle 4 £B,theangle -4 E Cisan out- ward angle: therefore itis greater then the angle B 4E (by the 16. propofition).And by the fame reafon,the angle 4 E B is greater then the angle C_4 E. Wherefore the an- gles at the point Z are greater then the whole angle at the pointe 4. But the angle BB Ais equal to the angle 84 E (by the 5, propofition) for e# Bis putto be equallto B é, Wherefore the angle remayning 4 E (‘is greater then the angle C_AE, Wherefore alfo the fide 4C is greater then the fide E (. But the fide_4 B is equall to the fide B E. Wherefore the fides 4 Bando C are greater then the fide BC, But if the triangle .4 5 C be.aScalenum, let the fide AB be the greateft, and let_dC be the meane, and BC the leait, Wherefore the greateft fide being added toa- ny one of the two fides muft nedes be greater then the third, For ofit felfe itis greater then any of them. But if 4 B being the ereateft,you would proue the fides AC and CB to be greater then it, Then as you didin the I- foceles triangle, cut of from the greateft a lineequall to one of them, and from the point .C to:the point of the interfection draw a rightline,and reafon as you did be “ fore by the outward angles of the triangle,and you jhal % haue your purpofe. This propofition may yet moreouer be demonftrated by.an argument lea- ding to an abfurditic, and chat after this manner. =. oo ans $$ SSS rer - -- —=—— - ~ ——aee ~ ee ‘ ro TSe “~ a > Ad Suppofe 42 Cto beatriangle. Then I fay that the As other demés fides 4 B and 4 C,are greater then the fide BC. For if fr ation leading | they be not greater,they are either equall or leffe. .Firt = fo an abjurditse | Jet them be equall,and from the line B C cut ofthe line & Eequallto the line 4 B( by the 3,propofition) wher- fore the refidue E Cis equall to 4C.Now forafmuch as > i) Lal ; fay! tht Ha ny i : I a ae ih it ag i at > Ail Bl - 44 B isequall to BE they fubtend equall angles. Like» . wife forafmuchas e4 Cis equall'to C Ethey fubtend e- *< qual angles,Wherfore the angles which areat the point \ £ are equall to the angles whiche are atthe pointee/, LZ N which is impoffible¢ by the 16. propofition ). ge Ae But now let the fides 4 B and AC belefle then the A fide B C,and from the line BC cut of ( by the 3.propofi- iy ae line B Dequail to the line 4 B,and likewife fro the famefitic B C cut of the line CE equall to the line 4 C, And forafmuch as 4 B ig equall.to BD, the angle B D Aalfoisequall tothe angle BAD (by the fifth pro- pofition) Againeforafmuchas «4C is equall to CE, therefore(by the fame) the angle CE is equall to tlie / afigle E eC: Wherefore thefe two angles B D Aand@ / € Aare eqnall to thefe two angles Red Dand £4 C, Agayne forafmuch as the angle’B D_4 is the outward : angle of the triangle .4 DC, therefoteiris ereater then D D E c the angle E AC.For itis createrthén the angle Det (by the 16-propofition ),And by the fame reafon,forafmuch as CE eis the the outward angle of the triangle 4B E, therefore itis greater then the anele B 4D (for it is greater then theangle B AE). ‘Wherfore the angles B.D _AandC E Aare ereater then thetwo angles B_4ADand E A C. But they were alfo proued equallwnto them: which is impoflible. Wherefore the fides 4B and4 Care neither equall to the fide B C, nor lefle thenit, but greater, And ‘fo alfo may itbeproued- ofthe refs = 20 2 | paar | Li. A T he firft Booke Wie teow. A man may alfomorebriefely demonftrate this propofition by Campanus Qration by the definition ofa right line, which as we haue before declared isthus: A right line ss cisfictrion ofa the fhorteft extenfion or draWwght that is or may be from one point to another. Wherfore any one weght line, {ide ofa triangle, for that itis aright liné drawen ftom fomé one point to fome other one point,is of neceflinie fhorter then the other twofides drawen from and tothe {fame pointes, 7 | Net all hinges Epicurus and {uch as followed him derided this propofition, not counting it enanifcfi tothe worthy to be added in the number ofptopofitions of Geometry for the cafiacs fefe.are UOT | ee SORA oe 7 oe { if ir Srey manifob re Tocteot, for that it is mantfeft evento the fenfe, But nor all thinges manifeft to recfinaxdGn- fenfe, are ftraight wayes manifeft to reafon and vaderftanding. It pertayneth to derifunding. one tharisa teacher of{ciences, by profe and demonftration to render.a cer- tayne and yndoubted realon, why itfo appeareth to thefenfe: andin that onely contifteth {cience. | Lhe 14. Theoreme. The 21.Propojition. Ff from the endes of one of the fides of a triangle; be drawen to. any pout witain the fayde triangle two night lines. thofe right lines fo dramen,fhalbe lefve theg the two other jides of the triangle, but {hall containe thg@gmmmer anole. * ct ‘ : RSr Sy | Dand CD to ¥ point D. ~~ a bd henk fayy'that the lines 2 ts BD and CDarelefse then the other fides of the B eat ‘ie triangle, namely, then the fides BLA and A Cand that the anole which they contayne, namely, BDC, isgreaterthen the angle BAC. Extend ( by the Demsnitratians fecond peticion ) the line BD to the point E, And forafmuch as (by the2o.proe pafition) in edery triangle the twofides are greater then the fide rémaynin A therefore the two fides of the triangle AB E, namely, the fides A Band A Et are greater then the fide E BD. Put the line EC common to.them. both, VVheree ‘fore the lines BA and 4 Cyare greater then the lines BE and EB CAsaine fors afindch-as (by the fame )in the triangle E D the two fides C E and E D, are Rreater then y/tde DC, put line D Bcommontothem both: wherfore 5 lines CE and B Bare greater thenthelinesC Dand DB. But itis proued that the lines‘ Aand AC,are greater then thelinesB E and ECVV herefore the lines 8 Aand AC aremnch greater then the lines BD and DC. Agayne forafmuch as ~ at 8a z - MPAs Se Be Sant eck Sy oe ee See of Egchide\Blembites Fol.30. as (by the 16, propo/ition )in enery eriangle, the ont wardiangless oredter then the invardand oppofite angle, therefare the outward anole of the tridngle C. DE, namely,BD C,is greater then the angle sD. ¥ i herefore alfo ¢ by the. fame) the outward angle of the triangle! A'B E, namely, the angle C'E Bis greater thenthe angle B AC. But it ss proked,that the angle B.D, Cis oreater then theangle CE B.VV berfore the angle: BOCs much greater then the ana gle BACVV herefore if from the endo te of the ji és ofa triangle; be'dra : wen to any point within the fayde triangletwo.right lines:,thofe ) right lines Jo. drawn fhalbe le[Se then the two othetfides'of the trian gle’; but fialtcontayne: the greater angle: which was required te be demonstrated, eeaiee ako CL ee . ae r : . . ae | ~s : Ma -bovuoty 9d o3 baitupot esw daidw +3 9 bai In this propofition is expreffed,that the two right lines drawen within the triangle,haue their beginning at the extremes of the fide’ pF the triangle. For tro the one extreme of the fide of the triangle,and fromtfome'one point of the fame fideymay be drawen two right lines within the triangte, which fhall be longer thé the two outward lines: which is wonderfull and feemerh{tranage, that tworight lines drawen vpon a parte of alinejfhould be grearérthtm two right lines'drawen vpon the shale line,And agayne itis poffibie trom the’oneextrenie of the fide ofa triangle,andfrom fome one pointof the fame fide to drawe two Tight ly hes withia the triangle which fhall containéan angleleite chen the angle contayned: < ‘ S& Din? $f ONE Th sre SAT ynder the two outward lines, ee SRE re AES | r - i ‘ Sinit QiryieSt:. FV Tf? cy “A999 * a ty 2siis sAULlst J: j ¢/ 5 VALE ws ~ -St3 & 2 Jaan JIUeh WiSIsIg As touching the firft part, € shiolo doinw -ofeasia wrens Suppofe 4B Cto bea reGdtel isboow bstnvo> 3090 Isud Mien eretiqocolig le triangle,whofe rightangle let a5,575hth 2t siors ant ,bot00 od OF ei S790 DLS eat the point 8. Andin the fide,.9 bsionsooidss bag giugh LoLiless3 5 _ 7 £ cake 8 pointat sladurotabee ag fans sait-anitel saei-cigus ohne > i Od iss {175 : Wiles Ve liis 3¢ bey ivisw s\ Jags’ « whichlet be D: and dtaw a rig S - nt Ae , D : ‘Ton 21} Gros “I Snifiq 2tti 2s 2 line fr6 4Ato’D Wherfore the hee 24 nro) 7 stq ad a A Dig greater then 3 tre AB wot sus 03 9102! : on a wT ee RS at , (by thet 9.propofitid) Fromtherl ysen overt bobir 910! § line AD cut of (by the thirde): gol s bus ‘o1uett2 nes 30 loasavalicashs.tincAy whdebi vgn aid mi apqebtl oui gis et be D E.And denidé the ine E ot pou into two equal! partes in the point F(by the to.propofition ) And drawalinefrd FrocC. Now forafmuch as 4 F Cisatriangle,therfore the lines 4 F and F Care greater thé the line <4 C( by the former propofit ):buteste is equal et Eanhosfart the right lines F E and F Care greapg othefiic.2 E‘Aindthe line DIE equaltto theline 4B. Wher- fore the right lines F Ca are drawen within the triangle 4,B C,the one fromione extreine df the fide BC and : He TIAA sie athe one fromone extreme Of thelide BC and the other roma poisiis thefamélid win Whidhewasrequiredto be proued, ‘ Laide rand shod ho ont Mtonodad easel sslosarss & spn Ot i) pan eas a feconde parte} 3 (Suppofe 2 B Eto Bee an Tolce- ég tflianele, and let the. afe thereof TA TAO then either.of the equall fydesi cand fra thetlyne'S Conte of a Tine equal tothe lynew4B (by the thirde propofition) whiche let bee BD: and drawealine from 4 to.D: andinthe we D 5 4 > * ~Is Tey a) . ' oO I. ij, line ° . : ° - ~ wert» ied a i =< = = a PFs ate = ~~- Vio. a Be es ~~ won Se ~— = . oa e he ee ~ > ~ So <=23 ete Kad SEP ene + mane a a Sa - See : << S SRR ceed) UB ei as se oe - - — - _ ty . ‘ . < . 3 - _ . . 7 . 9 TURES Seer ’ Peer teat i - € he adpeats se 7 me a baat ‘ ar — Grek aeons a =o = — -— - —- == ee <2 x - ee - _- - - Ps renee aed kg: pn & 5, 0d deena atyw wv rane apical ~~ ~ “ tz << : . n = ane t —__-- ; - A —— See a ST 4 see me ae - qd haar, ‘ : So. E dees tin i = = — = == .—— ee SS a — -: == ie = — a ~™ = Ss Bas. ~~, éca eee 7 . —— — : wane A= Sa = — : — - ~~ ~ — = ; SSS = : - a , oe = ae ~ — _ ne Fld = % ‘ar, —— . : ~ —_ i bn ren = ‘ \ 7 a= + Sate. as " — Ss. SS Aeon Si ae Z = og és GS SOE ee aS ee : — be pera Bc Be = = . -_~_——-— - + ~ TN Settee ~ <= SY ee Oe Replies a a ~ Sn a edo in . — SS Ss eee niente er we — ~—_ . —— . =~. er Se egeaper —— = —— ~~ + —" ~ pm eae oe a —— ——— - —— —_— _~-—-- =; - « na = ; <=. 5x = =- + ee : = = = SS = == a = — : = = ee a Pie sega o Reick mp dan a Bl cee i ~ = Se wy EOS. E 12 Fain a rs ee Ae 7 ~ « a — — es - —* ~~ = m — a = == > rk mr - ann + Pal . we. Ge “" — se — - itd a aad, — im ao = a a — = < ———< —— = = . = - — < ne SS a ee _ emsee nats: ~~ >. ~~ — = oe ote —— =-—< tee =~ -+=- _— 2 = er Sore — = = 3 7s = ——"" a ~~ — - ~ —_ — = - — — — —— = - — r. = >— = —_ = —— be - " 3 = “ ee a, a _ ~ - i = —_ * nt : _— = Se =— Se 3 = = : >= =s eS SSS SS SS cs EL eR Ss ee S>- === = 2 Bes + eS i = Se, Sip tise —— e a 7 a > sae ar —— — i a eet = = = — = " ‘ $= q tas = ? - * = — a a = Sr RSet » . vv" _ Sum ae oe aw linn > een eemee tee Pee ema a —— oe —— = = “ — — - > —— = oo a Gh sk ees ME SRG A ee

* ait; ciddiw aowshh2onil sfgit ows: mt 16113 bslloiqx> ei noitioqore zit a] = By: meanes ofthis, propsfie., m13x9 913 38 antniesd ried? ound. olensiaa lfo1s defcribed tha kynd.; = ee ON a ee ee cee ee a og OS SRO Ea Py th olometsy 9rit3 dylo sm2iIx5 ano 212 oftriay gles, wihiph contaynerh, tsb) w 29/7 jr « fourelid Psgfhs for, sop D be a llta: bnoyf 2: x sirhy :2Dnlbaty $0 OW2 303 Syacae Sets Var eee 1d o! ff one SHAR aod F. pen — end fewer i hi poche 19idMogeyatanyen shod ah the oad three omar! ah at spnnds See point h At “46 ae BS san ‘tbs the point C, VVhetefore this prefent figure ABC is a qua- drilacer triangle: which ofolde ® neq Priteds gnifovoteA F philofophers hath eucr bene counted wonderfull, : a1sodo7 DAK Sloqqu2 And here is to be noted, that there is difference Bes 'g18 igi slodwal, nei ofg twepe a three fided figure, and athree angled figure, °°) UPS 7mo7 3 7 ForSot cuery figure ‘hauing three angles hath alle.” ee ae oncly thee fides as itis plaine to fee in this ficur 1 Likewifeetepor alton ,afigare to haue tower fides, aaa fe andifower angles, Fora foure fided figure may hauie:- off 0} oncly thre anglesjas in ermer figure: anda foare bi: tay 1 4) Be a angled figure may hatte fie fidemasin inthis ma foldwil beh it el Pee A on 9fi3 ng, And fo Teh Act tinct Maia, miter SRP seaeete =. st tt 295TI58a 4 iLsidy 3 WI c7naith 2 a (nono oqord. Oo; ori xd )A 11108@ wot Dor. onion] s wetb bod anit ods $d1.1929919 978 45 ash. esail oils s1otau } vlgabiis 6219 AS 28 dommlarot rs Mt ttay ri “{ fr¢>? 1} CG } 4 rit: ref YOR. AVouil sigh wy &:Pro 228 epee Fd LOqO1g A9MIO1 IAT Y na © & onil® The CNG oh wd | ; ": ,* recy | a a nen hate a) ao »> PAX MJ WeEIS % ; WY miTi 29: it y — “ ulstaiy sdi nogy onbas. a 4 ] 3 33 A= igus neal sis7 oMy aidaiy .20ril bis WwIHO OF 7 “3F +o 575 Di bne 5, a 1c isha “OW brie & kw gout athe post 98 1 J Wiest cs au 1 3113 920% => ne bbr ht lines, which aed tothreripht-lines werk to make. a triangle. But it behoueth two ofthof lines which sot Wa foeuey ba takensto be greater'then the third: For thati ie “peony ess" which smo fr des foewer be Laken a p- ait tis ' Oo) = > fon sails Swe 3b = nee vu « bghea WS : aah aie it grea a ————— of Eucledes Eleméiites. Fol.31. greater then the fide remayningge®* >" A . * ma gary! MENG V ppofe that the three right selib ied) > NY lines gue be A,B,C:of which na alicaci * lec tha of them, which tho foeuer, be taken, be greaterthen the third, that 45, let the lines A,B be greater then the line C, and the lines as. then the line Byand the lines B, D C, thenthetine A. It is required of SarKE Hig tlines equall to the right A,B, 1S ef > Conftrudiom, lines A,'B,C ,to-mal eatriangle;t ake : ice aright line hauing an appointed ende pores ee A on the fide D, aud being infiniteon —S——— POW, yas Yetig plus: [Py Styinsybsyec Pighrhssc3 tOGO 7? 21s MRS RI the fide E, And¢ hy the 3. propefitie fs doisiw) soc 0 9d34no- ne ‘ op )pus vuse-tbe dine dan ehuellling, Sooo ee DF and put vnto the line Bah equill line FG and vnto ylineC,an equall line G FivAnd making the centre F and the {pice DF :deferibe (by the'z peticio) acircleD K_L. Agayne making the centre Gand the [pace G FL, déferibe> (by the fame )a circleH K Lianddet-the point of the sntex[ection-of the Jaya circles be Kand(by the firfi-peticiot draw a rivhe line from the point K.toy pamtF, ex an other from phe point K to the point G, Then FE fay, that of threvight lines equall tothe =o A,B,C,is made a trianole KF G:For forafmuch asthe poitl Demonstration F is the centre of the circleD AL therefore (bytheis; definttion')the line i D is equall tothe line F K. But the line Ais equal] t6 the line FD VV herfore fay thefir/Ecom mon fentence) the line F Kis équalltothe.line A, Agayne fot afmuch asthe point G,is the centre of the circle LRH stherefore( by the fame definition ) the line GKis equall to the line GAL But the line C is eqtiall ro the lineG BA: wherefore (by the first common fentence) the line. K.Gis equal lta the line C,But the line EG is by Juppofitton equal-to the line B: wherefére thefe three right lines G (A K, sand K 7,are equall to thefe three right lines A,B, C. VV herefore of three right lines, that is, K FBG; and.G ae which are equallto the thre right lines-genen-that is to A, Bi is made atrrangle K PG: Which was required tabedone. ~— eS jo | Agi other conftrution,and demonftrationafter Fluffates, fa. wi 7 See abe \ Suppofe that the three right lines bee 4,B,C. And.vnto fome one ofthem,namely, mame at ro C,put.an equall line D E,and(by the fecond propofition) from the point E,draw the “z.,onfration line E-Gsequall to the line B: and ( by the fame) ynto thepoint D put.the:line.D AH e- after Flufjates, qual to the line 4.And making the centre the point E, & the {pace E G3defcribe.a cit- cle FG: likewife making the centre the point D, andthefpace D A sdefcribe an o- ther circle 7 F;which circlestet.cutte-eHe One the other. in the point F. And. draw bs}: Tuy, thefe ) thefelines D F and E F. Then I faye that D F Eisa triangle defcribed of « 3. right lines equali to the right lines. “4,B,C.For fora{much as theline DH is equall to the right line 4, the line D F, thall aifo be equall to "the fame tight line 4,(For that thelines DA and D Fare drawen fré the centreto the circumference ).Likewife foraf- muchas EGis equallto E F(by the : 15, definition) and theright line 3 1s equall to the fame rightline E. G:ther fore therightlineE Fisequalltothe = © Aan supe © h rightlineB : but the rightline D E£, Ao Laainy(tt 2. KS satoael, D was putto bee quail to the right line ssitenn ©.Wherfore of three right lines ED,D F.And F ‘g} Which § are equal ene ight ties - geuen, ef, B,C, is Uptcribed a triangles which was tequired to be doney". deh UM ‘4 ns « a \ wm 1,94 ~ seh , & tt ‘ aRSSS aS ed FALa a Ltn TASS ‘aN HS sast - e Seffances in this In this propofition the EE paraditanany will cau har a ‘circle Prebleme, fhall notcut the one the other (which ching Exchde putteth them to do) But a ifthey cuttenot the one the other, either they téticli the one'the‘orhetjor they Are'diftaynts theone from the. othe . Fikfhifivbe poflible lec themroocheth® ane others as\in-che Breer pur Gbeisaniructia a wheceot tle econftrugyion ofZEuchde).) < oe Fee ae : Firit inflance, . And foramuchas Fis Peesn ise a eee ee "dre oF thé eitcle DK therforethe 241 \0 imo seeeke | jo a line D Fis equal tothe line FG ert 4 333 Mal LOM SIDS, 7 And fotafinuch asthe pointG,is. \- Ak 5 SS Pe Cee ees FS the centre of the circle ee Pe ee ae oS SENN fore the lite AG, isequallto the ™ AMINT RS pias Set, line G10) ‘Wherefore thefetwo alt orn Tr Tee Qt! rey } y; { one Wine; ‘namely, to FG. Bar aaa Rel Bid Bo they wete put to be greatet ‘chem? ti ths i a 3 OFS. ; ir: forthe lines D FFG, abd'G\ s\y.i5 sia H,. were put.to be. equall to the. ies “Eee eS ee _ lines 4,2, Greuety two of which* * Sa PRR Vooa ot Fe Pe eee se are fuppofedto be greater themsin “e sht Y aaal the thirdée s\wherefore: they) arei ony AN igaaye sah me both. greater; and alfo equi Hl, ene erat ig aiians Second inffance ‘hich isimpofiible, Agayneif it: * mp. SAB U7 BEB Pe AAU VOSS GN be poffible,let che circles pedi A ui indt .rsinbtdeois soads fo ovolyad WA fianrk thie aekcornt he oshats SI S. a1 sapeaeweasnt| - ce dy Wee arethecircles DKandHLAnd © YY “ om oe rr, foraf{muchas Fisthe centre of the circle D K, therfore the line _D Fisequalto the line FN,And forafinuch as Gis the centre of * ijsue, thecircle 2 A,thereforetheline | HGis equall'to the line G ALF. ’ _. wherefore the whole line F Gis ooo" Hs vetla greater then the two lines'D F, SEN (9EPStSAIYO) OM8 | i ee De andg a, , (forthe line F G,excee -* Lrnidkperis otinsosdi 2 | | seri! dethshe lines D F, and GH, by: a9 903 219099 sii watens slivy thc lineV 14 Jour's was GpOe> oils 900 Siete oud oidws iA slagipaeds of il. fed 104 SR = . . = rt wir * es xz —— “ , - . 4 “ . . 2 ata ‘ - Bra OF id ae Ke ‘ or . be en Ue Di d4 = = » ; ah staal eat BA noc bad es asc. < <> We ee se = ~ weaken = - ‘ os — —- — “i 2 ~ ae +e ora - . ot = - — nn S = = ; - = : . - . - x. — 5 & fs bare ~< ‘s 2 ‘ 7 - ry it. a7 ~ a _ 7 ion —- i : —23 WR Bio ees, - se : —% : ; Bee aA-es -S-5 ~. a Re oo = = ae ss : = a a SS . ay ie -¥v ? * : e - - . a - — . <= - - ~ ae a - ————— ee = ——— — - “ros Se — ar cokes = “= f * : a : - 5 - - —- ~ . — ‘ - e - — re : ~o - - = “ep pen a ana traz - eo = = - — - ——— - -- ~- <=> = ~—— — = ms : = 3 Se ga - 4 eink ett) ep > 5 oa lye : 2k > =e = —_ ee ee. - _ oa me % naa Seneca: aX — - ———, — = = SSS = ° i. ri it awe "a vwe> _ 2 Yo ape pie - > i. -- = ata a cn SS tee xq Ze 5 t= Se S wah hien “Ridge tyne, Y a mur < ~ a - poe _ : —-= ———_—— ee Oe tah ae a 7 = at eels —> - : ~ - < .— < : tee ; =. a Amy = “2 = ¥ r s 5 = = fe ne ve 2 =. aay « Se : =< — Ss - =. mee = S i ase es — * ° asain dees oN eS EEO Ce oe Ss a ee de = ———— 27: = = een eon eS a _ ee Se : = a a La ~~ — = — « S vine « - = —_ * i — - = = — ae = ~ == — - - * - — nT a ay oy ota = — anni . - — —* “ = a = =, —- = - —— —— >: = = = = ee == = a= a = ————— Actes LARS —— =z ae > = - . ar — Setenthnaegsnmn dh Senn : ; -— - -—-* . —— : ~ - —- = — > = —s = —= == = - ee + we se a —=——— a —- _ a oe Se SS Ss SSS Se — er —— ‘ ee ; : = = ——— = = 2 a eee eg ee ee = — By ey ~ Wate al . of Euchdes Filementes. Fol.32, fed that the lines D Fand #7 G ate greaterthen the line: FG: as alfo it was fuppofed.. that rhe lines 4 and C,were greater then the line B(for the lineD Fis put to be equall tothe line 4; aud theline FG to theline 2; and the line H G.to the line C._) Wherefore they are both greater and alfo equall: which is impoffible. Wherefore the circles ney- ther tooch the one the other, nor are diftant the one from the other. Wherefore of ne- ceflitic they cut the one the other: which was required to be proued. The 9.P rableme...The 23. Propofition. ‘Upon a right line geuen,and to a point in it geuen: tomake a rettiline angle equall to a retliline angle genen. i Sys Sy OK i V ppofey the right line ge- uebe AB, e-lety point im it geuen be A. Andlet alfo the reGiline angle ceue be DC H.Ieis required vpon the right line cenen A Band to the point mit geuen A,to, make 4 rechilineangle ep gual to therectiline angle geuen D CH, Take in-either of thétines CD and CHa point at alladuentares > eee let the fame be Dand K.dna( bythe: f 5 : ‘ first peticion) draw a right linefro b D to E,Andof thre right lines, AF, FG and G. A,which let be equall to ). ConftraBion. the threerighe lines geuen,that is,t0C D;D E,and EC, make (by.) propofition goyng before ja triangle and let'the fame be AFG: fo that let the lineCD bees quallto the line AF; and.the line C E to the line-A Gand moreouer the lye D E tothe line F G.And forafmuch as thefe twolmesD Cand C Bare equall to Demonfration, thefe two lines FA and AG,the one to the other and the bafe DE is equall to the bafe F G:therfore (by the 8. propofition).the angle D.C E is equall tothe anole FAG VV berfore vpon the right line geuen:A Band to the point inst ge0 uen namely Ais made areftiline angle F A G,eqnal to the'rettiline angle genen D CH: which wasrequired to be done. 5% —— S SS. \ | dj 1 ; | f { mit , : hel vn i } | { Wt hy ' 7 i ' 14 : # ij ' a ; ] i : y t 4 : 2 ; wn] i iz it al be \ hh ‘ | J H * Bhi ‘ b ' ; 4 7a ty 4 at i) A, i ‘ae } | qa ‘ | ‘ ¥ ’ A Hat - +] ., ts ae b '% ‘an |, 4 iy . . ih) ult 1) ie ue ag : I " rE { — be Va, a | ' VE iv ‘ i, iy i}! i 3 ip) ee t 4 : } Sop a - SE = ~ 2 nr en Seep eon S Sa SES er SS ee SS ’ ee SS - a a — + Ag otherconftruGidnand demonfteation after Proclus, ‘ Sippofe thar the right line geen be 4 B:& let the point mitgeuen be A,&letthe 4, opher con- rectiline angle geué be C D E, It is required vpé therightline geue A Bye to the point fruction and de in it geué 4,to make a reAtiline angle equal to the rectiiline angle gene C D E. Drawea monflration afa Ime froC to E-And produce the line 43 on either {td¢ to the points FandG.Andyn- ter Proclus. 7 Juli. t9 As ether Deo neanfzatios afe ter Pelitarixs. . EDjthen was'the angle gene artight 3 Theft Booke to the line CD, put the line F.4 equal,& vnto the line DE let the line 4B be equal,& vn- tothe line EC put the line BG equal.And making the cétre the point .4,& the {pace AF, deferibea circle K F.Andagayne making the centre the point B and the {pace BG de- {cribe an other circle G L: which thal of neceflitie cut the one the other.as we haue be- fore proued.Let them cut the one the other in theipoitites M & N And draw thef: right lines AN, AM, BN, and BM. And forafmuch as RAis equallto. A M:andalfoto AN ( by the definition ofa circle)but C Dis equall to. A,wherfore thelines AMand AN. areeche equall to the line D C.Agayne fora{much as'B-G,is equallto B M;and to BN, and BG is cquali to C E:therfore either of thefedines BM and BN is:equall to the line! CE. But the line B A is equalito the line D E,Wherfore'thefe twa lines BA & A M,are -~'=+ #4 3% ta t equallto thefetwo lines D Eand DC,the one to the other,and the bafé B Mis equal to the bafeC E.Whertore( by the 8:propofition)the angle M A B,is equall to the angle'at the point D.And by the fame reafon the angle N'A B,is equal! to the fameangle at the, point D.Wherfore ypon theright line geuen 4 Byand to the pointinit genen Ajis:de- {cribed a reGiline angle on either fide of the line 4B: namely,on one fide the re@iline angle N AB,and on the other fide the reGiline angle M AB. either of which is equallto the retiline angle geuen CD E: which was required:to be done, | -AnctherconftruQionalfo,and demontftration after Pelitarius.. . Suppofe thar the right line geusp be AB: andlet'the point init geuenbe C; and ferthereiline angle geucn be DEF, [tis aikedy poke line genen 4B;andto the point init peuenC,to defcribeia reGilingangleequall to the re‘tilineangle geuen DE # Produce the line F E tothe point G: and fromthe point E ere&(by the 1 t\propofi-: tion Jvato the'line GF a perpendiculer line £ A,whichifivexa@ly agree with the lyne angle.Wherfore if front the pointe Ors 644 6% sun od: querede aperpendiculerlinevnto, .. the'line 4B that fhalf bedone which\ was required to’be done But ifitdos» sy not,then from the point H, eretya-> \. to théline 7 £,a perpendiculer lyn@ f‘D , whiche being produced {hall (by the fifth peticion)concurre with the line E D being alfo produced: for the angle D £ # isieffe-then aright... -.!, jy, angle(when as GEHis arightangle), ae ee eae Gr E 1533 Mek och pation 5r a 4 oa . > a " E . a . _ \ of Euclides Elementes. Fol.33. the perpetidicular line EA (by the 3.propofition) sand from the point Kere& vnto the line K Cja perpendiculer line L,whiche let beequall to the perpendiculerlyne D. And draw alite from C'to Z:Thenf fay tharthe angle LZ C B,is equallto the angle ges uen-D-E'F.Forthe two triangléss£ D and KC L,ate (by the fourth propofition) ‘¢- qual, and equilater the oneto the other: and the two angles LCK and ‘DEH are equal, And the two angles\BCK and FEH are equal, for either of them is aright angle, Wher- fore ( by the 4icomimon fentence)the whole angle LG B,is equal to the whole angle D E F.Which was required to be done. ! And ifthe perpendiculer line chaunceto fall without the anglegeuen,namely ,if the angle geuetbean acute angle, thefelfefame manner of déemonftration will ferae: bur onely that in ftede ofthe fecontd'common fentence, mutt be vied the 3,common fentence. Appollonius putteth another conftrudion & demobnftrarion of this propofitid: which (though the demonftration thereof depende of propofitions put in the thirdbooke, yet for thatthe coaftruction is very good for him that wilredely, and mechanically, without démonftration, defcribe vpona line geuen, and toa point initgeuen, arectilineangle equall toa reQiline angle geuen) I thought not amiffe here to place it, And it is thus. , Suppofe that the rectiline angle geuen be C-D-Z, and let the right line genen be AB, andlet the point inat geuen bee-#. Take in the jine CD, a point at all ad+ uéntures, which let beF. And making. the ‘cétre the. ./ int D, and the {pace D - ¥, defcribe a circle F G,cut ting theline DE, in the pointG, and draw a ryght line from F toG, Likewife fromthe line 42, cut of aline equall to the line D F,whichletbe AH. And making the'cétreth¢point —, A,and the fpace dH de- \*'* —— ao ae {cribe a circle H K,and from thé point Hfubterid vnto the circumference of the circle a tight lyne equall to the rightline FG , whichelet bee HK: and drawea right lyne from Ato K. Then I fay thatthe angle 7-4 XK, is equall to the arigle CD E. The proofe whereof I now omitte for thatitdependeth ofthe's 8 and 27 propofitionis of the third booke. 49040 | sat)! But now,as I fayd, by thisydit may very redily; and mechanically, without demon firation, vpona line\genen, andto.a point initgeuen defcribea rectiline angle equall to arectiline angle geuen. For -_ | | inthe re@iline anglegené, you “ ** -D : A neede onely to marke the two ss)" I AY shotn s Pag.) pointes , where the circumfes, \ | = . ae mae 4 Ea i rence of the circle cutteth'the. , . > tea | — z, fines contaynhing the angleges °°) G ye pated nent asthe points Rand G; and\or PhP oo sia 2 eA likewife to markeintheline ge,;. \ / a! tien asin AB the point H, &fo “2 re B making thecefitre the point Adecording to the fpace 4M (which is put to be equal to FDP) defertbe a peece of a citcunferénce on tharfide that you wil haue the angle to be, as for example the circumference H K,and opening your compafle to the — from — K, 1; HAG An other com~ flvudtion EG dea monfiration nfs fer Appollonint, How to do this propofitson reds- by and mechant- cally, ——S 7 a ) t ie 1! ° ] i i ; 4 { } iW ( 1 tht + a : i” : d ' aT ae i } HO oa / nan watiil 1 "4 wit) } 4 } ath - | mi) yt oY iii | Me 1 4ii ay aah t Aili salt ‘ BH ; EEE ila a) "| BUT HEIR PI | HT 4 : it Lit ; fj StL eiTe be ih yi y ee hd } OATH } ‘ if itr | bat ' 1 il lb PE ate! { WRG aih Vaan et) Me! 1 Vi i thie 1} be ii \ THAW t BY LT iM) eae ue " wees) ee Uh) i) } } 4 Wy a | | W ne Vb BRL | ie ih . ion 5 the H 7 Hh Li } - f rH ae { Wh : WU ere FHA i) ’ Hai : | f : - 1 ty tn iy mH? iy % Bi : iH ie r 1 ; : | WEE ti bi) : | ane ' f i mi ; (i iy) be ' : i 4) DB TT ; ye : | alte fh} ale | + th} Ae + (BL y | - i ae j vit i} iy vi » OnE eet ve / pigeea tid tiple PH MGR oe eg call WEP Ree 4); ot Wea i } wr Hy th j 4 Mh Hea || (d4 Pe all | (ie a Hi Hat ai a j ie eae Aa Ont rue / LEGS pee lt We a ha at Se add detain meh EME TG VEE TRE a Ay | i eb iF) - EeGiunl ( peel MEE ual Ris ORL fr a Bal rhiel PML 1) rh \ “ih ete Bel i} yi a bf aay ‘ “aa: Dae al) s M4 i a ae HH y } ‘ hen TED an Pes tae o4 Sith (ie ies : 1b Bebe aiteee: 3 tg sO a '. i &#t)) eh ‘a ivy itr He eapea tt) Np. be mate ab *\ Ta faci) Pade a WA on uy} Pee eiivee yy | Pa eet Cease cehiare Pan PAE a Ri} y hay Ly ii ae thn t es! » -. - : Phd tela dee tay i / 1 if ant - t % " aa ; ; ne 4 3 | | : ‘ bait ’ ee i 3 dy 4, | er i) | 4; Weiner ae oh les fh KI ks ea a bt ‘ TT ai} " } . 4 : ih y ¥ i ; t , AP ai)? hf +4 ahi fl ay el pe a it : Ne i ih Yar pari ht vit tiki si 2 Ue "se } ir? and t : PAS AR f 1 oe ay any Se Hi | i All away os erage Fae Rises Vere mi = Se Se =o Bast... = —— = =: = his ait eee = aes =a ee ae So See =” Se ee or ee } ~ > me OTe. bene = ~ = > oe soled er .: ——— Pe ~ - i Ee z ; = RG 0 = a — a OS — . ee > as = — Sy a Se eT se nS Sewer = — - —> — = Seaewe = = ~ - i ae = 4 a = =< p> a Sng Se et a = cord aR = ~ a ——— ee eee a a ere. Denopsdes the Jitf tnuenter of Css propofit son. Cemltrudion, Demonftration, The frft Booke the point F,to the point G, fet one foote thereof fixed:in the point Hi, and marke che point where the other foote cutteth the fayde circumference, which point let be K, And from that point to the point e4,draw.a right line:and fo fhall you haue defcribed at the point 4,an angle equal to the angle at the pointD.Asin the figures in the end of the other fide put. Oenopides was the firft inuenter of this propofition as witnefleth Eudemius, . TLhers.Theoreme The 24..Propofition. Ff two triangles haue twofides ofthe one equall to rwo fides of the other, ech to his correfpondent fide,and if the angle cotat- ned under the equall fides of the one, be greater then the an» gle contayned under the equall ides of the other: the bafe alfo _ of thefame, foalbe greater then the bafe of the other. | QTE AV ppofe that there be two triangles ABC, and D EF, ZY GOD OEF | baning two fides of the-one, that is, AB,and AC, es WY ES \quall to two fides of the other, that isto D E, and D F. Leech to his correfpondent fide:thatis,the /ide AB, to the Qliide D Eyand the fide AC to the fide DF sand fuppofe F F\\ that the angle BAC be greater thenthe angle EDF. Sa) |Then I faye that the bafe BC, ts greater then the bafe BE, For forafmuch as the angle BAC is greater then thé angle ED F make (by the 23 .propofition vps | on the right line D E , and to the point init geuz D, anangleE DG equall to the angle geuen BAC. And toone of thefe lines that is, esther to. A Cor D F put an equall line D G, And( by the firft peticic) draw aright line from the poine Gyté the point EB. and an other from the point F, tothe pomtG, And © forafmuch asthe line A Bisequallto thelineD E,, | and the line AC to the line DG,the oneto other, © and the angle B.A Cis (by con/truction) equall to’: A D PBs angle ED G, therefore (by the'4, propofition)) the bale BC; is equal to te to the bafe EG. Agayne for as much asthe line DG is equall to the line D E,there (by the 5, propo/ition the angle DG Fis equall tothe angleD FGVV heres fore theangle DF Gis greater then the angle EGEVV herefore the angle E F Gis much greater then the angle EG F, And forafmuch as EFG isa triane gle, bauing the angle E FG greater then the ancle EG F and (by thé 18.pros pofition ) vader the greater angle is fubtended the ereater Jide. ‘therefore the fide EG isgreater thenthe fideE F,But the Jide E Gis equall to the fide BC. wherefore the fide BCis greater then the fide EF, Lf therefore two triangles of Buchdei Elementes. Fol.34. hane two fides of the one equall orwofidesof the other, eche to his correfpéme dent fide; and if the angle'contayned Ynder the eqnall fides of the one be greas ter then the angle contayned Vnder the equall fides of the other: the bafe alfo of. the fame fhalbe greater then the Lafe of the other : Which was required, to be proued. | Seadtae In. this Theoreme may be thrée cafes, For the angle €.D G, being put equallito.the 6 ee cafes nes = angle B.A C, and the line D G,being putequall tothe line 4 C,anda line being. drawen.. proposition from Eto G, the line EG shall either fall aboue theline.G F, or ypon it,or vnderit. -_ 7 tides demonftration ferueti, whén thé line GE falleth abdue the litle GF, as we hane: TetPeaieon. before manifeftly feetie.~ ~~ nee | IHOGOI 3 Butifit fall ypon it,as ia this fgure here put: PS 23 ASE Second cafes Then forafmuch as the two lines 42 and 4C,are equal to the twolinesD Zand DG the ohe to the. other, and they contayné equall ‘angles by coh- ftruGion: therefore (by the 4. propofition) the bafe BC, isequa'l tothe bafe EG. »but the bale B- G, is greater then the bafe E F : wherforealfothe bale B Cyis greater thenthe bafe EF ¥ whichwas: : requiredto be proued, \ ~ ee Butnow letthe line €G, fall vnder thetine &) e. F,as ini the figure here put.And forafmuch asthefe , twolines 48, and AC, are equall to thefetwo~- lines D Eand DG, the oneto the other, and they contayne equall angles, therefores(by the 4. pro-. ofition)the bafe B C,is equalto thebafe € G. And Franch as withinthe triangle DEG, the two linnes D F and F £,are fet vpomthé fide.D E: ther-, fore (by the 21.propofition the lines D F and F é areleffe then the outward lines DGand G E: but ~ the line D Gis equal to thelineé DF, Wherforethe®* line G Eis greaterthen the line FE, ButGe ise-- 5 quall to BC.Wherefore the line 8 Cis greater the. . theline E F,Which was required to be proued, . * Third tafe, This third cafe may alfo another way bedemonftrated.produce the lines DF and sao D G, which areequal, vnto the points ae —— cae K and H:and draw a line from F to G: A . he chindea( : wherefore (by the fecond part ‘of the fifth propofition)the angles KF. Gand F GH,whichare vnder the-bafe FG, are equall:therefore the angle E FG'is reater then the angle FG &,Wherfore Eby the 18 propofition) the fide E Gis greater then the fide EF: batthebafe _ B Cis cqual ynto the bafe E G;Where-" - fore the bafe BC, is greater then'the .“\' bafe £ F » Which was required. to.be.\ proued, Tt may peraduenture (eme; thar Euclide {hould herein this propofition haué proued: that not onely the bafes of the trianglesarevnequall,butalfo that thea¢ reas ofthe fameate ynequall: for fo inthe fourth pro pofition, after he had pro- Ku, ued Why Eachde C$. aeftee she 37. or Demonsfiration headsag to am Moedity. ) T he firft Booke’ uedthebafe tobe equall, he proved alfo the areas tobe equall, But hereto: may bean{wered, char in equallangles and bafes,and vnequall angles and bales, the confideration is not like, For theangles and bafes being equall, the triangles al- {o.fhall of neceflitiebe equall, burthe angles and bafes being vneguall, the areas fhall notofneceffitie be equall, Forthetriangles may both be equal! atid vne- quall : and that may be the greater,whiche hathe the greater angle,and the ores ‘ter bafe, and it may alfo be theleffe. And for that caufe Euclide made no mencis on ofthe comparifor-ofthe triangles, V Vhereofthis alfo moughtbe acaule,for thar'to the demonfttation thereofare required certay ne Propofitions concer- ning: parallel lines, which we arenotas yet comevnto, Howbeit after the 27. propofition of this booke you fhal find the comparifon of theareas of triangles, which haue their fides equall, and therebafes. and angles at the toppe vaeguall, The 16.T heoreme, © The2s.Propofition. If two triangles hae twofides of the one equall torwo fides of che other, eche to hiscorrefpondent fyde,and if the bafe of the one be greater then the bafe of the orber: the angle a lo of the fame cotayned under the-equall right lines, hall be erea- ter then the angle of the other. we | iV ppofe that there be two triancles => ei Hf EF haning tho hides a de ag A 45,4 Band AC equall totwo fides of the o ) | ther that is toD E, and 'D F, ech to his corre[pons dent fide namely ,the fide A B tothe fide DF, and the fide AC to the fydeD F. But let the bafe BC be greater then the bafeE F,The Tfay, that the ans le B ACis greater thenthe angle EDF, For if B | not, then ts ct either equall vnto it; or le(Se then it. Rass Bio But theangle B.A Cis not equall to the angle ED F: for if it were equall,the bale alfo BC fhould( by the 4.propofition) be equal tothe bafe EF: but by fuppoftionit is not. VV berfore the angle B A Cis not ee quail to theangle ED F Neither alfo ws the angle BAC le/Se then the angle E DF: for then fhould the bafe BC be le/se thé the bafe E F( by the former pros pofition ) But-by fuppofition tt is not VV berfore> anole B.ACisnot lefSe thes angle E DF And itis already proned,that it is not equal vato sd a babe yan gle BAC isgreater then the angleE DF, If therfore two triangles haue two [ides of the one equall to two fides of the other,echeto his corre/pondent [ide,ex if the bafe of the one be greater then the bafe of the other, the an ele alfo of the fame contayned vuder 5 equatright lines {hall be greater thé the angle of the oe ther: which was required to be proued, ) aa aes OST of Enclides Eleméntes. Fol.3s. This propofition is plaine oppofiterothe eight,8is the couerfe of the foure and twenty which went before,and itis proued(as common yal} conuerfes are) by arcaion leading to an abfutditic, But it may after Menelaus Alexand rinus be demonttrated direCtly after this maner, Suppofe that there be two triangles ef BCR D | E F:haning the two fides 4 B ande4 C equal tothe | A R two fides‘D EandD F,the one to the other:and let the bafe BC be greater then the bafe E F, Then #lay that the angle at the point 4,is greater thé the ane . gle at the point D.For from the bafe BC cut of (by. . the thirde ja lineeqnallto the bale € F, and let the fame be B G.And vpon the line GB and to the point B put(by the 23,propofition)an angle equal to the angle 'D E F: which let beG BH: andlettheline3 * =e H be equall to the line D E.And drawe a lyne from H toG,and produce it beyond the point G: whiche being produced fhall fal either vppon the angle 4, or vpon the line 4 B,or vpon the line AC, Firft let it fall vpon the angle 4.And forafmuch as thefe two lines BG and B Hare equall to thefe twolines EF and E D,the one to the other,and they contayne ¢- ~ H quall angles(by conftrution)namely,the angles @ ee B Hand D E F: therfore( by the 4.propofition the bafe G His equalltothe DF, and the anole B A Gto the angle E D F.Agayne forafmuch as the line B His equall to the jline B -A( for the line 4 B is fuppofed to be equal to theline DE, ynto which line the ine B H is put equal ) therfore( by the 5 .propofition the angle BH Ais equall tothe angle B.4H:wherfore alfo the angle E D Fis equal to theangle B 4 But the angle B A Cis greater then the angle B _4H:wherfore allo thean ele BAC is greater then the angleE DF. | | But now letit fall vpon the linee#B in the oint K, and drawealine from 4 to A, And for- afmuchas thefe two lines B G andB Hare equallto thefe twolines E Fand ED,the one to the other, & they containe equal angles(by conftruction) name- ly,the angles G BHand DEF: therfore (by the 4. _ propofition) the bafeG H is equall to the bafe D F, and theangle BHGtotheangle EDF,Agaynefor- 8B afmuch as in the triangle B A H,the fide BAis equal to the fideB H, therfore (by the 5 .propofition} the angle B A His equal to theangle B H A.But thean- gle BH Ais gteater then the angle B H G-wherfore alfo the angle BAH is greater then the angle BHG. Wherfore the angle B A Cis much greater then the angle B H G.Butitisproued that a. angle BHG is equall to the angle atthe point D. Wherefore the angle B A C is greater thentheangle at the pointe D: Which was required to be proued, | | CE F ‘But now fuppofe that the line H G beyng produced doo fallyppon thelinee4C, namely,in the point K.And agayne draw alfo.a line frome to H.And forafmuchas 8 G is equall to E F,and B H,to E D, therefore thefe two lines BGandB Hare equall to thefe two lines E F and E D,the one to the other,and (by conftruétion )they contayne equall angles,namely,the angles GBH and FED,Wherfore (by gees prope ) : di. ¢ wk conSerfes are commont} sudsrettly de~ monflrated. An other demo- frration after Menelaus Alen- andr suse, Diners cafes im this demonfira~ s10n, Fir cafe Second cafe, Third cafe. TA _ thebafe G His equal to the bale D F: & th’an ) ave nt gle BH Gis equalltoth’angle ED F,And for. - | A afmuchas GHisequall to.D:E,and.D F ise- | A TMT “oe quail to .4 C: therfore G H alfo is equall to 4 Ae A .. ©. Wherfore H K is greater then .4C, where- Lhd li , _ fore H K is much greater then A K.Wherfore MARTA (bythe 18. propofition) the angle KAH is A greater then the angle K HA. Agayne foraf- - BH aa much as B Ais equalto A Bi forB His pute- Di eh LB quall to D E,whichisby fuppofition equal to Ue A B) therfore(by the 5, propofition)\the an- He al gle B H Ais equall tothe angle B A HWher- ey A fore the whole angle BHK is leffe’then the Aue whole angle B AK, Butit hath bene proued, SME that the angle BH K is equall to thé angle at tee the point D,wherfore the angle B AC is grea- ALAR ap est i terthen the angle at the point D, which was Be iia tequired to be proued, eee | | Hero Mechanicus alfo demonftrateth ican other way, and that by a direct ae * demeonitration, | sate tM ont aye Suppole that there be two triangles ABC } ( ig i! am patton es mi ' and DEF, hauyne the two fides_48,and A : A {Biases tf chasicus, &sequalltothetwofidesDE,&DF,theone _ nee ae ee nn toithe other, andletthe bafe BC, be greater oe : ality tin theathe bafe £ F.Then I fay,that the angle ar Bye the. peint,4, is greater.then the angle atthe 2 7 a Lamers! point.D.For forafmuchas BC, is greater thé | ate £ fF produceé F to the point G , and put the | HOR ee Hh line EG, equail to the line BC. Likewife pro- ! Hue et . duce the line DE to the point H, and put the a line D #7, equalltotheline D F, Wherefore UP Ht et making the centre the point D,and the {pace Cah, D Fdefcribe a circle,and it thall paffe alfo by rar re the point .Let the famecircle be F K H.And Bee ae forafmuch as 4Cand AB are greater thé BC oo wie (by the 20. propofiti6)& the lines AB & AC, Re | ar equal to the line €#/,& the line BC is equal EB cia at to the line E G. Therefore the line E His grea Li ae Ht) ter then theline EG, Wherefore making the Pepe Re ey centre the point € and the fpace EG defcribe AY a circle,and it fhall cut the line E A, Let the Fy aa fame circle be GK: and from thé common pee ti fetid of the circles, which let be the’point K, pe Te draw thefe right lines K Dand KE. And for- : eae afmuch as the point Dis the centre of the cir- | sane ah cle H K F, therefore( by the 15. definition )theline D K,is equall to the line D A, thar Leta aie is vate the line 4C. Agayne forafmuch as E is the centre of the circleG KX, therefore ae, the line E Kis equal to the line EG, that is,to the line BC. And forafmuch as thefe two eG lines 4B and 4 C,are cquall to thefe two lines DE and D K,and the bafe B Cis equal a are to the bafe E K(for E Kisequallto EG (by ther 5 definition) & EG is put to be equal eaties li to BC’). Whereforé(by the 4-propofition )theangleB ACis equal to the ancle DK, | Pa a But the angle E D Kis greater then the angle ED F:wherefore alfo the atigle B AC,i$s | ea) gteaterthen rhe angle € D F: which was required to be proned. is : | Fe The of Enclides Elementes. Fol.36. The17.T heoreme. The2x6;P ropofition. Ffewo triangles hane two angles of the one equall totwo an- _ gles of the other, ech to hs correfpondent angle,and bane alfo one fide of the one equal to one Ie of the other, either that fide which lieth betwem the equall angles, or that whichis fubtended under one ofthe equall angles: the other fides alfo ‘ofthe one,{halbeequailto the other fides of the other, eche to bis correfpondent fide, and the otber angle of the one fhalbe 7 equall to the otber angle of the other. TV ppofe that there be tho triangles A'B e & wu) Cand D E F hauing ‘wo angles of the | A D IKI one, thatis the angles £BC,and BC A, Cy” ? | | equal totwo angles of he other, thatis, to the angles D E F and E: F Dych to bis correfpo dentangle that is,the angle ABC, to the angle D HG EL ap EE ,and the angle BC A to the amle EFD and one fide of the one equal to one fide of) other, firft that f ide which lieth bet wene the equal angles , thatis, the fide BC,tothe fide EE, The Lay that the other fides alfo of the one fhalbe equal » the other fides of the othersech to bis corres Jpondent fide,that isthe fide AB, tothe fide D B,and the fide AC,to the fide DF .and the other angle of the ore,to the other anole of the other that ts, the angle BAC tothe angle ED F For if the fide A B be not equal to the fide D E,the one of thetnis greater, Letthe /yde A B be greater: and (by the 3 prope fition) vato the line D E; put an quall ineGB,and drab a right line from the point G,torhe point C. Now foraimuch as the le GB, ts equall to the line D E andthe line BC to the line E 7 therefore thele two lines GB and BC, are equall to these tivo lines D E ant E F,the one to the other ,and the angle GBC is (by fuppofition) equall tothe aigle D E FV V berefore ( by the 4. propelt- tion) the bafe G Cis equal to thebafé D and the triangle GCB ss equall to the triangle D E F,and the angls remayning are equallto the angles remays ning vnder which are fubtended qual fydes VV herefore the angle GC Bis es qual totbeangle DF E-But theangle DFE ts Juppofed to be bates to thean gleBC AVIV herefore( by the fift common fentence )the angle BC Gis equal to the angle BC A, the lefse ange to the oreater: which is impofstble VV bere- fore the line A Bis not ynequall o the line DEWVV berefore st is equall And the she line BC is equall to the tine £ F:now therefore me are to [ydes A B — : SEES JY. | Densenfiration leading to am a bfsr dit ae + —— ; ,) SO A : ' : : ; : aH Ay 1 Hi t 1H) ‘ 1y! He Hf ii ; ti 1 iti | Hai & i! i M I ! Mi | Wi : eis. : nis | ee . | ‘HERE. : i ii} i Hi} nh) | Hil ik iis i 1 \ ii a r : ; ia ‘] ‘| & ‘i a : - } ih Hy ‘ Vo 41] | | i] ie vig : Hh! Reh! ii ’ if iat) B i "| Lae ry . Kg ie. Lae ‘ 7h! hi : Mh, f ia ; 1 » if i} - : i} = —— = —— —— a ee. =——— —— ————————————e = ae — ee ee OS - : < — + eats a tied nf = ae se aa a =a , - = - — _—- ——_. - ~~ ~ - —— ~ = =< - = = —— ——-———— = - eT ; adeeb = 7 ———— — — ———— a — —- a ———————— = u = a aamcihanaee == ~ > 7 - . - - > - - nt Des ae en —_ — ———— — SSS == < Per - —~ = > aa = s.=- a — == —— _ - _—— = SSS t ~ =~ Se - ees ee | — - a ~ = = _ = = —_ a -' a SSS eS : ~ - Stes 05 wae <3 = — A — ~ eS ee eee Pe arenes - SN a NRCS ae =e > ‘ ~ EF a eae —_- SS es ee es ee — = — : EP sae hn + eet nae ——Ss ~~ RY: ‘ Lhefrst Booke BCequall to two fydesD Eand EF, the one tothe other.and the angle ABC, tsequall to the angleD EF VV herefore( by the 4-propofition )the bafe AC is equal to the base DF ,and theangle remayning BAC is equall tothe angle re mayning EDF, | _* Agayne fuppofe that the fydes fubtending the equall angles be equall the one to the other let the fyde Lfay A Boe equal to the fyde D E. Then agayne f fay,that the other fydes of the one are equal to the other fydes of the other, ech to his corre/pondent [yde,that is the /yde AC to the fyde D Fyand the fydeBC to the fyde EF: and moreouer the.anglerema yning, namely, BAC, is equall to the angle remayning, that is, tothe anole E DE. For ifthe fyde' BC be not equal to the fydeE F the one of them is greater: let the /yde BC,if it be pofsie ble , be greater. And (bythe third propofytion,) vuto the'line EF, put an equailline B Hand drawea right line from the point |4 tothe point H, And forafmuch asthe line B EZ isequall to the line E-F , aud theline A B tothe line D E,therefore thefetwofydes AB and B Fi, are equallto thefe two fydes D Eand EF, the one to the othersand they containe | | equallanglesVV berefore (bythe a.propofition) she A D vafe A Lis equall to the bale DF and the triangle. : 4B Fi,isequall to the triangle DE Fyand the ans gles remayning are equall totheangles remaynin 1 vader which ar fubteded equalfydesVV berfore the _ ingle BEL Aisequall tothe anole RFD. But the ingle EF Dis equall to the‘angle BCA, VV heres jore.the angle BH dis equal tothe angle BC A. 2 i (V herefore the outward anole of jtriangle.A HC, | samely,the angle BH Ais equall tothe inward and oppofite angle namely sto ihe.angle ICA, which(by the iO\propo/ition) is impo/Stble.V. Vherfore the Iyde Ei Fis not vnegtall to.the Pde Be svherefore itis equall. Andthe Ayde ABis ‘quail to y /yde'D E:whereforethe/e. two fydes A Band BC are equal to thefe wafydes D Is and E: F the oneto the other, and they contayue eguall angles: ’ CK berfore( by the 4.pr opo/ition )tbe bale AC is equall to thebale D.F-and the iriangle AB Cis equall.to the triangle D E Foand the angle remayning names yisthe angle B AGis qua Ul fot heangle r emayning, that £5 ,t0 theangle E DE if therefore two triangles hane to an gles of the one eguall to.two.anoles of the (ther, ech to his corre/pondent angle and haye alfo one fyde of the one equall to que [yde of. the other, esther that fyde which lerh betwene the equal angles. or ibat which is {abtended vnder one of the equall angles: the other Jydes alfoof the.one fhalbe equall tothe other Sydes of the other, eche to his conre/pondent fde,and the other augleof the one fpalbe equaltto the other angle of the other: vhich was required to be proted.~ =. 4, .V Vhetcas inthis propofition itis fayde, e | : that triangles are edijall which ' hog! in , Fairing twoangles af the one equall to two angles ofthe other,the od tothe o. ther, oe eae of Exuchdes Elementes. Fol.37. ther, haaé alfo one fide ofthe one equall to one fide of theother,cither chat fide which licth betwene the equall angles, or that fide which fubtendeth one of the equallangles:this is .obe noted that without that caution touching the equal, fide,the propofition fhall not alway es be true.As for example. Suppofe that there be a rectangle triangle 4B C,whole right-angle let be at, the point B,& let the fide B Cbe greater thé the fide B_A:and producethe line 4 &,tré ths point B to the point D,And vp6 the right line &C & tothe point in it C, make vnto the angle c BAC an equal angle(by the 23. propofition), | which let be BCD,& let the lines BD & CD,be | ing produceéd cocurre itt the point D.Now thé there are two triangles 4 B C,and BCD,which hauc two angles of the one equall to two an- Se gles of the other,the one to the other,namely, TH SPR the angleedBCtotheangle DBC (for they 4 B F “B are both right angles ), & the angle B_4 C, to the angle BC D( by conftruGion Jand haue al- fo one fide of the one equall to one fide of the other, namely ,the fide BC, whichis cé. mon to them both,And yet notwithftanding the triangles are not equall : for the tri- angle B D C,is greater then the triangle 4 B C.For vpon the right lineBC, and to the point in it C,defcribe an angle equall to the angle 4C B:whichlet be FC Bi by the 23, propofition ),And forafmuch as the fide B C was {uppofed to be greater then the fide, 4 B, therefore (by the 18,propofition) the angle B.A Cis greater then the angle BC A, wherefore alfo the angle BC Dis greater then the angle BC F, Wherefore the tri- angle BC Dis greater then the triangle B (-F. Agayne forafmuch as there are two tri- angles 4B (and 2 CF hauingtwo angles of the one equal totwo angles of the other, the one to the other,namely,theangle 42 Ctothe angle F 8 C(for they are bothright angles)andthe angle AC B tothe angle F CB(by Perrision) and one fide of the one is equall to one fide of the other, namely,that fide which lieth betwene the equall an- gles,thatis,the fide B C which is common to both triangles. Wherefore (by this pro- pofition)the triangles 4B Cand FB Care equal.But the triangle DBC is greater ‘thé the triangle F BC, Wherefore alfo the triangle D B (“is greater then the triangle 4 B C.Wherefore the triangles ABC and DBC,are not equall:notwithftanding they haue two angles of the one equall to two angles of the other,the one tothe other, and one fide of the one equall to one fide of the other. The reafon wherofis,for thatthe equal fide in onetriangle, fubtédeth one of the equall angles,and in the other lieth betwene the equal angles.So that you fee that itis of necefficie that the equall fide do ia both triangles ;either{ubtend one of the equall angles,or lie betwene the equall angles, Ofthis propofition was Thales Milefias the inuentor, as witneffeth Eude- mus in his booke.of Geometrical] enarrations; | Ther8. Theoreme, Ther 7-Propofition, Ifaricht line falling upon two right lines,do make the alter- nate angles equall the one to the otber = thofe two right lines are parallels the one to the other. es : a es FI ——— Suppofe Thales Milefins the inuentcr of thes proposition, ere ee eS ee — ~r — — a Se EEN oe ae - - = SSS z STS a 2 = = , = wack a “ = + os 2 > 7 { ; i HH i} ; d{ \ The first Boke AGRA ppofe that the right line E Ffalling vppon thefe two right lines A B Rey ioand C D,do make the alternate angles,namely,the angles AE Ex E ASSOLE’ 7) equal the one to the other. Then I fay that A Bisa parallel line to. CD. Forif not, thenthefe lines produced fhall mete together either onthe fide of Band D,or.on Densonfration the fyde of Ac C.Let them be produced therfore, aie f eating Fea" and let them mete ifit be posible on the [ydeof.B. = siete Mat Asse, : ; . ; and Din the point G. VV berfore in the triangle fe es G EF the outwardangle AE Fis equal to the in- iys ward and oppofiteangle E FG, which (by theo...“ JF lak propofition)tsimpo/sible,VV berfore the lines AB ; andUDbeyne produced onthe fide of B and D, fhallnot meete.In like forte alfo may it be proned that they fhall not mete on the frde of AandC, But lines whiche being produced onno fyde meete together,are parrallell lines( by thelaft definition: )wherfore A Bis a parrallel line toC D If therfore.a right line falling vpontworight lines,do make the alternate angles equallthe oneto the other: thofe cworight lines are parrallels the one tothe o» ther: which was required to be demonstrated. Ghiswordeal- .-- This worde alrernate is ofEuclide in divers places diuerfly taken: fomtimes Rerun forakind offituation in placejand fomrime for an order in proportion, in which fignification he'vfeth it in the v.booke,and in his bokes ofnumbers. And inthe iG fitftiignification he vfeth it here inthis ee and generally inallhys other Mirhieplice. bokes ,hauing to do with lines & figures,And thofe two.angles he calleth alter- Whicoenglee mnateswhich beyng both contayned withintwo parallel or equidiftant lynes are . aecaliedaleers geitherangles inorder, nor are onthe one and felfe fame fide, but are feperaged the one fromthe other by the: line which falleth onthe t wo lines: the one angle bey ng aboue,and the other beneath. ) The19.T heoreme. The 28.P ropofition. : Ff aright line falling vpon two right lines make the outward angle equall to the inward and oppofite anvle on one and the fame {yde,or the invrarde angles on one and the fame [yde, e- quall torworight angles:tbofe rwaright lines {ball be paral - lels the one tothe other. | | IY | ppofe that the right line EF, fallyng yppon thefe tho right lines COQ 4 BandC D,do make the outward angle LGB equal tothe inward IQ and oppofite angle G Fi 'D, or do make the inward anzleson oneand ra et the of Euchides Elementes. Fol.38. the fame fide,that ts, the angles BG H andGHD equalltotwo right angles.T hen I fay that the lyne A Bisa parallel line to the lyne CD For forafmachas the angle EG Bis( by fuppofition )equall to the ane gleG HD,and the angle EG B ts( by the 1.5. proe pofition equall tothe angle AGH: therfore the ane gle AG Hiseguall tothe angleG. HD: and they are alternate angles.VV berfore( by the 2°7.propofis tion) A Bis a parallel line to C D, Fr Agayne forafmuch as the angles BG Ff andG H Dare( by [uppofition jes quallto.tworight angles ex (by thex.propofition the angles A G Hand BG H, are alfoequall to two right angles, wherefore the angles AG H and B G Fi, are equall to the angles BG HandG HD: take away the angle BG H Which 1s commonto them both VV herfore the angleremainyng namely, AG H is equall to the angle remayning namely.toG HD, And they are alternate an gles. VV berfore( by the former propofition) A Bisa parallell line to CD, If therfore aright line fallyng vpontwo right lines,domake the outward angle es quall to the inward and oppofite angle on one and the fame fide,or the inwarde angles on one and the fame [ide ,equall to two right angles ,thofe two right lines hall be parallels the one to the otber:which was required to be proned, Demonftration Prolomeus demonftrateth the fecond part of this propofition, namély, that the two inward angles on one and thefame fide being equall,the right liges are C An other demi parcilels,after this manner, fi ration of the ; fecond part of Suppofe that there be two right linési4 Band C Dyand let a certayne ri oht line E *#sspropofitsom FG H-cuttethemin fuche forte, chat. . 3 after Mt obomess itmake the angles BF G andF GD. e- quall to tworight ‘angles. Then I fay, thattheferight linese# Band CD are parallellines,thatis they thall not con- curre,For if it be poffible,let thelines B Fand G D being produced concurre in ‘ the pointe K.\ Noweforafmucheas the, | right line E Fitandethvppon the right line.4 B,therfore(by the.13.propofiti- es 5 on )itmaketh the angles e4 F E,and' B | ae F Eequallto tworightangles: likewifeforafmuch as the line FG. ftandeth vp6.the line CD,therfore( by thefame propofition it maketh the angles CG F and DG F equall to two right angles, Wherfote the foure angles B F E,e 4 F E,CGF,and DGF are equal to foure right angles: of which'the two angles B FG anid FG Dare(by fuppofition) e- quall to two right ariglesswherfore the anglesremaining,aamely,\e 4 F.G and CGF are alfo.cquall to. two rightangles,If therfore the right lines F B and GD being produced (the inward angles being equall to tworitht angles Jdo concurre, then {hall che lynes F Aand G C being produced concurre. Forthe angles 4 F GandCG F 4re'equall'to two'rightangles,For either the tightJinés fhallconcurteomeither fide, orels-on nei= ther fide. For that on either: fide.the angles are equall to,two right atfeles, Wherefore let the right lines F “ahd G C concirte inthe point L,Wherefore the two right lines LAFKandLCGK do comprehenda fpace, which PY the 6, peticion ) is — saljs Ce T he firft Booke ble.Wherfore itis not poffible that the inward angles being equal to two right angles, the right lines fhould concurre, Wherefore they are parallels: which was required to be proued. The20.Theoreme. The29.Propofition. eA right line line falling vppontwo parallel right lines = ma- keth the alternate angles equal the one tothe other: and al- Jothe outwarde angle equall to the inwarde and oppoyjite an- leon one and the fame fide: and moreouer the inwarde an- gles on one and the fame fide equall ta two right angles, IV ppofe that vponthefe parallel lines A BandC Ddo fal 4 ithe right lneEP,\Then I fay that the alternate.an- : les which it maketh; namely, the anoles AG H and s c Hi D,are equall the one to the otber:andy the out: f SA ward anole EGB is equal | y) tothe inwarde'and oppoe fiteangle on the fame fide, ¥ bes > m ~ 5 arazed ‘Rights ae 5 pr angle G FD,the one of them is greater Let the angle AG Hi be greater. And abjurdity. ~ forafmuch as theangle AG Fis greater then theangleG F1D, put the angle Frit part. ae «Bae and. of Euchdes Elementes. Fol.9. and BG FZ are(by the 13, propofition equall.to tio rightaneles, V/ Vherefore theangles BG H andGH® are alfo equal tatwo right angles, Ifa lyne therfore do fall vpo two parallel right lines:it maketh thealternate angles equal 4 the one to the other:and alfo the outward an ole equall to theinward wid oppos fite angle on one and the fame fide: and moreoner the inward angles onone and th ony fyde equall to two right angles: whiche was required to be demon? rated, This propofition is the conuerfe ofthe two propofitions next going before. For,that which in either of them is the thing fought, or coclufionyis in this the thing geuen,or {uppofirion, Andcontrariwife the thinges which in them wete geuen orfuppofitions are inthis proued,arid ate conclufions, Pelitarius after this propofition addeth this witty, conclufion, Lf two right lines which cut two parallel lines ,do betwene the fayde parallel lines concurre in a point , and make the alternate angles equal, or the outWard angle equailto theinyard and oppofite angle onthe fame fide,or finally the twoinward angles on one andthe Jelfefame fide equallto twa rive angles :thofe to right lines aré drawen directly and make one right line, Suppofe that there be two right lines e4B and C B, which let cuttwo parallel lines D and FG: andlet 42 cuttheline D Ein the point A:and ket C Bcutthe line BG in the point K:& let the lines 4 B.& C B,concurre betwene the two parallel lines DE & FGinthe point 2: andlet theangle DA ZB be e- qual to the angle B K G:orlettheangle 4 A Dbe equall to theangle B K F; orfinally let the'ahgles 6 Hf Dand BKF beequal totworight angles, The DP \ Ft a: Ifay thatthetwolines.ABandBC are drawendi rectly, anddo.make oneright line, For if they be notsthen produce 4 B ynti' it cutE Gin the point . L,and let AL be one right line,and {0 thal bemade csi a the triangle B L K, Now then (by the firftpartof. F sf ha G this 29, propofition)the angle ‘DH B thalbe equal ce. to the alternate angle GDB: but(by fuppofition ) the angle D H Bis equall to the angle B KG: Wherefore the angle BL Gis equall to the angle B KL, namely, the outward angleto the inwarde anid oppofite angle:which (by the 16.propofition )is impoffible. | Morcouer ( by the fecéd part of this 29. propolitié )the angle AH D thalbe equal to the angle B L K, namely, the outward angleto the inward: and. oppofite angle on oneand the fame fide, But the fame angle 4H Dis {uppofed to be equall tothe angle B K F: wherefore the angle B K Fis equall to the angle BL K,Which (by the felfe fame 16. propofition) is impoffible, ALeal 33 Laftly forafmuch as the angles BD and B K Fare fuppofed to be equall to two right angles, & the angles BH D & BL K arealfo by the laft parvofthis 29 propofiti- on cqual to two right angles,therefore the angle B K Fifhalbe equal to the angle B LK: which agayne by the felfe fame 16 ‘propofitionis impotfible, a Ther1.Theoreme The 30.P ropofition. » Rightlines which are parallels to one and the jevfe fame right line:are alfo parrallel lines the one tothe other. L at, S upp ofe T bss propofitson ss the cony erfe of the two for- ET Propofitsoss An addition of Demonftration leading fo an abfurdities F, se ff pare. Second pars. T hira part 5. 4 - aa el iy |! Hida fi Hii i 1 Hi) I, 7 i f Hi} i} EY PE + Bt Bi aii} ht HM 1) MEPL ark tt i AY iii . iH ; i 4] 1} i} | ; 4) BR SEL Ria oe Wa i} As ang int hAgas ' a8 | va ete hee . ' pide ‘ TEE iuael HATH Demonftration. WG Lea - WATE HATE HY : i ia} Hy iff : + id hy : iy AR yt HEE Re Vien ai) il t i } ii i ita | } i }\ Sat 7 4 aia aaa a ae ii ; ; 4 ran 1} 44) et 7 ae fs , : BET oi 1S bai | P Pigat hy 4947 ijt} 7 / : phd 4 ai 7% it : rhea il; it ih | Ay saa aeaay Mh tS Lise bod - : i b ee ie if bath Wall TN vat fq 4 , a 1 : i aa ili * iad mA 4 Ware | tii! ty it} = Pavecttt ti Awe) eae dl mere Wot he ean | ELH ; a 4 | we 3 ae OAT ALY ary :7 y 4 ” Tei 44'S, tt RN ah wk hi: Bad} an ie ey 4 ‘3 F3) 1% ee! AEE had - : vot, : i - ‘y 4 ‘ bop Ui dae ARTE LOMe IRL if 7} 1 Le din jot d ' 4 4 oy i * haa + PRG ‘ : rs oe <= fee! fee: , f, ® ; } ; ; Te VRE Ass other cafein As le Al Tee te, ' Aips P. é rick stss Probleme, wv) “4 ae Tee Pee ~ 3 oes | Hy : " WA ls 1a bret ‘he, ae : uy tama i) fd Bett i \ We I "7 4p x ’ ti ¢ ss wah if aff 4] hf 7) Eee | | Me | We) ay Bia nts wt f bg i a | _ bal , , ie) 3 Na ’ 4 * We a ) | 4 ’ tt obse 1078. Ager, Jay, that the line A Bisa parallel line Lhefirft Booke {dV ppofe that thefe right lines A; 4 BandCD,be parallel lines = totheright line EF. Thenl —_—-= toC D,Let there fall vpon thefe thre lines a right lneG LHLK, And forafs - much as the right lineGHK, falleth vppon thefe parallell right lines AB and E Fyherfore( by the propofition going before )the angle AG His es quall to toe angle GEAF. Agayne fore afmuch a: the right lineG K_falleth vppon thie parallelt right lines E F and CD therefore (by the fame) the ae angle GHF is equallto the angleG K D.Now then it proned that the angle A G Hiseauall toy angle G FF, andy the angleG KD isequall to theangle G HLF .VVherfore the angle AG K is equall to the angle G KD, And they are alternate angles:wherfore AB isa parallellineto.C D.Right lines therfore whichart parallels to one and the felfe fame right line,areal/o parallel lines the one to the other: which was required tobe proued. | Euclide in the demonftration of this propofition, ferteth the two parallel lines which are compared to one,in the extremes, and the parrallel towhome they are compared,he placeth in themiddle, for the ¢afier demonfttation, It may alfobe proued even by aprincipleonely, Foritthey fhouldeconcurre on any onefde,thcy fhould concurrealfo with the middleline,and{o fhould they not be parallels vnto it,which yet they are fuppofed to be, | But iilyou will altertheir pofition and placing andfet thatline to which you will cOpare the other two,aboue,or beneath: y ou may vfethe {ame demonitra- tioa which-you had before,As for example, : Suppofe that the lines 4 BandC Dbe parallels tothe line € F;:-ahdlet both the lines 4 Bind C D., be aboue, and Jetthe line EF bebeneath,and notinthe middeft. Vponwhich-et the rightJine.G AK fall. And forafnuch as either of the angles KH D.and-H ¢ Bis equall tothe ‘angle HK Bj. © (forthey er alternateangles )therfore theys i J are (by the firft common, fentence Jequalk +... the one tothe other,Wherefore( by the 28 propofition )the right lines 4B and €F, are parallels. | . But here ifa man wilhobie& thar.the linesEK andK Fj.are parallels vnto the line C Djand therefore are parallels the one totheother, V Ve will an{were that the linesE K and K F are pattes ofone parallel line,and ate nottwo parallel lines, of Euclides Elementes . Fdl.4.0. lides,For parallel lines ar'vaderftanded tobe produced infinitly Bur EK being 27e/lmes producedifallerh vpon K F,Wherefore its oneand the felfe fame, with it, and are Gnderfian- ded to be prodx~ not an other, Wherefore all the partes ofa parrallelline are parallels, bath tothe ced infinstely. right line vato which the whole parallel linetsa parallel,and alfo to al rhe parts of the {ame right line, As the lineE K 1s a parallel vnto WD,and the lire K Eto the line C H, Foriithey be produced infiaitly they will neuerconcurte, Howbeit there are fome which like nor, thattwo ditting parellel lines, fiould be-taken and counted for one parallel line: for that the continuell quan- tity namely ,theline is cutafonder,andceffeth to be one, V V herefore hey fay, that there ought to be two diftin& parallel lines compared to one. Anctherfore they adde to the propofition acorrection, in this maner,T wo lines beingparallels to one line: are either parallels the one the other,or els the one ts fet dirett ly agaifte the othe fo that if they be produced they fhould make one right line._As for example, Suppofe thatthe lines CD and & F be parallels to one and the felfe fame line 42, and let them not be parallels the one to the other. Then I fay,that thetwo lines CD & E F,are direily fetthe one to the.other. For for as muchas theyare not parallel lines, Jet them concurrein the point G, And from the point G draw aline cutting the line 42 in the point H.Now by the former propofition the angles 4 HG & HGC are equall to tworight angles, but by the fame propofitid, the angle HG, is equall to the alternate angle HGF, Wherefore the angles G Cand 4G Fare equal to two rightangles. Wherefore (by the 14 propofition Jthe lines C G arid FG are drawen dire@ly and make one right line, Wherforeal- > ce u fo the lines C Dand E F aré fet directly the on¢to the other: and being prodicedithey will make one right line. ST he 10.Probleme. The 31.P ropofition. tne. SiG V ppofe that the point genen be | as yA A andlet the right line geuen be. + xX BC. Ir is required by the point geuen namely A,todram vato the right ‘line B Coa parallel line: Take in the line “BC a point at alladnentures,and let the ‘fame be D.and (by the firft peticio draw ‘aright line from the point A; tothe pome ~. D, And (by the 23 .propofition) vpon the. right line geuen A D,and to the pomt'm =~ ‘it geen A make anangleD AE, equallto Bsvak the angle'genen 2D Cy And { by the rg. -propofition) put puto the tine AE the line A Fdirectly, in fuch fortethat they La. ‘By a point geuen,to draw vntoaright line geuen, aparallel Constralin. Cc both Aes te lg ee ee —— — ss >= - mb ‘ T= SS Se ——————— ~ rs eo ———————————— = —~ ites ~ —$—$—$—$——— — - > Ss —=— = —— So —————— 1 ie i] i: - ni} ii Hi: | a Hy Th ni ” 2 SS ee SS === —— a . ang - “a y Canft. fa CE i0%s “a Dewmenftration. - —_— ae ans = = a £ an’: = ee We le RE ‘ = acme wh Sas SE Ss x * = Sz 7 Pita 8s . wae ee er ae ae eon : as ee en i : 5 — = 7 = : =} - oe a * — a —_-- SS > hen x a a J Se > ; =i = x = "% . » a i Reet ERE EE Ry ES ee EE A ek —_—e fue. a a —? yaa as. —- af = on wan ene a : a ee Sa a —— = = os = on. - aaa a eer eeieeereen a awe eee — mae wad ote — wit Some era . - . P +: ORB Me eee be toy) bebe. ‘ ll . - r 7 T he fr St Booke both make one right line. And forafmuch as the right line A D falling vpou the right lines BC and E F doth make the alternate angles namely, EAD, and A DCequall 5 one to the other ,therfore(by the 2°7,propofition)E Fis a parallel line to B CVV herfore by the point geuen,namely A,is drawne to the right line Seuen BC a parallelline E AF: which wasrequiredtobe done, This propofition is to be vnderftanded ofa point geuen without the line ge- uen,and in {uch forte alfo,chat the fame line geuen being produced,doo not fall vppon the pointe geuen, Lhe22.Theoreme. The32,Propofition. Ff one of the fydes of any triangle be produced : the outwarde angle that st maketh,is equal to the two inward and oppojite angles.eAnd the three inwarde angles ofa triangle are equall — to two right angles. | VV ppofe ABC be a triangle. Z| pale one of y fides share A | RE | r,,| ofnamely ,CBtothe pointe D. : pe Then I fay, that the out warde | angle ACD iséqualltothetwoinwarde - and oppofiteanglesC A Bex ABC:and J the three inwarde angles of the-triangle, that isthe angles ABC, BC A\andC A B are equall to two right angles. For(by the propofition going before )rayfe vpfro the point Ca parallel tothe right line A 2 e > Band let the fame be C EF, And forafmuch | as A Bisa parallel toC E,and vpon them falleth the right line AC; therefore the alternate angles BA Cand AC Eare equall the one to the other. Agayne forafmuch as A Bisa parallel yntoC E,and vpon them falleth the right lne'B D, therfore the outward angle EC Dis (by the 29. propofition) equall to the inward and oppofite angle A BC, Andit ts proued that the angle AC. Eis equal to the angle BAC: wherfore the whole outwarde angle AC Disequall to the Wo inward and oppofite angles thatis,tothe angles BAC and-A BC.Put the angle AC Bcommon to them bothVVherfore the angles AC Dand AC Bare equall to thefe three angles ABC, BG A.andB.A c But the angles ACD em AC Bare equall to tworight angles( by the 13 propo/ition): wherfore the angles ACB, CB A, and C A Bareequalltotwo rightangles, If therfore one of the fides of any triangle be produced, the outward angle that it maketh, ts equall to the two inward and oppofite angles. And the three inward angles of 4 triangle are of Euchdes Elementes. Fol... dre equalltotwo right angles: which was required to be demonftrated, Euclide demonftrateth either part ofthis compofed Theoreme, by drawyne fd oneangle ofthe trianglea parrallel line to one ofthe fides of the fame triangle, withoutthe triangle, Erther part therofmay alfo be proned without drawyng of a parallel line without the triangle, only chaunging the order ofthe thinges re- quired or conclufions. For Buclide firftproueth that the outwarde angle of atri. angle(one ofhis fides beyng-produced)is equallto the two inwardeand oppofite angles: and by. that he proueth thetecond party namely ;that the 3.inward angles ofatriangleare equall totworight angles, But here it is contrariwife: For firft is prouedthat the threeinwardangles ofa triangle are equall to two right angles, and by that is proued the other part ofthe Theoreme, namely, that one fide ofa triagle beyng produced, theoutwardangleis equal to the two inward and oppo- fite angles,And that afrerthis maner, Suppofe thatthere bea triangle 4BC,and producethe fide BC to the point E-And take in the line B Ca point at alaunentures which let be F:: & draw aline from 4 to F.And by the point F drawe A D vnto the line _4 Ba parallelline (by the former pro- ' pofition) which let be F D, Now forafmuch as F Dis a paralleltvnto 4 Band ypon them falleth the right line A F,and alfo the right line BC, therfore the alter- nate angles are equall,and alfothe outward angle is equall to the inward angle, Wherefore the whole an- J ele AF Cis equallto the angles FAB and ABF.And by the fame reafon(ifby the point wedrawaparal-"" 3 E C z lel line to the line 4 C) may we proue that the angle AF B is equalito the angles FA Cyand 4 C F:Wherfore the two angles 4 FB& AFC are equall to the three angles of the triangle 42 C.Butthetwoangles AFB & AFC are(by the 13.propofition )equallto two right angles, Wherforealfo the three angles of thetriangle 4 B Care equalltotworight angles, But theangles 4C F and¢4C Earealfo (by the 13. propofition) equall to two right angles. Take away the angle 4 C F which is common, wherfore the angle remai- ning,namely,the outward angle 4 C E 1s equall to the two angles remaining, namely, tothe ove inwarde and oppofite angles 48 CandC 4B: which was required to be proued, . Eudemus affirmeth,that the latter part ofthis Theoreme, The three angles of a triangle are equall to tWoright angles,was firft foundout by Pithagoras,whofe demon: {tration thereof is thus, Suppofe that there bea triangle 4 B C:and by the point 4, draw (by the former propofition)vnto the line B C,a parallel line, which let be D E.And forafmuch as the rightlines B Cand D Eare parallelsjand vpon them falleth the right lines 4 B and 4C,therefore(by the 29. propofiti- D A —— & on) the alternate angles are equall. Wherefore the angle D A Bis equalltothe angle 4.BC,and theangle E 4Cto the angled CB, Adde the angle B_.4 C common-Wherefore the angles D 4B,BAC,C AE, thatis,theanglesD AB andB B , Cc A E,namely,two angles equal to two right angles,are equal to the thre angles of the trianglee 4 B C. Wherfore the thre angles of a triangle are ¢e- quall to two rightangles;which was required to be proued. The conuerfe ofthis propofition is thus, Mj. If An other de- monstratiows The latter part of this Theo - reme fir? found ont by l’sthago- YAS, T he demonftra- tion thereof af= ter him, T he fir/t Booke If the outvyard angle of a triangle be equall-te ther woimward angles oppofite againft stzone of the fides of the triangle is produced,and the line without the triangle, is drawen direttly to the fide of the triangle, maketh one right line- with st.And if the thre smWward angles of a rettiline fi igure be equal to two right angles the fame rectsliue figure is a triangle. The conuer fe of this proposition. Demonftration Suppofe that there be atriangle 4BC-and letthe outward angle AC D be equal to of the fir part the two inward & oppofite angles 4B Cand C.48.Then ofthe conwerfée ¥ fay that the fide BCis produced to the poynt D. And A thar BC Dis one rightline,For forafmuch as the angle ACD is equal to the two inward & oppofite angles ,adde the angle _4 CB common, Wherefore the angles CD and _4C B areequal tothe three angles of the triangle 4 B C.But the three angles of the triangle .4 BC are equall to two right angles. Wherefore alfo the two angles ACD and 4C Barerqualltotwo rightangles. But if vnto' a right line,andto a pointin the fame line be drawen two’ B» ips? right lines,no: both on one and the fame fide, making the fide angles equal totwo right angles:thofe tworightlines {hal be drawé dire@ly, and make oneright line(by the 14, propofitton:,) Wherefore the right line BC is -dra- wen direétly to the line C D, and foisB¢ D onerightline:which was required to be sroued. Te 3: : ta Sane Agayne fuppofe that there be acertayneretilinefigure 4B C,haning onely three of thefecond ang’es, namely, at the pointes 4,8,C>which angleslet beequal to two right angles pert of the con- Then I fay that e4 BC isa triangle. Firft 4 C is oné right | ys werfe. line. Fordraw theline SD. And foraf{muth asateither | - disugs of the triangles 4B Dand DBC, the three angles are é- ? qual to two right angles,of which theangles at the paints e4,B,Cyare ecual to two right angles, Wherefore the an- gles remayning, namely, 4 D B and C DB aré equal to D two right angles. Wherefore( bythe 14:propofition ) the line’ DC is fet direétly to the line D 4, Wherefore the fide ACis onerigat line.And in like fort may we prone that the fidec4 Bis one right line,and alfo that thefide BC is 5 o one riglitline, Wherefore the figure ef 2 Cis. atrianele: which wasrequired to be proued. | By the fecod part ofthis 29.propofitio nam ely ,three angles of atriangle are equal to tworight angks, may eafely be knowen, tohow many right angles, theangles within any fizure hauing right lines andmany anglesare equall. Asare figures of fower angles of fiue angles,offixe angles,and fo confequently: and infinitly, And this is to be noced ,that euery rightlined figure is refolued into triangle, Evergrighth- Vor thata triangle is the firft ofall figures. Fortwo lines accomplith no figures ene nicerte V Vherfore how many fides the figure hath, into fo many triangles may it bere- Lae pee folued,fauing two,As ifthe figure haue fower fides, itis refolued into two trian- Atriangleis gles, if it haue fiue fides, into 3.triangles:if 6 fides into 4, triangles, and fo con- she firfofalf- {equently and infinitly. And it is proued that the three angles ofeuery triangle ee many aC Cqualltotwo right angles. V Vhercfore ifyou multiply the number of the trianglesafi- ttiangles,into which thefigure is refolued, by two, you fhall haue the num- gure maybere- ler of tightangles, to which theangles of the figureare equall, Sothe angles of “anni cuery qua irangled figureareequallto 4.rightangles,For itis compofed oftwo triangles, And the angles ofa fiue angled figureare equal to 6.rightangles, forit is compofed of three triangles and fo forth in like order. The redieft and apteft manerto reduce any retiline figure into triangles, is thus AC. errollary : of Enchdes Elementes. Fol.42. thus.From any oneangl¢ af thé figure ro euéry otherangle (ofthe ame), beyng oppofice ynto it; drawe aright line;fo fhall.you haue'all thetriangles of that fi- gure defcribed, : lnaquadragle, »: from:one angle you-can drawe but onelyne to theoppofite an gle,sby which it is deuided into two triangles only, Inapen- tagon figure, from oneangle youmay draw lines to two op pofitcangles, and fo you fhal | haue three triangles.In an Hex'agon,you may from one angle drawlinestothre oppofiteangles,and fo fhall you haue 4,triangles. Iman hepragon, ‘rom onc an- glé may be drawne lines to foure oppofiteangles,and fo fhal therebe fiue trians gle.Aand fo confequently ofthe reft,As you fee inthe figures here fer, This thing may alfo bethus expreffed. In any figure of many fides,thenum- ber of the angles of the figure doubled; is the niiber of the right angles to which the angles ofthe figure are equall fauing foure,As for example, : Let thete be an hexagon figure A B C D EF,and withinittakca poinratall auentures,namely,G, And draw from thefame point . ) to cuery one ofthe anglesa right line,&-fo thal there . 7 A be comprehended inthe figurefo many-triangles,:as | there are angles inthe fame. V Vhereforeby this 32, , propofition all the angles of thefe triangles takentos' C | B gether,are equall to double fo many right angles, as there beangles in the fizure, V Vherefore forafmuch \ J as thereare fixe triangles, there are twelue right ans : gles. Butall the angles atthe point G are equallto 4, E right angles by the 13,pcopofition, V Vhereforetake away foure out oftwelue,and there reft eight, V Vher fore thefixe angles in the Hexagonfigiire are equall rocightright angles, By that which hathnow benedeclared,it folowetl tharall the angles ofany fi- gure hauing many-fides;také together, are equal to rwife fo many light angles, asthe figure is inthe reaw or orderof figures,a triagle is the firtt figire in order, & his angles are equal totwo rightangles, which aretwife one,A quadrangle ts the fecond figure in onddei 7 heeianlils angles. are equal ro fowet ght angles, which are wile two, The order offigures is gathered-of the fides, For if you take rwo fromi the number of the fides ota figure,the number of the fidesremayning, is theaumber of the order of the figure. As ify ouwill know,how miny inorder 1s. a figure offixe fides: from fix(which.is the number of his fides) take away tWOy and there will remaine foure, V Vherforea figure of fixe fidesis the burth figure , one M.i1, in An other way to: know the ni- ber of right An- gles Gro which the angles of a- ny figure are é- qual. Another exprefs fion of the for- mer Corollary . A triangle the jirft figure in order. AA quadrangle the fecoxd, and fo confequent ly, How the order* of ficures Pty fa~ thered, . Ses einnh Dn —e_ = Snape = - - - == =. er ee SE ro = 5 Ses —— ao a = = SS = = al ~ ; — - - ~ ti " ve x Im . denials a peed . ; f ee Pe — —--= ee — — it ——— oer re eel — Se ~ === a el rt = = _ - = . ~ _ - => SSS SSS ee “ - ~~ n 7 rr ta ’ i . - 2 ‘ os a ‘. pores = SSS — =¥ Py eM. Sa net - ee i >? : ; ey = - & ery 2S - a a Qa—re wh other C. Ofr~ Tolar}. Ans other Cats rellary. ‘hn ather Core volar}. An ether Core vallary, ThefrftBooke inthe order of figures, Then double foure;fo fhall you haue eight, V Vhereforé the angles therof are equall ro eight right anigles,And fo ofall other figures, Hereby alfo it is manifeft,chacthe outward angles of any figureof many fides taken together,are equall to foure right angles.For the inwarde angles: together with the outward angles,are equall to twife fo many right angles,as there bean gles in the figure(by che 13,propofition) But the inward angles areequalto rwife {fo many rightangles, asthere be angles in the figure, fauyng foure: as it was be- fore declared, V V herfore the outward angles are always equal to foure right ans gles, As for example, | Suppofethat there be a pentagon,A B C DE, And produce the fiue fides ther- ofco the points F,G,H,K ,L,.Now (by the1x3, | propofitid the two angles at the point A fhall be equalto two rightangles .and(by the fame) the two anglesat the pointe B fhall be alfo e. quall totwo rightangles, And fotaking cuery cwo angles they fhall be in all equall to tenne rightangles. V Vherfore rakyng away the in- ward angles , whiche(as hatlybetore bene pro- ucd) are equalleo fixe righteangles, the, out~ wardangles fhall be equallto fower right. ans gles,And {9 ofall other figutes, Ki Itisalfo manifeft,chat every pentagd,which is fo defcribed,that ech fidetherof deuideth two of the ocher fides, hath his fiue angles equall to two right angles, For fuppofe that ABCDE: be fuch a pentagon asis there required fo that let the fide AC cut the fide B E in the point G:& let the fide AD, cut A the fame fide B & in the peint F. Now thé by this propo- fition the angle A F G fhalbe equall tothe two angles at G/ \F the point B and D:namely,the outward angletothetwo 8 inward and oppofite angles.And by the fame reafon the angle F G Ais equal to the angles at the points Cand E which arein the triangle CE G.But the two angles AFG and FG A,together with the angle at the point 4, are e- quail to two right angles( by this propofition ).Where« fore the fower angles at the pointes, B,C, D,E, together with the angle at the point 4,areequaltotworight an- 9 ¢ gles:which was required to be proued, Dp By this propofition alfo itis manifelt,thac cucry angle ofan equiface trian= gic is two chird partes ofa right angle,And that in a triangle of two equall fides auing aright angle at checoppe,etther ofthe two angles at the bafe is the halte ofarightangle. And ina triangle called Scalenum,fuch a Scalenir( I fay ) which is made by the drawght ofa perpendicular line from any one ofthe angles of an equilater triagleto the oppofice fide therof,oneangle isa tight angle,an other is two third parts ofa right angle,namely ,chat angle which wasalfoan angleof the equilater triangle, wherfore of neceflity the angleremaining ts one third part of a rightangle.For the three angles ofa tridgte muft be equalt to two rightangles, Morcouet by this propofition ic is manifeft chat 1fthere be two triangles, and iftwo angles of the one be equal to rwo angles of the other:the angle remat« ‘ping we ye of Euchdes Elements. Fol.43. ring fhall alfo be equall coche angle remayning,F or forafitiuchas three angles ofany triangle are equal tothree angles ofany other triangle(forthacin ech the three angles are equal so two rightangles): Iffrom ech triangle be taken. away the cwo equall angles,the angle remay ning fhall (by the 3, common fentence)be equrall co the angle remaynimg. And here I thinke 11 good ro fhew how to dewide arightangte into three e- quall partes, forthae che demonftrasion thereof dependeth.ofthis propofition, i a Suppofe thatthere be aright angle 4BC,contayned of the right lines AB and BC2& wow rs dewiden intheline BC,takea pointatall aduentures,which let be | vight angle into D.Anw vypon the line BD deferibe( by the firkt)an equila- A three equall ter triangle BD E,And(by the 9.propofition)deuide the partes, angle.D & Eintotwo equall partes by theright line BF. Then / fay that the right angle_4 B-Cisdeuided into thre: equal parts bythe right lines B E and B F.For forafmuch as E B Dis an equilater triangle,therfore as hath before bene declared, the angle € B Dis two thirde partes of a rightangle. But the wholeangle 43 C,.is.a rightangle.. Wherfote the angle remaining,namely, ABE is one third partofa right angle.Again forafmuch as the angle EBD, is two third partes of a right angle and itis deurded into twoequall-parts by therighe ine BF thereforeeither of thefetwo angles EB F & FB Disonethird part of aright angie.Wherefore the three angles 4B €,2B Fand FB Dare equall'the onetothe o- ther. Wherefore theright angle 42C is deuided into'three-equall partes by theright lines BE and B F:which was required to be done. The23,.T heoreme. The33.Propofition. Tro rightlines toyning together on one and the fame fide, two equall parallel lines:arealfothem felues ezuall the. one to the other,and aljo parallels. 55 V ppofe that A 'B and €D: be S2| right lines equal, and parallels: <\and let thefe two rightlines A ='C and BD ioyne thé togerber, the one on the one fide,and the other ony other fide.Then I fay that the lines AC er B D are both equall,cx alfo parallels. Draw (by the first pettcion) aright line from the point Bro the point C, And fore afmuchas A Bis a parallelto€ D, and’ bpothem falleth the right line BC, there 2 inca ane fore'the alternateangles A BC and BC | Dareequall the one to the other (by the 29. ah porns er forafmuch as the line A Bis equall to the lineC D,and the line BC is commonto them both, M Jit theres i ——— —— —-~ — -- Fai 2: - 7 in wee = = = . = Es = ——- iia ——- —— : =: —— —= 1) ee 2 ti q ; —— E ~ —— nas - Or 6 ee SS SS a — ~ == - = SO EO SS or ~~ -_ - - -_—_— —— a cee etn a — == SS — SaaS SSS ’ Pore — + ~— = = 3 = — = = = ~- YS a SS oe ee eee ———— - — SS SS ee es, ee ——— “a = os a a —— — = 2 >- F S362 4 a — ~ : a = 9 ~ + oe. : .< “ot - . - - . _ — . e *, ca 4 line_4 C,may make an angle equal to the angle ge K | nen E.Now then ypon ete line e7C and tb ihe i Conftru sem, pointinite4,make anangle equal! to theangle E geué E(-by the 23,propofition), which let be C4 F.And produce the line F Aon the otherfideof ¢ pe Se the point Ato the point G:and let_4 G be equall to the line geuen D ( by the 3. propofition ), And | by the point G,draw (by the 3 1.propofition) a parallel line to the line ef C, which let be G Hand produce it vntil it concurre with the line 4 B: which concurfelet bein the point 7. And agayne by the point A draw the line HK parallel vnto the line GF:which let cut the lince #C in the point X, Then I fay that the line H XK is placed betwene the lines 4B & AC & is equall to the line D.And that the angle at the point K is equall to the angle geuen E, For forafmuch as(by conftru@iion)_4G HX isa parallelogramme 2emonfraston the line K A is equall to the line _4G(by this propofition ), Wherefore alfo it is equall tothe line D, And forafmuch as the line 4 Kfalleth vppon the two parallel lines, FG and K H,therfore the angle 4 K His equal to the angle F .A K(by the 29. propofitid) for that they are alternate angles. Wherfore alfo the Eos angle at the point K is equal to theangle geuen &.Wherefore the line A K being placed betwene the two lines 4B and C,and being equall to the line D, maketh the angle at the point K equall to the angle geuen E: which was required to be done. Though this addition of Pelitarius be not (o muche pertayning to the N.i, propo DewsonPration. Three cafes in this propofition, The firft cafe. TL hefrft Booke propofition:yerbecauleicis witty and femechfomewhat difficult, I choughr it good here to anexe it. The25.T heoreme.. The 35. ropofition. ‘Parallelogrammes confifling uppon one and the fame bafe, and in the felfe fame parallel lines, areequall the one to the other. G Vppofe that thefe paralleloe _ x grammes ABCD and EB 4 CF do éonfift vpon one and |) === the fame bafe,that is, vppon BC,and mn the felfe fame parallel lines, thatis AF,and BC.Then I fay, that the parallelograme ABCD isequal tothe parallelograme EBC F.For forafmuch © asA BC Disa parallelogramme, ther fore (by the 3 4.propofition ) the fide A Dis equallto the fide BC, and by the fame reafonalfo the fide EF isequallto {/_ the fide BC, wherfore A Dis equallto .B c E FandD Eis common to them both. | : VV berfore the whole line A E is equall tothe whole lineD F_And the fide A'B is equall toy fide D Cwherfore thefe two EA and ABare equall té thefe to F'D and DC, the one to the other:and 9 angle FDC ts equal to the angle EAB namely the outward angle toy im Ward angle( by 5 29,propo/itio): wherfore( by } 4. propofition )the bafe EB ts equall to.the Eafe FC, and the triangle EA Bis equallto thetriangle FD C.Take away the triangle DG E, which is common to them both,VV herefore the refidue,namely,the trapefium ABG Dis equall to the refidue, that is,to the trapefium.EG CF, Pat the triangleG BC commo to them both VV berefore the whole parallelogramme A BC Dis equalltothe whole parallelogramme E BCEVV, herefore parallelo grammes confifting vps on one and the fame bafeyandin the felfe fame parallel ii nes,are equal the one.to the otber:which was required to be demonjtrated. | ; Parallelogrammes ate {ayde to be in the {elfe fame parallel lines, when therr bafes,andthe oppofite fides vatochcm, are onecand the felfe fame lynes wyth the parallels, | 3 Inthis propofition are three cafes, ForthelineBE may cutte the birresilh F, either beyond the point D,or1n the point D,or on this fide the point D , When oe ae + _- of Euchides Elementes. Fol.4.6. itcutteth the line A F beyond the point Dthe demonftration before put fers ueth, ButifthelineBEdo cutte the line 4Fin - Sed The fecond cafe. the point D,then forafmuch as ( by the former ropofition )the triangle B CDor BC Eisthe hal e of either of thefe para‘lelogrammes 4 BC Dand EBCF ( for inthe parallelogramme _4 BCD the diameter BD maketh the triangle B DC the halfe of the fame parallelogramme,and in the parallelogramme E BCF the diameter / ECorDC maketh the felfe fame triangle BD Cthe halfe of the parallelograme EBC F )ther- / fore(by the 7.common fentence )the parallelo- grammes 4 BC DandE2CF areequall. / | > th” Gy ‘3 Butif theliue BE do cutte the lynee#R > The shird cafe. on this fide the point.D, then forafmuchias ey- ther of the lines AD and E Fis equall to the line B C,therefore by the firft common fentence they are equall the one to the other, Whercfore taking away ED, which is'‘commonto both, the refidue ef E fhalbeequall to the refidue:D F. Agayne forafmuch as (by the 3 4.propofiti6) thelinee 4B is equall tothe line CD}; and. (by the 27. propofition)the angle E 4 Bis equalto the angle F DC: therfore (by the 4. propofi- tion) the triangles EB AandFCD are equal, Adde the trapefium C D E B common tothem both : and fo (by the fecondé common fen- tence )the two parallelogrammese7B CD and EBC F thalbe equall; which was required. to be proued. bo) Q) > wy oO xy 5 Cc Se The26.Theoreme. The36.Propofition. Parallelogrammes conjifting upon equal bafes, and in the Selfe Jame parallel lines are equall the one to the other, RG V ppofe that thefe parallelogrammes. ABC Dand EFG Hi do confift BSS" ypon equall ba/es chat is, ypon BCand F G,and in the Selfe fame paral- =H! Jel lines that is,A Hand BG, ThenI fay,that the parallelogramme A BCD is equall to the parallelogramme EF G H.Drawa right line from the Corfe point B to the point E,and an other from the point C to the point FZ, And fore Demonfrarien afmuch as B C is equall to G,but F Gis equall to E A, therfore BCalfo ise quallto B Fd, and they are parallel lines, and the linesB E andC A joyne them together. but two right lynes ioynyng together two equall right NN}. lines Three cafes rr this propojition, The firft cafe. Exery cafe may happen feuen diuers WAyes» Ze Lhe frft Booke. 3. lines being parallels, are themfelues | alfo (by the 32 propofition) equall A D E the one tothe other, and parallels. VV berforeE BC Hts a parallelos.,.. gramme and is equall tothe parallez lograme ABCD: for they haue both © one andy fame bafe,that 1s,BC.And._. are iny felfe fame parallel lines that is, BC ¢» EH, And by) fame reafon- | alfo the parallelograme EFG His ex..4.-. J qual to the parallelograme EBCH; VV berforethe parallelograme AB We Pi CDisequal to the parallelogrameE B Cc F G FG H.VV berfore parallelogrames.. 22:1 © < confifting vppon equall bafes,and in the felfe fame parrallel lines, are equall the one tothe other: which was required to be proued, Ta this propofirion alfo are three cafes,Forthe equall bafes may either. be ve~ terly feperatedafonder: or they mayitouche.at one of theendes: or they may haue one part common to them both, ?'o°™ 1S ¥G)eR E i9ien Euclides demonftration ferueth when the bafesbe ytterly feperated a fonder, V Vhich yet may happea feuen diuets wayes,For che bafes being feperated af6- der,their oppofite fides alfo may be veterly feperated a fonder beyond the point D,as the fides A D and E H inthe firlt figure, | ; Orthey may touche together in one ofthe endes, andthe whole fide may. be beyond the point D,as the fides A DandE H do in the fecond figure, Or one part may be beyond the point Dj andan other part commonto them A A Eovw¥# ee ee ee. ~ of Enclides Elementes. Fol. 4.7. both ,asin the third figure,the fides A D and E Hhauethe part ED common to them both, | -----~ 2" es Or they may iuftly agree the one with theorher,thari s,the pointes A andD 4, may fall vponthe poinces Eand H: as in the fourth figure. Or the fide A D being produced on this fide the poine A,part ofthe oppofite 5. fide vnto the bafeF G may be on this fidethe point A,andan other pate may be common with the lineA D,as inthe fifth figure, | Or oneende ofthe fideE H may light vpon the pointe A, and the whole fide 6. on this fide ofit: As in thefixt figure, © | : Or the faidfide E Hmay vtterly be feperateda fonder onthis fide the pointe _ A,as in the feuenth figure, AnaneSe? 5 8 A tare The lske Garse- And the two othe? cafes'alfo may inlike maner havé {elien varieties: as in ty in ech of the the figures here yndernerh and on the other fide of this leafe fetitismanifeft. °*ertmocafes, : : . | : ‘ i Exclides con- and here is to be noted,thac in thefethree cafes and imall their varieties alfo,the frudion sail conftrucion & demonftration put by Euclide( namely the drawingoflines fd ‘demonfration the point B tothe point E gz from the pointe C to the point Hand fo prouing / ATel line CF.VVberefore EB C A. and D B( Fare parallelogrammes, And the parallelogramme E B( A, E A D is (bythe 35. propofition) equal to the parallelogramme DBCF, For they confist vppon one and the felfe fame bafe, namely, BC, and are in the felfe fame parallel lines, that is, B (and E F, But the triangle ABC is( by the 3 4. propofitio )the balfe of the parallelogramme E BCA, for \ the diameter AB deuideth it into two equall parts:¢o(by thefame)the trie B C angle D B Cis the halfe of the pae rallelogramme D BC F, for the diameter D (deuideth it into two equall parts: but the baines of thinges equallare alfo equalltheoneto the other ( by the 7, common fentence ), wherefore ebe triangle A Bis equall to the triancle DB (VV bevefore triangles confisting vpon one and the felfe fame bafe,and in the Jelfe fame parallelszare equall the one to the other: which was required to be demonstrated. | Conftrudion, Thofe of Euilides Elementes. Fol.4.8. Thofe triangles are faide tobe contay ned within the (elfe (ame parallel lines, which having theitbafes in one ofthe parallel lines haue their coppes in the other. “Hereas I promifed will I thew out of procius the compartfon of two trian- gles, which hauing their fides equall ,hane rhe bafes and angles atthe toppes vues quall, Andfirft I fay thatchevoequall angles at the roppe being equall to two right angles the criangles fhalbe equall.As for example, ‘Suppofe that thefe twotrianzles.48 Cand DE F hauetwo fides of the one,name- ly, ABand AC,equall to two fides of the other, namely, to.D Eand DF, echeto his correfpondent fide;thatis, 4B to DE,and.4CtoD F,andletthe bale BChe greater then the bafe EF : and let the angle atthe pointe be greater then the angle at the SPO the fayde angles atthe pointes AandD, A -f,- fois age A gat | t ¢ equall to two right angles. Then I fay that the triangles 4ABCandD E Fare equall.For forafmuch as the angle B A Cis greater then the angle E D F,vpon the line E D,and to the point D defcribe (by the 13. propofizion) an angle equall to the angle Bd C,whichlet beED G:and purthe line D G equalltothe line AC: and draw a line from Eto G,and an other from F toG: and produce the lines ED & F D beyond the poynt D to the pointes Hand K.Now for a{much as the angle B ACis equall to the angle ED G, and the angles BA Cand ED F are equall totworight an- gles,therefore the angles EDGand EDF are equall to tworightangles.But the angles EDGand K DG, arealfo equal to two right angles:take avay the angle F DG com- mon to them both: wherefore the angle remayning EDF Bak is equali to the angleremayning G D.K.Butthe angle ED Fis equall tothe angleH DK (by thea § -propofition)for Es they are hed angles. Wherefore the angle GD K is equall | totheangle H D X.Andforafmuch asin the triangle G D F the outward angle GDH is(by the 32.propofition) equal to the two ts ith oppofite angles st the points G and F:whichtwo.anglesalfo are(by the 5-propofition) equall the one tothe other: for the line D Gis by conftruStion equall to the line <4 C, namely, to theline DP, Wherefore the angle GD H is double both to the angle at the pointG,and to the ane gle atthe point F,Butthe angle GD His alfo double tothe angleG DK (for the an- gi¢G D Kis prouedto be equallto the angle K DH) wherefore the angle at DGF isequall.to.the angle G DK: and they are alternate angles, Wherefore (by the 27, propolition) theline D Eis parallel ro the line F G- Wherefore the triangles GDE and FD Eare vppononeand thefelfe fame bale, name! y, DE, and in the felfe fame paralicHlinesD Eand GF.Wherefore by this propofition they are equall. But the tri- angle G D EisbyconfiruGion equall to the triangle 4 BC.Wherefore alfo the tri- angle D E Fis equallto the triangle e4 BC : which was required to be proued. But now let the angles B 4Cand ED F be greater then two rightangles: & let the angle at the point 4 be greater then the angle at the point D,asit was before. Thé Alay thatthe triangle 4B Cis letiethen the triangle D E F. Let the fame coufiuGion behere that was in the former. And forafmuch as the angles B AC and éD F, that is, theangles E DG and ED Fare greater then: two right angles, but the angles E DG and G D Kare equallto two rightangles: takeaway the angle FD G which is common to them both. Wherefore the angle remayning,namely,EDF is greater thé the angle remayning,namely,then GD K: thatis,the angle K ‘DA which by thers, propofition is equall tothe angle ED F jis greater then the angle G DK, whereforethe angle GD H 1s more then donhletothe angle GD Ky burthe angle GD His doubleto the an- “ eae N.U). gle How triangles are ayde tobe ' sathe felfe fase parallel lines, Comparsfon of two triangles whofe fides be- tng equal, thesr bafesand angles at the toppe are Gneguall, When the twe angles at the toppes are canal to tte right An~ gies. When they are greater the twe rig ht angtes. When they are deffe then twe right angles. T he first Booke gle DG F, as was before proued. Wherefore the + angle GD Kis leffe then the angle DGF. Vnto the | angle G D K put(by the 23.propofition) the an- 293 gle DG Lequall: and produce thelineG L tillit concurre with theline E F inthe pointe Z. And. B draw aline from ‘Dto ZL. Wherefore(by the 27, propofition)G L isa parallel line to D £,for that H K the alternate angles DG Land G D Kare equal. , Wherfore the triangles GD Eand L D Eare (by this propofition Jequal(for they confift vpon one and the felf fame bafe,namely, DE, and are in the S felfe fame parallel lines,namely,é D andG Z)But D the triangle Z D E is leffe then the triangle FDE, Wherfore alfo the triangle G D E is lefle then;the triangle F D E.Burthe triangle GD Eis equal to the triangle 4 BC. Wherfore the triangle ABC is leffe then the triangle ‘D E F; which was requi* E red to be proued. an e But now Iet the angles B A Cand EDF beleffe A then two right angles: and agayne let the angle at the pointe.4 be greater then the angle at the G point D.Then / fay that the triagle 4 B Cis grea- ter then the triangle D E F.Letthe fame conftra- Ke &i6 be alfo here that wasin the twoformer,And © ) forafmuch asthe angles B 4C and E D F,thatis, the angles EDG & ED F,areleffe then two right angles,but the angles EDG and GD Kare equal to two right angles, take away the angle FDG which is common to them both, wherefore the angle remayning,namely, ED Fis leffe then the angle perpayung amely,then G D X:thatis,the angle H D K(which by the 15. propofition is e- qualiltotheangleEDF)islefflethentheangle G ¢ F D K.Wherfore the whole angle G D His lefle then double to the angle G DK, But itis double to the angle D G F(as before it was proued ): wherfore the angle GD Kis grea- ter then the angle D GF. Put theangle D G Lequall tothe angle GD K (by the 23. - propofition Jand produce the line G Z till it concurre with the line € F alfo produced, & let the concurfe bein the point L.And draw aline from Dto Z.And for as much as the angle DG Lis equallto the angle G D K, and they arealternate angles, therefore the line G Lis aparallel to D E(by the 27. propofition ).Wherefore(by this propofiti- on )the triangles G D Eand LD Eare equal: butthe triangle L D Eis greater then the triangle F DE,andthetriangle GD E is equall tothetriangle_4 BC, Wherefore the triangle e-f BC is greater then the triangle D E F: which was required to be proued. The28.Theoreme. The38-Propofstions Triangles which confi/t vppon equall bafes, and in thefelfe Jame parallel lines are equall the one to the other. i An V ppafe that thefe tr tangles A BCand D EFdo confist vponequal kee bo es that is,vpon BC and E F andinthe felfe fame parallel lines thatis ee BF and A D.Then I fay that the triangle AB Cis equall to the trians 3 gle i a a of Euchdes Elementes. Fol.49. gle ABC is equallto the triangle D E EF. Produce (Ly thefecond peticion) the ConPrudion, line AD on eche /ide to the pointes G and H, And the pot B drawe vnto C Aa parale lelline BG,and( by the fame) by the pointe F drawe vnto VE a parallel line F H. VV berforeG BC A and DEF Hare parallelogrammes.But the parallelograme GBCA is (by the 3 6 propofrtion )equal to the parallee logramme D EF H., for they confist vponequall bafes that is,BCand E F, andareinthe felfe fame parallel lines thatis,BF andG H, But (by * the 34.propofition ) the triangle AB Cis the balfe of the parallelogramme G BCA, for the diameter A B deuis deth it into two equal partes:and the triangle D E F is( by the fame )the halfe of the parallelogramme D EF Fi, for the diameter F D denideth it into two ee quall partes,But the halues of thinges equall are (4y the 7,common fentence )e» quall the one to the other.VV berfore the triangle AB Cis equall tothe triane gleD EF. VV berefore triangles which confist vppon equall bafes, and in the felfe fame parallel lines are equall the one to the other: which was required to be proued. In this propofitton are three cafes. For the bafes ofthe triangles either haue One partcommon to them both or the bafe ofthe one toucheth the bale of the other onely ina point: or theirbafes are ytterly feuereda funder, And ech of thefe cafes may alfobe diuerfly as we before haue fenein parallelogrammes con fifting on equall bafes,and being inthe felfe game parallellines,So that he which diligently noteth the variety that was chzre put touching them, may alfo eafely frame the fame varietie to ech cafe in this propofition, V Vherefore I thinke it nedeles hereto repeate the {ame agayne: for how foeuer the bafes be put, orthe toppes,the manner of conftruion and démonftration here put by Euclide will ferue:namely ,to:draw parallel lines to'the fides. : (by the 3 1.propofition) by pe. \ . H \ “An addition of Pelitarius, T 0 deuidea triangle geuen into tWo equall partes. Suppofe that the triangle geuen to be deuided in totwo , equall partes be 42C.Deuide one ofthe fides therof; / \ namely,B C into two cquall partes (by the 10. propo- | fition jin the point D, And draw a line fromthe point D to the point 4,Thé I fay that the two triangles ABD ‘& ACD,areequal;whichiseafy to proue( by the 38. pro- pofition)if by the point_Awe drawevnto'the line Ca paral'elline (by the 3 1.propofition ),which let by H K: for fo the triangles 4B D and WDC, confifting vppon equal bafes BD & DC,and being in the felfe fame paral- lellines /7 Kand BC are of neceffitie equall.The felfe 6B O.1. fame Demonstration Thre cafes in this propofitsoxs, Ech of thefe Ca- fes alfo may be asuerfly, 4s addition of Velstarius,to deusde atriat- gle into te e- gual partes. Note. ets other addi- floes of Pelsta- C4868, ContiruFson, Demonfrration T he firft Booke fame thing alfo wil happenif the fide B 4 be deuided into two equall parts in the point E,and fo be drawen a rightline from the point £,to the point C. Orifthe fide 4C be deuided into two equall partes in the point F,and fo be drawen aright line from the point F tothe point B: which is in like manner proued by drawing parallel lines by the pointes B,andC, to the lines B 4 and AC, And fo by this you may deuideany triangle into fo many partes asare fig- nified by any number that is euenly euen; as into14.,16.32.6448Cc. Another addition of Pclitarius, From any point geuen in one of the fides of atriangle,to draw a line which fhal dentde the trian- gle snto two equail partes. Let the triangle geuen be BC D:and let the point genen in the fide BC be J. Itis required from the point 4 to draw aline which thal deuide the triangle B CD into two equall partes. Deuide the fide B C into two@quall partes in the point £. And drawea right line from the point 4 to the point D.And(by the 3 1.propofition ) by the point E draw vnto theline AD a parallelline EF : which letcutte the fide D C inthe point F,And draw aline from the point 4to the point £, Then I fay that the linee 4 F denideth the triangle B C D into two equall partes: namely, the trapeftum A B D Fisequallto the triangle 4 C F.For draw aline from Eto D,cuttingtheline AF inthe pointG.Nowthenit is manifeft ( by the 38 ‘Seipoktion that the two trian- glesBEDandCé D are equall(ifwevnderttand aline to be drawen by the point D parallel to the line B C,for the bafes B Eand E Careequal).The twotriangles alfo DEF and AEF are (by the 37.propofition) equall:for they confift ypon one and the felfe fame bafe E F, andare in the felfe fame parallel lines 4 D and E F. Wherefore taking away the triangle EF G whichis com6 to thé both, the triangle 4 E G fhalbe equall to the triangle DFG swher fore ynto either of thé adde the trapefitt C F G E,and the triangle ACF fhalbe equal to the triangle DEC, But the triangle ‘D E Cis the halfe part of the whole triangle BCD whereforethe triangle _4C Fis the halfe part of the fame triangle B (‘D.Wherfore the refidue,namely,the trapefium 4B F D isthe other halfe of the fame triangle. Where- fore the line A F deuideth the wholetmangle BC D into two equall partes: which was required to bedone. SThe29.Theoreme. The 39.Propofition. Equal triangles confifting ypon one and the fame bafe, and on one and the fame fide: are alfo inthe felfe fame parallel lines. eV ppofe that thefe two equall triangles A BC andD BCdoconfift vp. pon oneand the fame bafe,namely, B Cand on one andthe fame fide. The] fay that they are in the felfe ame parallel lines. Drawe a right line from the point A tothe point DNow I fay that ADisa parallel line to B C. For tfnot , then (by the 31. propofition) by the point A drawe vuto the right line B Ca parallet line A E,and draw aright line from the point Eto the point ee ee aa: " ' A) y _ thatir falleth not within. Wherfore the line.4 Disa pa- of Euclides Elementes. Fol.so. point C, VVberfore} triangle E B Cis (by 5 37,propofitio equal to the triangle ABC, for they confift vpon one and the RR D Jelfe Jame bafe namely, BC and are iny ae felfe fame parallels, thatis, AE and B oN C, But the triangle D BC is (by fuppos fition ) equall to the triangle ABC, C VV berfore the triangle DB Cis equal to the triangle E BC, the greater ynto the le/Se. which is impofsible. VV heree foretheline AE, isnot aparallelto | the line B C,And inlike forte may it be proned that no other line befides AD is =iSaio aparallellinetoBC, wherefore ADis 5 a parallel line toBC.VV berfore equall triangles confifting vpon one and the fame bafe, and on one and the fame fide, are alfo inthe felfe fame parallel lines:which was required to be proned, © This propofition is the conuerfe of the 37,propofition,And here isto be noted thatifby the point A,you draw ynto the line BC a parallel line,the fame thal of nece(fitie either light vp6 the point D,or vader it,or aboue it, If it light vpé it, then rs that manifeft which is required: but ifit light ynder it,chen foloweth that ab{urditiewhich Euclide putteth namely, that the greater triangle is equall to the lefle: which felte fame abfurditiealfo will follow,ificfall abouethe point D. As for example, Suppofe that thefe equall triangles 4B Cand D B Cdoconfift yppon one and the felfe {ame bafe B C,and on one and the fame fide. Then J fay,that they arein the felfe fame parallel lines,and that a right line ioyning together their toppes is a parallel to the bafe B C.Draw aright linefro_4 to D.Now ifthis be nota parallel tothe bafeBC,let 4E be aparallel yntoit, andlet 4 € fall without theline.4D, And produce the line B D tillit concurre with the line 4 Ein the pointe E and draw aline from E to C.Wherfore the triangle e4 B Cis equal to thetriangle E B C:but the triangle 4 B Cis equall to the triangle D BC:Wherfore the triangle € BC is equall to the triangle D BC. Namely.the whole to the part:which is impoflible.Wherfore the parallel line fal- leth not without the line 4 D.And Euclide hath proued rallel ynto the line B C.Wherfore equalktriangles which are on the felfe fame fide, and on one and the felfe fame bafe,are alfo in the felfe {ame parallel lines: which was required to be proued. An addition of Fluffates, This Theoreme the conuerfe of the 37 -prepefi- 610% The felfe fame alfo followeth in parallelogrames.Forifvponthebafe ABbe sn addition of fet on one & thefame fide thefe equal parallelogrames ABCD & ABG E,they éafares. fhall of receffitie be in the felfe fame parallel lines,For ifnot,but one of them is O,i1, fer ail a triangle haue one ¢> the felfe fame »»\bafe; and bein the felfe fame parallel lines : the parallelo- grame fhalbe double to the triangle. \ ©, , } OSA V ppofe that the parallélos ¢ 7 NJ crame ABC Dand the trie 0 oa. Di | y pointe CVV berfore (by the 37. propos Oui, ition . Confirudion, -. Demonffration leading to an ablurdstie, This propofition $3 the conuerfe of the 3 8, Dre pofitses. Demon rations T he firSt Booke fition) the triangle ABCis equallto E£ the triangle EBC; for they are vppon one and the felfe fame bafe BC, andin the felfe Jame parallel lines BC and E A: but the parallelograme ABCD ts double to the triangle ABC( by the 34, propo/ition )for the diameter thereof 4 C denideth it into two equal parts:wher fore the parrallelogramme ABC D is double to the triangle EB Cif therfore a parallelogramme and a triangle bane c B : | oneandthe felfe fame bafe, and be in the felfe fame parallels , the parallelo- | grame fhallbe double to the triangle: hich was required to be proued. | she aa a This propofition hath two cafes For the bafe beyng one,the triangle may ebeprepenr” ane his toppe withoutthe parallelograme, or within, The firftcafeis demon- {trated of the author, The fecond cafe is thus, Suppofe that there be a parallelocrame ABC D,andiet ¥_ & the triangle be EC D, either of which let haue oneand the felfe fame bafe, namely, CD, andlet thembe in the felfe {ame parallel linesC D and A B, and let the toppe of the triangle & C D,namely,the point £,be within the paralle- lograme A BC D,Then / fay that the parallelograme 4B C ‘Dis double to the triangle ECD. Draw aright line frd the point 4 to the point D.Now forafmuchas the paral- lelograme A BC Dis double to the triangle .4¢ D (by the 34-propofition)- and the triangle e4 DC is equall tothe \ triangle € D C( by the 37.propofition ). Therfore the pa- rallelogramme 4B C Dis double to the triangle €C D: C D which was required to be proued. > A corollary. By this propoficion it is manifett chat if the bafe be doubled, the triangle e- rected vppon it fhalbe equall to the parallelogramme. The [rife fame And ifthe bafes ofthe triangle and of the parallelogramme be equall, the —— et felfe fame demonftration will ferueif you drawe the diameter of the paralleloe a ak = she tame, For the triangles being equal, the parallelogramme which is double to paraltclograme the one, fhal alfo bedouble co the other, And the triangles muft nedes be equall beSpomequall (by the 38. propofition for that their bafes are equal and for that they are in the Seite felfe fame parallel lines, The conuerfe of this propofitionis thus, If a parallelogramme anda treangle haue one and the felfe fame bafe,or equal bafes the oneto the other,and be defevibedon one andthe fame fide of the bafe : sf the parallelogramme be double to the triangle,they fhalbe in the Selfe fame parallel lines. Theconuerfe of _ For ifthey be not,the whole fhalbeequall to his parte, For then thet . shispropeftvem. ofthe triangle muft nedes fall either withinthe parallel lines or without, a whether | of Euclides Elementes. Fol.52. whether of both foeuer ir do, one and the felfe fame impoffibilitie will follow, ufby the coppe of the triangle be drawen vato the bale a parallel line. An orherconuerfe ofche Gme propofition, If a parallelogramme bet he double of 2 triangle, being both within the felfe fame parallel lines: Ax other con- then are they upon one and the felfe fame bafe,or vpon equall bafes.For ifn that cafetheirbas ¥er/< of rhe S fes fhould be vnequal, chen might ftraight way be proucd, that the whole is e- fame propefitite quall to his pare: which 1s impoffible, A trapefium hauing two fides onely parallel lines,is ey ther more then dou- Comparifon of a ble, or lefle then double to a triangle contay ned within the felfe fame parallel *riangle anda lines and hauing one and the felfe fame bafe with the trapefium,or table, Iuft the 8 SS dou ble itcannot be,for then ir fhould bea parallelogramme. A trapefium haz folfe feiee bus, uing two fides parallels hath ofneceflitie the oneofthem longer thenche other: <#d sx rhe felfe for if they were equa!l then fhould the other two fides enclofing them be allo e- a parallel quall (by the 33,propofition,) Ifthe greater fide of the trapefium be the bafeof “”“” the triangle,then fhal the crapefium be leffe then the double ofthe triangle And if the lefie fide of the trapefiumbe the bafe ofthe triangle then fhall the trapeti- um be greater then the triangle, er we a: —— = = rr nes ee For fuppofe that 4B C D bea trapefium,and let two fides thereof, namely, 4B and CD be parallel lines,and let the fide 4B be leffe then the fide C D, & produce the fide 4B infinitlye on the fide Bto the point F,And let the triangle EC D haue one and the When the grea- ser fide of the trapefism 33 the bale of the sri angle, i - = i t | tf . ) Si ih = << Peas {elfe fame bafe with the trapefium, namely, the line CD,Then I fay that the trapefium 4 BC D is leffe thé the double of the triangle E CD.For put theline AF equall to the line C D ( by the 3.propofitid)and draw aline from D to F. Wherefore ACD Fisa parallelo- gramme (by the 33. propofition). Wherefore ( by | the 34 propofition) itis double to the triangle ECD. But the trapefium 4 B C Disa part of the parallelo- gramme AC D F.Wherefore the trapefium ABCD islefle then the double of the triangle ECD: which was tequited to beproued. ~~ See SS Fe Tae : a = . = SS er a ren ee er = . .* jaw re aa When the leffe [ie 68 the bafe, Agayne let the triangle haue to his bafe the fide A8.then I fay that the trapefium_4 BC Dis grea- ter then the double of the trianglee 4 E B.For from the fide CD cutof the line C F equall to the line_AB (by the 3. propofition).And drawa line from B to F,Wherfore (by the 33.propofition )«73BC Fisa parallelogramme: and thereforeis (by the 34.pro- pofition ) double to the triangle «4 EB, Where- fore the trapefium e4 BC Dis more then the dou. ble ofthe triangle ef EB: which was required to be proued., OE ROS IG eB ee == > eee - aah . — . ae = ee ~ = Po = Oui. + The The first Booke The .Probleme. The 42.propofition. Vito a triangle geuen,to make aparallelograme equal, whofe angle fhall be equail to a rethline angle geuen. it 4 ppofe that the triangle genen be A'B C, and let the peti angle geuen be D, It ts required that vnto the We LR Dey raansie 4B C there be made a parallelograme equal, TCAD NZ 2 whofe angle fhal be equall to the retisline angle genen, : in % HN 4 5 Conftration. YE inamely,to the angle D.Deuide( by the 1o,propofitio )the BEF FS line B C into two equall partes in the pointe E. And(by eS I SE 7} the first peticion) draw a right line from the point A to ee yi ‘the point E. And( by the 23. propofition) vpon the right line genen E Cand to the point in it geuen E,make the angleC E Fequal to the angle D, And( by the 31. proe con pofition) by the point A draw vnto the line EC a parallel line A H:and let the line Ei Feut the lne AH in - the point F.and( by the fame )by the point C, drawevntothe ime EFa parallel line CG.VVherfore FECG __ 4§aparallelograme, And fora/muche Pemerfirat® “4s B Eis equall toE C,therfore (by the 28.propofition ) the triangle A V5 B Eis equall to the triangle AEC, for they-confist vpo equall bafes that is B Eand EC, andare inthe felfe fame parallel lines,namely, BCand A H. VV berfore the triangle A BCis double to the trianole AEC,And the paralles .bogrameC EF Gis alfo double to the triangle A EC: for they haue one ¢> the -feife fame baje,namely,E C: and are in the felfe fame parallel lines,that is,EC and AH, VV berfore the parallelograme REC Gis equall to the triangle A B Cand bath the angle C E Fequall to theanole geuenD, VV hereforevuto the triangle geuen AB Cis made an equad parallelocrame namely, FECG, whofe angleC E Fis equall to the angle geuen.D: which was required to be done, : r T he conuerfe ofthis propofition after Pelitarius. 1) é “his pees Unto a parallelogramme geuen, to make atriangle equall, hauyng an angle equall to a rettiline pofition. angle genen. Suppofe that the parallelograme geuen be 4 BCD, and let the angle genenbe E. Itis required ynto the parallelograme 4 BCD to make a triangle equall hauyng an | angle angle equal.to the angle E, Vpon the line C.D and to.the pointe in ‘ “ it Codeferibe (bythe23: propo- Au Bo B pofition) an angleeqnallto the .\0\ (9254) i angle E: whichlet be-DC F; aud let the line C-F cut the line <4 B being produced, in the point F: and produce the line C D(which is parallel to the line 4 F).tothe point G,And puttheline D Ge- qualltothelineC Dand draw a line from FtoG, Then/fay that Vf Be ae thetriangleC FGisequaltothe 0) py G parallelograme 4BCD.Forfor-9 pei oY . afmuch as(by the 38. propofition ) the whole triangle C F’'Gis double to the triangle CDF. Aud(by the 41.propofition thé parallelograme ABCD is double to the fame triangle CD F: therfore the parallelograme'z4.B C D andthe triangle C F G are equall the one to the other: which. was required to bedone,. >. 2620. g of Euchides Elementes. Fol.53. The 32.T heoreme. ' The 43.Propofition. co fnenery poe pars entes of thofe parallelo-. grammes which are about the diameter are ezuall the one to the other. 2 aD * ppofetbat.A BC D bea paralelograme and let the diameter thers RN of be A Cand about the diameter A Clet thee parallelogrames EH RX and G Fonfist; and let the fupplementes be 'B K and KD, Then I ————— fay that the jupplement B Kis equall to the fupplement KD. For fora/muchas ABCD isa parallelograme and the diameter therof is AC, therfore (by the 3 4.propo/ition ) the triangle A BCisequalltothetriangle ADC, Aa gayne forafmuch as AEK Fi isa pae rallelograme,and the diameter therof is AK, therfore (by the fame) the trians gle AE K is equallto the triangle AH K.And by the fame reafon alfo the tri-- angle K FC is equallto the triangle K: G C.And forafmuch as the triangle AB Kis equall tothe triangle AH K ,and the triangle KFC tothe triangleNG . ¢ G C, therfore the triangles AE K and K G Care eqnall to the triangles AH K and K FC: and the whole triangle A BCis equall tothe whole triangle AD | Pj. C > [= Pe SS Se at — = i = 1 i is | At Bape Bh | : ’ n ‘ Re ; 4 et, Ph! } 's I - MF i i — 1 a HW ut) ih ian \ a ; 4a} ee i ' } f ik ne ATs ots T hefirftBooke “ C: wherfore the refidue namely,the fupplement B K is (by the 2, common fene tence) equall to the refidae namely to the fupplement K D . VV herefore ine uery parallelogramme, the fupplementes of thofe paraltelogrammes whiche are about the diameter, are equall the one to the other: whiche was required to be proutd,~ ai ets | How parallelo- Thofe parallelogrames are fayde to confit abouradiameter which haueto ee we, their diameters partof the diameter ofthe whole and great parallelograme;as faydeto confifte - ; ey ‘ aboutadia. ithe exampleofEuclide, Andfuchparallelo. _ “a ‘ wseter. grames which haue notto their diameters pare Pe poe ca 7 ofthe diameter ofthe greater parallelograme, | rat are fayde not to confiftabontthe diameter,For | thé che diameter of the greater. parallglograme ..|, cutterh the fide ofthe leflecétaynedwythimies » |: As in the parallelogramme 4 B,thediametet@Dy:.° : cutteth the fide E Hof the parallelogramme CE. {~ Wherefore the parallelogramme C Eis not about p B one and the felfe fame diameter with the parallelo- erammeCD, __ ba Anahi © 4 sie : 7 a Ree oS . se > 4 27 a “ PA wat! « On es wssiw 3 Supplementes or C omplementes are thofe figures which withthe two pa- ; tallelogrammes accomplifhthe whole\patallelograminie. Although Pélitarius E for diftin tien fake putteth a difference betwene Supplementes and Comple- ' mentes, The parallelogrammies about the diameter he calleth Complementes, | the other two figures he calleth Supplementes, | Supplements €F Complementes. Three cafesin OS This theoreme. hath three cafes onely, For the parallelogrammes which rhisTheoreme, eonliftaboutthe diameter, eythet touch the one the otherin.a point: or by acer tay ne parte of the diameterare feuered the one from the other: or els they cutte T hefrft cafe. the one. the orher. For the firft cafeis the example ofEuclide before fet, The fe- cond cafeds thus: 2 po A ; 4 Suppofe that 4B be aparallelograme: So aati whofe Eine let be CD P and abate the : fame diameter let thefe parallelogrammesC’ ~ <4 Kand D Lconfift:and betwene thé let there: be acertayne part ofthe diameter, namely, LK. Then Tfay that the two fupplementcs 4 GLKEXBFKL Hare equall. For wemay as before(by the 34,propofition )proue that ’ the triangle AC D, is equall tothe triangle: * BC D,and the triangle E C K tothe triangle KC F, and alfo the triangle D G L to the tri- angle D L, Wherfore therefidue,namely, the fiuefided figure AGL KE is equall to the refidue,namely, to the fiue fided figure B ai FKL #3 thatis, the onefupplement tothe -_D ii. B other:which wasrequired to be proued, ~_ | The third cafe. But now fuppofee 4B to bea parallelogramme,and let the diameter thereof be C D:and let the one of the parallelogrammes about itbeEC FL, and let the otherbe of Euclides Elementes. Fol.54. DG K H,ofwhich let the one-eut the others*-¢ Then fay that the fupplementes:F Gand@EA:: 4 P io tle are equall.For forafmuch as the: wholetridn- : gle ‘DGK is equal to the whole triangle DAK ( by the 34. propofition ),and partalfo. ofthe ; one,namely, the triangle K LU7is equalito partof the other,namely,to the triangle K.L N(by the fame), forLKis a parallélegrame’ . therefore the refidue,namelyjthe Trapefium«:: DI F is equall to.the refidue,nanielys to: the trapefii D L 44 Gtbut the triangle ADC. is equaltothe triangle BC D,and inthe pas - rallelograme E F, the triangle CL is equalh tothe triangle EC L,and the*trapefium:D.G: ——— ph steps. MVis(as it hath bene-proued)iequalhtoithe join, - 202: 8 too ARe ys B trapefiumD H NL. Wherefore therefidue, x 3 namely, the quadrilater figure G F is equall to the refidue, namely, to the quadrilater figure E #,that is,the one fupplement to the other:which was required to be proned. This is tobe noredythat in ech.ofthofe three cafes it may fo happen,thac the parallelogrammes aboute the diameter fhall not haue one angle common wyth the whole parallelogramme,as they hauein the formerfigures,But yet though they havenot, che felfefame demonftration wil ferue, as tris playneto fee in the figures heré'ynderneath put,Foralwayes,.iffromthinges-equall be taken.away thinges equall the refidue fhalbe equall. This propofition P elitarius calleth Gnomicall, and mifticall,for that of it (fayth he) {pring infinice demon ftrations jand vfes in geometry. And he putteth the conuerfe thereof after this manner, Ifa parallelogramme be deuided into two equall fupplementes and into t¥o consplements What- foener: the asametcr of the two complementes [hal be fet directly, and make one diameter of the whole parallelogramme. . ban Here isto be notedas 1 before admonifhed that pelitarius for diftin@ion fake putteth a difference betwene {upplementes and complementes, which diffe- rence,for that I haue before-declared, I fhallnot neede hereto repete agayne, Suppofe that there be a parallelogramme 4B CD, whofe two equall fupplements let be 4 €F GandF HD K,andletthe two.complementes thereof be GF C K and EB F H:whofe diameterslet beC F and FB. Then Lay that C F Bis one right line, and is the diameter of the whole parallelogramme 42CD; forifitbenot, thenis iy an Pay, other This propofities called Gnomscal and miftical, The comuerfe of this propofition. = a as = ab tudes tite . _ == —— = 2 ~ = = = == : EE Ed a ne = . _—— et - ‘ » — PP we we oe, — ~ 7 = - n> aa — = a — a —-— —e ~s ~ — ~ —_ - ~ " a a” - ee — eo Fe ead > 1 = St = = = = SS Se o> shee = epee ay)! q : / Wika Wi: He OBGFEL j a) ee TREE i Hu i | 4) stat, a a Pare eshe? jill \ ; mianl + a | 4 s yt ai it | 4 4 i Lt eal) iS Ef ita Higa { Hilti aay! BE ie { REE PRR i! wie ; | i 1a) ii | ] | } aT : i? 1] ! - 4 | Consitrudson. | Lhefirft Booke other diameter ofthe whole parrallelogrammie,which let be CLB being drawen vnder the diametersC Fand F B, and cutting the line GH in the in the pointL,And( by the 3 1.propofition ) by the point L,draw vntothe line ef C a parallel line ML N.And fo are therein the whole paral- lelogramme 4 BC Dtwofupplements AMG Land LH N D,which by this propofition fhalbe equafl the one to the other. For that they are about the diamerer CL By : But the fupplemented E F Gis (by fuppofition) equal” to the fupplement F H D K:and forafmuchas- FH D Kis ° greater thenLH DN,AEFG alfo fhalbegreaterthen' A’ M GL,namely,the part greater thenthe wholé:which is . impoflible. And by the fame reafo may it be proued,that » the diameter cannot be drawen aboue the diameters C Fo"? - : if and F B. Wherefore CF Bis one diameter ofthe whole paralielogramme ef BCD: which was required to be proned. - Jobst ish yy - 15 Ste’ Theiz: Probleme. The 4.4.Propofitions °“Oppon a right line geuen, ta apple a parallelograme equall ow-toa triangle genen, and contayning an angle equall to a rece tiline angle geuen, | | V ppofechat the right line genen be A B,and let the triangle genen fs be C,and let the rectiline angle genen be D.It is required ‘vpon the ON) right line geuen AB, to applyea parallelogramme equal to the trians — gle genenC, andcontayning an angle equall to the rectiline angle ge» ue D.Defcribe( by the 4.4.propo/ition ) 5 paralleligrame BG EF equall to the tris Fre K angle C,andhaumg-the angle BG Fee quall to theangle D.And vnto the line E Bioyne the line AB in fuch fort that they make both one right line. And extend the line FG Leyond the point G to the poynte i ZX. aul . Hi And( by the 31,propofition by the pomt H A L C Aaa , itil A drawe'to either of thefe lines BG and E Fa parallelline A FZ. And (by the fir/t peticion \draw aright line from the point FZ to the point B. And fora/much as vpon , the parallel lines A Fi and E F falleth a certayne right line HF, therefore(by the 39 propofition the angles A FF and HF Eare equall totwo rightangles: wherefore theangles BH Gand GF Eareleffe thentwo right angles : but if vpon two right lines fall a right line making the inward angles on one andthe fame” of Euchides Elementes. Fol.s5. fame fide lefJe then two right angles, thofe right lanes being Teube produced shall at the length mete on that fide in which are the angles leffe then two right angles( py the 5, peticion). VV herfore the lines A Band FE being infinutly produced will at the length mete,Let them be producedco let them mete inthe point K.And (bythe 31 propofition ) by the point K draw to either of thefe lines E A and F Ha parallel line K_L.And (by the 2.peticion) extend the lines H A and GB till they cocurre With the line K_L in the pointes L and MVV heres fore HLK Fis a parallelogramme and the diameter thereof is HK: and a bout the diameter H Kare the pargllelogrammes AG and ME, and the fup: plementes are B and BIywherefore (by the 4.3, propufitron )the fupplement L Bis equallto the fupplement BP: but by conftruction the parallelograme BF is equallto the triangle C: wherefore alfo the parallelogramme L Bis equall to the triangle C,And forafmach asthe-line F His a parallel to the line K,L, and vpon them lighteth the line GM) therefore (by the 27, propofition) the angle G Bis equall to the angle BML But the angle FG Bis equallto the angle therfore the angle BML ts equal tothe angle D.VV berfore vpo the right line geuen A Bisapplied the parrallelograme L, Byequal to the triangle genen C,and contayning the angle B ML equal to the re€étilme angle genen D: which was required to be done. Applications of {paces or figures to lines with exeeffes or wantes is (fayth Eudemus ) an auncient inuention of Pithagoras. V Vhenthefpace or figure is ioyned to the whole line,thé is the figure fayd to be appliedto theline. Bur ifthe length ofthe {pace be longer then the line,thé irusfaydeto.exceede: and ifthe length of the eats be fhorter then the line, fo that pate of theline remayneth without the figure defcribed, then is it fayde co want. Inthis probleme are three thinges geuen,A right line to which the applica- tion is made,which here mutft be the one fide ofthe parallelogramme applied.A triangle whereunto the parallelogramme applied muft bee equall : and aa angle wherunto the angle of the parallelograme applied muft be equall, And ifthe an- gle geucn be a rightangle,thé fhal the parallelograme applied be either 2 fquare, or a figure on the one fide longer, But ifthe angle geuenbeanobtufe or an acute angle, then fhall the parallelograme appliedbe a Rhombus or diamond fi- gure,or cls a Rhomboides or diamondlike figure, The conuerfe of this propofition after Pelitarius, Upon a right line genen,te applie unto a parallelograme genen an equall triangle hauyng An Ane gle equak te an angle gener. Suppofe that the right line geuen bee 4 B,and let the parallelograme geyen be C D € F,and let the angle geuen be G It is required vpon the line 4B to defcribe a tria- gle equallto the parallelograme C D E F,hauing anangle equall to theangle G. Drawe the.diameter C F & produce CD beyend-the point Dto the point H. And put the hoe | P.iij. D Demonffration Applications of Jpaces with ex- ceffes or Wants an auncicnt thea wentsion of Pi~ thagoras How a fi ure bf fajde to be ap- plied to a ine, Three thinges genen sn thes propofiteon, The conuerfe of thss propofitten. T he first Booke D Hequallto the lineC D. And Bhs WAY draw aline from F to 2. New thé (by the 41-propofition) the tria- gleC H Fisequallto the paralle- lograme CD EF. And(by this propofition)vppon thelineed B defcribe a parallelograme ABKL equall to the triangle C @ F,ha- uing the angle 4B Lequaltothe anglegeuen G: and produce the ¢ ny: eae sa Aa LE genh line B L.beyonde the pointe Lto be ey fae ee the point M.And put the line LM equall to the line B L,and draw a line from A to’ Ms Then I faythat vpon the line 4 Bis.defcribed:the triangle_4 2 M,which is fuchactrian gle as is required. For(by the.41.propofition)thetriangle 4B Mis equal to the paral« lelogramme 4 B K L(for that they are betwene two parallellines B Mand 4K, & the bafe ot the triangle is double to the bafe of the parallelogramme):but 4B K Lis:by* conftrucion equall to the triangle CH F:and the ‘triangle C & Fis equall to: theipa- rallelograme.C D E F.Wherfore (by the firft common fentence) the triangle 4:2 M is- equall to the parallelograme geuen CD E F;and hath his angle 4 B M equal to the an- gle geuen G: which was required to be done. *-—* ead Gls bk eh The 13.Probleme. The 45 Propofition. Todefcribea parallelograme equal toany revliline fioure ge. uen,and contayning an angle equall to a rettiline angle geue. =4 V ppofe that the rectiline figure geuenbe A BCD,and let the reEtiline SSV8 angle gent be Et is required to defcribe a paralleloovame equall to the =! rettiline figure genen ABCD, and contayuing an angle equal to the rea conpirusion, Biline angle geuen E.\Draw( by the fir? peticion)a right line fro the point D to the point B. And (by the 4.2.propofition)yntothe triangle AB D defcribe an ¢« qual parallelograme F F1,hauing bis angle F KH equall to the angle E, And (bythe 44.0f the firft) vpothe right — } line G ELapply the parallelogramme ° alas 9 G Mequal to the triangle DBC haz | | uing bis angleG HIM equallto the | Demonttrariom angle E. And fora/much as eyther of - thofe angles H K,Fand GHM 1s | equall to the angle E. therefore the B C828 \e angle HK Fis equallto the angle. G FM: put the angleK HG come _ monto them both, wherfore the ane gles FK HandK AGare equall to the anglesK HG and GH™M. but theanglesF K H and KHG - are (by thez9,propofition) equall to | upasls tworight angles VV berfore the angles K FG andG Ff Mare equall to two right a ae ow ee _ thegreater../t is of Euchdes Filéméntes. Fol.56. . rightangles.Now thin yntoa richt-line G Hand toa pointinthe fame Hare drawen two right lines K Hand FP M not both on one anil the fame ‘fide, mas king the fide angles equallto two right angles VV her fore( by the 14.propofiti- on) thé lines K, Eland H M make diretity one right line, And fora jmuchas v2 pon the parallel lines K.M and F G falleth the right line FLG, therefore the alternate aaglesM FLG and AGF are by the 29. propofition equall the one tothe other: put theangle 1G L common to them both, VV herfore the angles MITC and 1G Lare equall tothe angles 1G F and 1G L. But the angles MHGer HG Lare ne to two right angles( by 9-29.propo/ition), V7 ber- forealfo the angles AGF and HG L areequall totwo right angles. VV bers fore (by the 14,propofition) the lines GC and G L make directly one right line. And forafmuch as the line K Fis (by the 24.propofition) equal to the lyne HG, and st is alfo parallel vnto it;and the line L1.Gis( by the fame) equall to the line ML, therfore (by the first common fentence) the line F K is equall to the lyne ML and alfoa parallel vato it (by the'2.0.propofition), But the right lynes K Mand FL ioyne them together VV herfore (by the 3 3 propofition) the lines K _ Mand F Lare equalltheon to the other and parallel lines VV her fore KELM is a parallelograme. And fora/much as the triangle A B Dis equal to the paral: lelograme F H,and the triangle D B C to the parallelogrammeG M:; therfore the whole rectiline figure ABC Dis equall to the whole parallelograme KFL M.VV berfore tothe rettiline figure geuen AB C Dis made an equall paralles grame K FL M,whofe angle F K Mis equalto the angle genen,namely,to E: which was required to be done. The reGiline figure geué is inthe example of Euclide.s a parallelograme,But ifthe reétiline figure be of many fides,as of5.6.ormo,thé muft you refolue the figure into his triangles,as. hath bene before taught inthe 32, propolition. And thé.apply:aparallelograme equal to eucry triangle vponalinegeué,as before in the example ofthe author.And the fame:kind ot reatoning wil ferue that was be fore,only by reafo ofthe mulritude oftriangles,you fhall haue neede of oftener repeticid ofthe 2 9,and 14,,propofitids to proue chat che bafes ofal the parallelo- grames madeequall toall the triangles make one right line, and ‘fo alfo of the toppes of the faid parallelogrames,Pelitarius addeth vnto this propofition this Probleme following, . T wo unequall rettiline fuperficieces beyng geuen,to find ont the exceffe of the greater aboue the leffe. Suppofe that there be two vne- quall reGtiline fu- perficieces 4 & B of which let Abe required to finde out the exceffe of the fiiperficies 4 aboue the fuper- . ficieces B . De- {cribe(by the 44. A P. iii}. pro- An addition of Pelitarius. T hefirSt Booke *: propofition)the parallelograme C D E F equalito theredtiline figure 4,contayning @ right angle.And produce the line C D beyond the point D te the pointg : & put the line D G equall to the line C D.Andagaine (oy the 44.propofition ) ypon the liné DG defcribe the parallelograme D G H K eqhall to the rectiline figure 8, and hauyng the angle D G Ka right angle.And produce the line K H beyond the point H,vntill it cutte the line C E inthe point L,Then I fay that HLEF, is the excefle of the redtiline figure ef aboue the rectiline figure B.For firft that C GK L isaparallelogrammeit is mani- feft. neither nedeth it to be demontftrated. And foraf{much as thelines CD and DG ate by {uppofition equal and either of them is a parallel to K L,therfore(by the 3 6,propo- fition )thetwe parallelogramesC Hand D K are equall.And forafmuchas D Kis{up- pofed to be equall to the re@iline figure B,C Af alfo fhall be‘equall-to the fame re@iline figure B.Wherforeforafinuch asthe whole parallelograme F is equall to the re@iline figure e4,andL Fis the excefleof C Fabone DLor DK, it followeth that L F is the ex- ceffe of the reGtiline figure e abone the reGiline figure 2 : whiche was required to be done. | | | | | a | An other moreredy way, . Let the parallelograme C D E F remayneequall to the reGtiline figure 4, & produce the line C D bevond the point D to the pointe G.. And ypon the line D G defcribe the parallelograme DG H K equal! to there@iline figure B. And produce the lines EC: & H K beyond the points Cand K till they.concurrein the poiut L. And by the pointe D draw the diame- eadt on Ou | ter LDM, which es on SN letcuttetheline “ile ig BOEE PA SUED DANE, 3 oe HG beyngpro- [== duced beyonde © the pointe G m x, |—— thepoint M, & by the pointe M drawe vnto the E Fr line.H La paral- ae | lel M'N cuttyng the line EL in the pointe N: and by that meanes is H L M Na paralle- lograme. Then J fay that N Fis the evceile of the re@iline figure ef aboue the reGiline figure B.For forafmuch as the parallelograme H Dis equall to the re@iline figure B, 8 the fupplementes H DandD N ar(by the 43.propofition) equall : therforeD Nalfo is equall to the reGiline figureB,which re@tiline figure D N being taken away fro the parallelograme C F (which is fuppofed to be equall to the reatiline figure 4) the refi- due N F thall be the ezceffe of the reGiline figure 4 aboue the re@iline figure B:which was rejuired to be done, , The 14.Probleme...T he 46.Propofition, Uppon a right line geuen,to de[cribe a fquare. AY ppofe that the right line geuen be A BI eis required vpo the right £1 line A B,to defcribe a JquareV pon the right line AB, and from a iq Nr point in it geuen,namely, A, rayfe vp (by the 11.propofition) a pers ~ pendiculer lin: AC, And (by the 3 propo/ition) vnto AB putan ec quallline AD. And (by the3:.propofition) by the point D drawe vnto A Ba parallel line D B, And (by the fame) by the point B drawe vuto.A Da parallel line of Euchdes Elementes. Fol.5'7. line BE,VVherefore ADEB is apas ¢ rallelogrammeVV herefore the line AB is equall to the line D Fi andthelned pp D totheline BE: but the line ABise: quali to the line A Dwkerefore the/e fos wer lines BA,AD,D E,E 5B, are equall the one to the other. VV berefore the pae rallelogramme ADE Bconfifteth of e qual fides , I fay alfo that it is re€tangle, For forafmuch as vpon the parallel lines A A Band DE falleth a right line AD:theres fore( by thea 9 prapaicion) the angle: B A Dand AD Eare equal totworight angles:but the angle BA Disa right angle. VV herfore the angle AD Ealfois a right angle, But in parallelogrames the fides and angles which are oppofite are equall the one to the other(by the 3 4 propofition), VV berefore the to oppo/ite angles A B Eand BED are ech of thema right angle.VV berefore the parallelograme ABE Dis rectangle:e> it is.al/o proned that it 1s equilater,VV herfore st ts a/quare,eo it is defcribed vps onthe right line geuen A B; which was required to be done, B Fo defcribea Square mecha- nicely, . This is to be noted that if you will mechanically and redily, not regarding dem0. ftration defcribe a {quare vpon a line geuen,as vpon theline ef B, after that you haue erected the perpendiculer line C_Avpontheline 4B,and 9 yc put the line 4 £equall to the line 4 2: then open your | compatieto the wydth ofthe line 42 or 4E,& fetone. +E foote thereofin the point Z,and defcribe a peece ofthe » circumference of a circle:and againe make the centre the point Z,and defcribe alfo a piece of the circumference of a circle,namely,in fuch fort that the peece of the circum- feréce of the one may cut the peece of the circumference of the other,as inthe point D: and from the point of the interfection,draw vnto the points E & Bright lines: & fo thatbe defcribed.a {quareAs in this figure here put,wher- A B in/ hane not drawenthe lines E D and D B, thatthe pee- ces of the circumference cutting the one the other might the plainlier be fene, SG, Anadadition of Proclus. If the lines upon Which the fquares be defcribed be equall,the fquares alfo are equal, = 3 a | tn addition of Precis, Suppofe that thefe right lines ABand €D be equall, & ypon the line 42 defcribe a fquare_4 BE G:and vpon theline CD de- fcribe afquareC DHF, Then J fay that the two {quares 4 BE G&CDHF are equal, For draw thefe rightlines GBand HD. And forasmuch asthe right lines 4 Band€ Dare-equall,& thelinese4 Gand HC are alfo equall,and they contayne eqaul The co ~wnerfe thereof. Conftradion, The firft Booke angles,namély,right aneles (by the definition of a {quare)therefore( by the 4. propos fition jthe bale B Gis equall to the bafeD.And the triangle .42.G isequall egthe triangle C D H.Wherefore the doubles of the faide triangles are equal]. Wheréfore the {quare 4 Eis equall to the {quare CF: which was required to be proued. | The conuerfe thereofis thus, Ifthe fguares be equall: the lines alfouppon which they are defcribed are equall. Suppofe that there be two equall {quares 4 F and CG defcribed vpon the lines 4 B&B. Thél fay, that the lines 4 B and B Caréequall.Put theline 4B dire@ly to the line BC,that they both make onrightline.And | forafmuch as the angles are right angles,ther- fore alfo( by the 14.propofition) the right line FBis fet direGly to the right line BG. Drawe thefe right lines FC, AG, AF, and CG.Now for afmuch as the {quare 4F is equal to the {quare CG,the triangle alfo 4 F B thalbe equall to the triangle CB G:put the triangle BC F cOmon to them both, Wherfore the whole triangle 4C F is equal! to the whole triangle C F G.Where. fore the line .4G isa parallel yvntothe lineC F (by the 3 8.propofitis): forthe triangles confift vpon one and the felfe fame bafe, namely C F, Againe forafinuch as either of thefe angles 4F G & CBG isthe halfe ofaright angle, therfore y, (by the 27.propofition )the line AF is a paral- Cc lel to the line C G.Wherfore. the right line-4F is equal to the right line C G(for the op- polite fides ofa parallelograme are equall ), And forafmuch as there are two triangles AB F and BCG,whole alternate angtes are equall, namely, the angle 4 F 2 tothean- gle BGC, andthe angle B AF to the angle BC G,and one fide of the one is equall to one fide of the other,namely,the fide which lieth betwene the equal angles, that is, che fide 4 Fto the fide CG, therefore (by the 26.propofition) the fide 4 Bis equal tothe fide B Cand the fide B F to the fide BG, Wherefore itis proned thar the {quares of the lines - F andC G being equall, their fides alfo fhalbe equall: which was required to be proued. | | egies Ste The 33. T beoreme. The 4.7. Propofition. Jn rectangle triangles, the Jquare whicheis made of the fide that ubtendcth the right angle,is equal to the fquares which are made of the fides contayning the right angle. | YT ACS EF ts ee ‘ A ene < > ~~ £ xa >, xX , a ALS o\ = S77 ve ey ~~ Ab ~~ hg j te nr rR | LJ —— Li Pps TI Vv Regen eas of Exnchiles Elementes. Fol.58. line AL, And (by the firSt peticion) drawairight-lyne from the point A tothe ~ point D,and an other from the point Cro the point E-And forafmuch as the an= Demonstratiam gles B ACandB AGare right angles,therfore vnto aright line B.A,and toa point init geuen A, aredrawen BQ. oye copy pe sight lines A Cand AG, not both on | one and the fame fide, makyng the G two fide angles equall to twa,right A> reh angles. wher fore(by the 14. propofte it ola iyp tion)the lines AC and AG make dts F Li ) reébly one right line.And by the fame Blatt \ C reafon the. lines B A and A Huiake alfo dire€tly one right line, And fore afmuch asthe angle DB Cis equal 3h to theangleF B A (for either of ae iden | isa right angle)put the ancle ABC | | prebline to pate Sbevon the poses : wholeangleD BZ is equall tothe whole angle F BC And fora/much as thefe twolines A Band B D are equal to thefe two lines B Fand B C,the one to the other,and the angle D B Ais equal tothe angle BC: therfore( by the 4,propo/ition the bafe A Dis equall to the bafe F Cand the triangle A B Dis equall tothe triangle F BC, But(by the 3. ae mane parallelogramme B L is double to the triangle AB D, for they haue both one aud the fame bafe, namely, BD, and arein the felfe fame parals lel lynes, that is, BD and AL and (by the fame ).the fquare G Bis double to the triangle FBC, for they bane both one and the felfe [ame bafe, thatis, B F, and are in the flfe fame parallel lynes, that is, FB andGC, But the dous bles of thinges equal, are (by the fixte common fentence) equall the one to the other. VV herfore the parallelograme BL is equall to rhe (quareG B, And in like forte if (by the firft peticion) there be drawen aright line from the point Ato the point E,and an other from the point B to the point Ky we may proue y the parallelogrameC L is equal to the fquare HCVVherfore the whole fquare BD E Cis equall tothe two fquaresGB and AC Dut the fquare BD E Cis defcribed vpon the line B Cand the fquares.G Band HC are de(cribed vppon the lines BAg> AC. wherfore the {quare of the fide BC is equal tothe fquares of the fides B.A and AC VV berefore in rectangle triangles, the {quare whiche is made of the fide that fubtendeth the right angle isequal to the /quares which are made of the fides contayning the right angle: which was required to be des monjtrated. | } This moft excellent and notable Theoreme was firft inuented of the Greate Lishagoras the philofopher Pithagoras,who forthe exceeding ioy conceiued of the invention ff imuenrer of therof,offered in facrifice an Oxe,as recorde Hrerone, Proclus, Lycius, s Via-. error _ truurus,And ithath benecomm@ly called ofbarbarous writers ofthe latter time Dulcarnon, i Q.ii. An ——— aa$ . - — = - ss =— —— = - = ne en ate oe - —- -=—-- Ss = sa LSS a i — —— - 2 - = =: =r SS -~ ee = — SS —— ———— ~a >= eae -_— ———— e _ — <= > + <= ee : a 2 = = See = SS = a - tli — ~~ . ° we : —— — — a at =——- <= ——== =< = ‘ é + in - - - —— ’ | el “ - = das - — — — — © Be. “ a 7 om : at ne . ne = ee aE anit a A ‘ , ——, - - ed A 7 5 — — - : : ; - Az = Ss = Soe foes * <= =sS SS a a 2 ee ~— = ce Phe m7 «a 5 Beery «Telia Booke An additi bat Rose | An additiomof P elicarins, © tit a To reduce LWO ynequall{quares to.tworequall {quares, Suppofe that the fquares of the linese 7B and 4 C be vnequal!.Itis required to ree duce them to two equallf{quares./oynethetw6 linese4 Bandcie€ at theirendes 1m fuch fort that they make arightangle B 4 C. And draw 3 line fronvBto C.- Then vppo the two endes# and C make two angles eche of.which may be equal tohalfearight an- gle (This is done by ereGting vpon theline B C perpe- | diculer lines,from the pointes Band Cand fo (bythe ~ D 9. propofition,) deuiding eche of the rightanglesinto, A two equall partes): andlettheanglesBCDandCBD | _- be either of thé halfe of a right angle, And let the lines % DandCD concurre in the point D. Then /fay that * the two {quares of the fides BD and C D,are equall ta— the two fquares ofthe fides 4B and 4C.For(by the 6. 2 | wae : propofition )the two fides D Band D Care equall,and the angle at the pointe Dis (by the 32, propofition) a: . | right angle. Wherefore the {quare of thefide BC ig e- qual to the fquares of the two fides D Band DE (by the 47.propofition) :butitis alfo equall to the fquares of the two fides 4 3 and AC by thefelf fame propofition) wher- fore’ by the common fentence )the fquares of the'two fides B Dand D Care equal! to’ the {quares of the two fides 4 B and AC: which was:required to he done. oe An otheraddition of Pelitarius, An other aditia ° of biledhiag. Ff rWo right angled triangles hane equal bafes.the Squares of the two fides of the one are equall ; tothe fanares ofthe two fides of the ather. .., his is. manifelt by. the formerconftrudtionand demonftration, her adding SO Av othet additiondf Belitarins, ; Another addits .. pe i seh axis 4: om hte: : 3 . ox of Pelitarius, Tworwmequall nied being Speen sila bow _ the fquare of - ome is greater then the Squars of theorher, Suppofe that there be two ynequal lines 4B and BC:ofwhich let 4B be the grease ter.Ftis required'to fearch ont héw mn uch the fquare of 4 B excedeth the {quare of B €.Thatis {wil finde out thefquare; which with the fquare of the line BC thalbe equal to the {quare oftheline 4 B, Put the lines 4). | : Band BC dire@ly, that they make both.one . rightline-and niaking the centre thé point B, © and the {pace B Adefcribea dircle ADE,And produce the line e4C to the:circumfereace, and letitconcurre with it inthe point €,And_ ’/ vpon the lyne E and fré the point C ere@ (by the 11,propofition) 4 perpendiculerline. ‘ C D,which produce tillit concurre with the |. circumference in the point D: & draw aline from B to Di Then J fay,that the (quare ofthe line CD,is the excefle ofthe fquare of theline- 48 aboue the {quare of the line B C. For forz a{much asin the triangle BCD,the angle at the point Cisa right angle, the fynareofthe © - bafe BD ts equall to.the {quaresof thetwo. arts sitthg fides BC'and C D(by this.47 -propofition).Wherefore alfo the {quare of the line 4B is equal! to the felfe fame (quares of thelines BC and CD. Wherefore the fquare-of the line BC is fo much leffe then the {quare of the line 4 B,as isthe {quare.o theline C D;which was required to fearch ont, —Aa of Euchides Elementes. Fol.59. » | . An othet addition of Pelitarius: 13) T he diameter of 4 fguare being geuenstogene the Square thereof. An other aditiz ; | of Pelstarins, This is eafie to be done, For if vpon the rwo eade’ of the line Be drawen two halfe right angles,and fo be made perfe@ the triangle then fhalbe defcribed half: of the {quare;the other halfe whereotalfo isafter the (ame manner eafie to be des feribed, ty | Flereby it is manifeft that the fquare of the diameter is double to that [qnare whofe diameter it + 4 Copp, ary, The34.Theoreme. The 48. Propofition. lfthe fquare whichis made ofone of the fides ofa triangle, be equall to the fquares which are made of the two other fides of the fame triangle: the angle comprehended under thofe two other fides is a right angle. Rigi, ppofethat A BC be a triangle and let the [quare whichis made of one 1 &sn¢, of the fides there namely of the fide BC,be equall to the {quares which pe are made of the fides BA and A C.Then I fay that t he angle BAC isa right angle, Ray/e vp( by the 11 propofitio) from ibe point A nto the right line ACa perpendicular line AD, And (by the thirde propofition) vnto the line A B put an equall line\A D, And by che first peticion draw aright line from the point D to’ the potat C. And fora much as the line D Ais equall to the line A B, the Jquare which is made of theline D A is ée quallto the {quare whiche is made of the line AB Put the fquare of the line AC, common to them both. VVherefore the Squares of the lines DA and AC are equal. © + to the fquares of the lines B.A and AC, But (by the propofition going before) the fquare of the line DC is equal toy fquares of the lines. A D and AC, (For'the angle B D ACis arightanole) andthe /quare of BC is (by /uppofition) equall to thefquares of ABand ACV herefore the Square of D Cis equall to the fquare of BG wherefore the fide DCs equall to the fide BC, And fora/muchas A Bis equallto A Dand AC is common to them both, therefore thefe rwo fides D Aand AC are equall to thefe two fides BA and AC, the one tothe other, and the bale D C is equall to the bafe B C-where fore (bythe 8, propofition )the an le D AC is equall tothe angle BA C.But the angle D AC isa right angle whe refore alfothe angle BAC ata right ang le, Ff SY. theres ed ~ a te - a Lee Sem SS ea : SS — a = ; ———— > " a = — SS - =S-ae <= a re aioe ~ eed —s oe ee aa <> eeey Saree Oo rt Se an aS Se 7 a — Soe = oz \oe = : = ; = = ray Poe gee > SON —o. = Z - —_ ~~ -- ~ a — - ~ . pr - at ain _— “ « — sh em — - < . = a 3 < tend co 2 5 5 tthe wren rm : on Se ee =< ~ =e a a nS — =a. a ee ee > = = So = = —— -—~-—- —— eee ee ee eae . = anaes — — =- = “ ae iano io — eww ek i ———— ets = - ——- - ; aa —— SS — ————_ — .- SS ee at ha c = = : = = == => ee LI es IS eee > any eens en ee ee a ~~ - — - 3 ——2” > =z —s - = 1 eee —— : — ~ = = 1 ene $ ——— o. = This propofition és the conuerfe of the former. An other De- monftration af- ter Pelitarius. The first Booke eherefore the {quare which is made of one of the fides of atriangle, be equal to the fquares which are made of the two other fides of the fame triangle, the ans gle comprehended vuder thofe two other fidesis a right angle. which was ree quired to be proued. : This propofition is the conuerfe ofthe former,a:d is of Pelitarius demon- ftrated by an argument leading to an impoffibilitie after this maner. Suppofe that ABC bea triangle: & let the {quare of the fide AC,be equal to the {quares of the two fides 4 Band BC. Then/ fay thatthe angle at the point 2, which is oppofite to the fide AC, isa right angle.For if the angle atthe point | B be notaright angle, then thal it be eyther greater or leffe théarightangle. Firft let it be isgreater, Andilet the angle DBCbe aright angle, by erecting from the point Ba per- pendicular line vnto theline BC (by the 11.propofition) which let be BD: and puttheline B D equall to.the lyne #4 B (by the thirde propofition ), And drawe’a line from to D. Now (by the former propofition) the fquare of the fide C D thalbe equall to the fquares of.the two fides BD and BC: wherefore alfoto the fquares of the two fides B 4Aand BC. Wherefore the bafe CD fhalbe equall to the bafe C_A, when as their fquares are equall: which is con- trary to the 24. propofition.For forafmuch asthe angleA 4 | B B Cis greater then the angle D BC, and the two fides AB. et and BC are equall to the two fides D Band BC, theonetothe other, the bafe C A thall be greater then the bafe CD, Itis alfo contrary to the 7-propofition, for from the two endes ofone & the fame line, namely, fré the points B & Cfhould be drawn on ~ one and the fame fidetwo lines B Dand DC ending at the pointe.D, e- quall totwo other lines BA and 4 Cdrawen fromthe fameendes | and ending at an other point, namely,at A,which isim pof- fible.By the fame reafon alfo may we proue that the whole angle at the pointe B isnot leffe then a right angle. Wherforeitisa right angle: which was required to be proued. e The ende ofthe fit booke of Euclides Elementes, q lhefecondbookcofEu- «: clides Elementes. iY Ww ay \O es * nae: PH alfo out of this booke gathereth may dious rules of | A ea SSF ny compendious rules of reckoning,and many rules reckoning ga- 4 whe : “tise ei. sito si Aleabaiah the equatids cael et. The *hered ch of groundes alfo of rhoferules are forthe moftpartby this fecond booke demon. : “ booke,and ftrated, This booke moreouer contayneth two wonderfull propofitions, one of haieok i ie an obtufe angled triangle, andthe other ofan acute:which with the aydeof the fy,, . 4-7,propofition ofthe firft booke of Euclide, which isofa re@angle triangle,of Two wender. how great force and profite they are in matters ofaftronomy,they knowe which ful propofitis haue trauayled inthat arte, V Vherefore ifthis booke had none other profite be se m this fide, onely forthefe2, propolitions fake it were diligeatly to be embraced and 00k. ftudied. | T be definitions. Firft defini« 1. Eueryreitangled parallelogramme, is fayde to be contayned sion, under two right Lines comprebending aright angle. A parallelogramme is a figure offower fides, whofe two oppofite or contras What «pas ry fides are equall the one to the other. There are of parallelogrammes lower ee kyndes,a {quare, a figure of one fide longer,a Rombus or diamond, and a Rom- aa kindes boides or diamondlike figure,as before was fay de in the 33.definition of thefirft of parallela- booke. Of thefe fower fortes, the {quare and the figure of one fide longer are premmes, onely right angled Parallclogrammes: for thar all their angles are right angles, And either ofthem is contayned (according to this definition ) vnder two right lyrics which concurre together andicauferhe right angle,and containethe fame. Of. whichtwo lines theone:is the length of the figure, & the osherthe. breadth. The perallclogramme is imaginedto be made by thedraught of motion ofone ofthelines into the length ofthe other,Asifcwo numbers fhoulde be mulripli- edhe! one into the other; As thefigure A BC D.isa parallelograme, and 13 faydeto be contay ned ynderthetwo tight lines A Band A C,which contayne therightangleB A C,or vnder the two right lines A Cand: 4 a C D, for they likewifecontayne the right angle A C Drof | Tey which'2,lmes the one,namely,A B is the length, andthe o- | ther,namely ,AC is the breadth. Andifwemmagmetheline c D AC tobe drawen ormoueddireatly according to the légth Quail. of = SS ss — A SS = ees oie SSS RR > = . 5B SSS Se 72 I Ls A ae ‘ a aes , 7 Pe > _ - . < encased ee ee te «. to rene = joe gbawt ral adie ye. ea ng ba. Bi : met 0 is a8 ae - ee ———— - =") = nena -Soeweie 2 2 (osgeian naps seen _ mt o> 5 > = r sie ts PE --ws a. [RZ = = oe oe = a — = a = Se - it~ ; ° so S --> ne ~ == -: ae = SS ‘3 —S— = < Ss =e SS = =~ <= =— Se ee ' Tree ee ‘me an : - ~~ = . — fa. Tas : —— =—— a SSS “ -- ~~ a Se a Kk ee = 2 : ” Se erry = oie a : : : M - = A 2 - a ay <== = ~ = ? oS == Ss on. : : ieonih 7. age 3a = eS Ae ~ Sn se = : —- a > aon A Se - ex — ™ an on Gage. S-4 = = ~ ~ itccnastionl a oe Ast ~~ men TEA ~ =e = machina patente nani — = = ——— ~ at - > eet = - 5 arm : 2. =~ es b> 4 — oa eee —— : = S22: : 2 —— ~ _* a < . e ~ren *,~ Sh a “3 oe ee ot axe —-. = > glad we eee e po — oe —— a, * =~ =a. Fnac = tS VSS 2 of Sere —— : : es - — ~ ee a a SS ~—. ~~ —- _ = a - - —n? = -s = ee 4. a aa : ~ a = 1 Se SS = = = ——— —_— ‘= So = = so) = > = Sn = —— = = _ = = weit —— — - —- = aml — me ae ——= ——— Se S eS eS SS — yy: Z ~ = 3 a as + Sa SLPR eS >t BET ; a — = fo - — == 2a, = “es a SS TS = —_— SS ee ee 2+ ee - = = —_ a — —" A = Ene ee —<—— 7 — —- . SS — = — —S=_— —— = = : —= == = a are _ 2 Sens ee apwoas —_ —- oo. ve os ss >> ay = aie —— ne i > - “ * a — == : + a SSS -—- —_ a o —— tae _— ———= — a — > : -— ————— — —— ESS = == = Sars == , 5 -_-. F P..4,$ | D lelogramme. As C K and K B the perticuler diameters, are partes of the:line C Bywhich is thie diameteriofthe whole parallelogramme, And thetwo paralle- logrammes AEG K and K HF D,are fupplementes,becaufe they are-wythout the diameter ofthe whole parailelogramme,namiely,C B,Now anyone ofthofe pactiall parallelogrammes about the diameter together with chetwo {upple- mentes make agnomon, As the paralelogrameE BK Hi) with the two fupple- mestes AEG KandK:- HF Dmakethe gaomon PGEH. Likewife the paral- lelogramme G K CE with the fame two = erg the: gnomon EH f G,And this diffinition ofa gnomon extendeth it felfe; and is generail:'ro:all ky andes of parallelogrammes,whetherthey be {quares or figures of onefidelon- ger or Rhombus or Romboaides; To be fhorte,if youtakeaway from thewhole parallelogramme one ofthe partiall parallelogrammes which are about the di- amacter » aa San of Euchdes Elementes. Fol.61. ameter whether ye will,thereft oftheficureisagnomon. 5) OY s-- & , \Campa: ¢ after the lat propofition of the firftbooke addeth this pr opolitid, T ipo Squares being geuen, to adioyne to one of them z Gnombh'equall to the other '{quare-which, for that as then it.wasnot taught whata Gnomon ss|.ther e,omitted, thinking, that it might more aptly be placcdhere, The doingatid demonftration whereof, is thus, 345 NH3 UI yas ANN Sieh Suppofe that there be two fquarés 4B and C D:wnto one of which, namely ;vnto A B,itis required to addeaGnomomequall:tothe other {quare,namely, to C D,. Pro- duce the fide B Fofthe {quare 4B di- rectly to the hare E:andputthelineF gy p Eequallto the fide of the fquare CD. Sees “ And draw aline ftom Eto A.Now then; © heeal a | forafmuch as E F-disare@angle trian-, ' gle,therefore(by the.47. ofthe frft)the {quare of the line, EA is equall to:the )\>.4 {quares of the lines E F & F_A.Butthe. - ° yt yeh fquare of the line EF is equalkto the —+—— 3 {quare CD, & the {quate of thefide FA Bo AS £ena E is the {quare_4 5.Whereforethefquare sss 9h of the line 4 Eis equall to thetwo fquares C_D.and.4B,But the fides EF and F Aare (by the 21.of the firit) longer then the fide 4 E,and the fide F A is equall to the fide FB. Wherfore thefides EF and F Pare ton gerthé the fide 4 E, Wherefore the whole line B E is longer then theline.4.£.From the dine # Ecutofa line equallto the line 4 E,whichlet be 8 C.And (by. the.46 -propofition,). ypon.theline B.C defcribe a {quare, which let be: BCGH: which thalbe equal to the fquare of the line -4 E, but the {quareof the line 4 Eis equal to the two fquares .4 B and D C.Whetefore the fquare BCGA is | equal to the fame-fquares. Wherfore forafinuch as the fquare BCG His compofed 6f the {quare cf B and of the gnomonF G 4H, the fayde gnomon fhalbe equall ynto the {quare CD: which was required to be done. | Anothermore redy way-after Pelitarius, Suppofe that there betwo fquares,whofe fides let be e fB and BC,Itis required ynto the {quare of the line e41B to adde a §nomon equall to the fquare of the line BC.Set the lines &-4 Band B Cin fuch fort that they makea right angle 4BC, And draw aline frd 4 to C.And vpotheline 4B defcribe afquare whichlet be 4 BD E.And produce the line B A to the point F,and putthelineB F equallto the line 4C; And vpon the line B F defcribea {quare which letbe B EGH :. which’ fhalbe equal to the fquare of the line 4 C,whé as the lines B Fand.A CD H Care equaltand therefore it is equal to the {quaresofthetwo ® lines 4B and.BC. Now -forafinuch 4s the fquare B F G His made complete by the fyuare 4 BD Eand by the gnomon Fé GD,the genomon F EGD fhalbe equal to the fquare of theline BC; which was required to be done. AA bropofition added by (ame pane after the laft propo/fiti- on of the firit booke, Demouftratio lines DK, EL andC H.Now then the parallelo- T he fecond Booke $a» The 1.Theoreme. The1.Propofition. Ff there betwo right lines, andifthe one of them be deuided into partes bowe many foeuer : the retlangle fioure compre- bended under the tworight lines is equall to the rectangle fi _ gures whiche are comprebended under the line wndeuided. and under enery one of the partes of the other line. ERGO? Uppofe that there be rwo right Ines «A ; x NEA and B Cand let one of them, namely, BC be deuts SNORE ded at alladuentures inthe pointesDand E.Then he fy) I fay that the Pie fame comprehended vne fC 58 equall vnto:the refans CSET “ der the lines A,and B rN HC FG ae : ‘gle figure comprehended vnder the lines 4 and B WALI Mia Der ynto the rectangle figure which is co prebens Sa ia\\ ded vader the lines dand D Ey and alfovnto the DBE rettangle figure which is comprebénded vnder the lines Aand EC, For fromthe pointe B rayle vp (by the of the firft) vuto the right line BCa perpendiculer line BF ¢o vuto the peas origepa i BS call, line A (by the third of the first) put the line BG es quall, and by the point G (by the'r, of the fir/t) draw 4 parallel line vnto the right line BCand let the fame be G Mand by the felfe fame )by 'y points D,F,andC, draw vnto the line BG the/e parallel grame B Fis equallto thefe parallelogrammes B K,OL,andE H.But the parallelograme B His equall ynto that which is contayned vnder the lines A and BC, (For it is coms prebeded vnder the linesGB.ce BC,and the line G Bis equall ynto the line A) And the parallelograme BK is equal to that which is contayned vnder the lines AandB D: (for it is comprehended vnder theline GR and BD and BG is es quall vnto.A).And a D Lis equall to that which is conta ned vnder the lines Aand D E( for the line D K., that is,B Gis equal vnto A) And moreouer like Wife the parallelograme E His equall to that which is contained vader the lines Azg ECVV berfore that which is compreheded vnder 5 lines A e> BC is equad to that whichis comprehended vnder the lines A c> B D,c> ve 10 y whichis comprebeded vuder the lines A and D EF) and moreoner vnto that which ts com prebended vnder the lines Aand E. C If therfore there be two vi cbt unes,and if the one of them be denided into partes how many jocucr the reGan: gle of Euchides Elementes. Fol.62. gle figure comprebended vnder the two right lines,is equall to the reCtangle fie gures which are comprehended vnder the line yndenided and ynder euery one of the partes of the other line: which was required to be demonstrated. “Becaufe thatall the Propofitions of this fecond booke forthe moft part are true borh in lines and in numbers, and may bedeclaredby bach: therefore haue I haue added to euery Propofitionconuenient aumbers for the manifettatioa of the fame. Andro the ead the ftudiousand diligent reader may the more fully perceaue and vnderftand the agrementofthis arcotGsomercry with the{cience of Arithmetique,andhow nere & deare fifters they are together,fo rhat the one cannot without great blemith be withour the other, 1 haue here alfo toyned a little booke of Arithmetique written by one Barlaam, a Greeke authour aman of greate knowledge, In whiche booke are by the authour demonttrated many of the felfe fame proprieties andpaffions in number, which Exchide in this his fecondboke hath demonftraredin magaicude,namely the ficft ten pro. pofitions as they follow ia order. V Vhich is vadoubtedly gear pleature to co fider,alfo great increafe & furniture ofkno wledge, V Vhofe P ropofitids are fer orderly after the propofitids of Euclide, eucry one of Barlawm corre(pdent to the fame of Euchide. And doubtles itis wonderful to fechowthefe two cotrary kynds of quantity quantity difcrete or number,and quantity continual or magnitude (whicharethe fubie&es.or matters of Arithmitique and Geometry ) thoulde haue tn them oneand the fame proprietiescommonto them borh in very ma- ny pomnts,and afe@ions,although not inall.Foraline may in fuch fort be de. uided, that what proportion the whole hathto che greater parte the fame {hall the grearsr part haue to the leffe, But chatcan not be in number. For a number cannot to bedeuided,thatthe whole number t» the gceacer part thereof, {hall haue that proportion which the greater parthath othe leffe, as Lordaney ery play.ncly.proueth in his.booke of Arithmertke , which thynge Campane alfo (as we fhall afterwardinthe 9, booke after the I§. propofition fee) proueth, Andas touching thefe teane firlte propofitionsof the feconde booke of Eu. clide,demonftratedby Barlaaminnwnbers they are alfo demdftrated of Cam- pins after the 15,propoficion ofthe 9. booke, whofe demoaftrations [ my ade y Gods helpe to fet forth wheal {hal come to the place. They are alfo dem5- {trated of lordane that excellet learned aurhout in the frit booke of his Arith- metike. Inthe meane ty me | thouzhrit not amiffe here to fet forth che demon- itrattons of Barlaam, for that they geue- great lightro the fecondebooke of Eu- clide, befides the ineftimable pleafuve;which they bring to the ftudtous confide- rer, Andnowto declare the firft Propofition by numbers. EF haue put this exam- ple following. | | Take two numbers the one vndeunidedias 74,:the other deuided into what partes and how many you lift,as 37. detiidedvinto'z0. 10,5. aud’ 2:which altogether make the whole 37. Then ifyou multiply the number vndeuided, namely, 74; into all the partes of the number deuided asinto 20. ¥o. s.and 2. you fhall produce 1480, 740, 37° .148.which added together make 273 8: which felf numberis alfo produced if you multiplye the two numbers firtt geuen the oneinto the other. As you feein the exam- ple en the other fide fet. } ; Rit, Mule 1 HL Barlaam, Barlsam, The fecond Booke. 74 37 Multiplication of the whole 1480 20 naber vndeuided into the Se ne oe 10 artes of the whole num- 370 5 er deuided. | 148 2 2738 } thenumber produced of theone | whole number into the partes of the other whole number Multiplication of the one | 74 | : whole number into the o- 37. . cequallto ther. mer EL aa L 2738 the number produced of the ~~~ fame whole into the other whole So by the aide of this Propofition is gotten a com pendious way of multiplication by. breaking of one of the numbers into his partes: which oftentimes ferueth to great vie in working chiefly in the rule of proportions. The demonftration of which propofition followeth in Barlaam.Butfirft are put of the author thefe principles following, 7 q Principles. | 1. eA aumberis faydto multiply an other number: when the number multiplied is fo oftentymes added to.it felfe,as there be unities in the number,\which muleiplieth: wherby is produced a ctrtaine number which the number muleiplied meafureth by the unities which are in the number which mul- tiplieth, | Bz : 1 = And the number produced of that a multiplication is callod a plasne or fuperficiall number. 3. ef fquare uumber is that which ts produced of the multiplicatian of any. number into it felfe. 4. €xery leffenumber compared to a greater is faydto be apart of the greater whether the leffe meae. farethe greater ,or meafure it not. | 3 ries 5. Numbers, whome one and the felfe fame number meafureth equally, that 3s, by one and the felfe fame number are equall the one to the other, ei w 657 6. Numbers that are equemultiplices to one and the felfe famenumber, and the fame number equally and alike,are equall the one to the other, The first Propo/ition, T wonumbers beyng genen,if the one of them be deuided into any numbers how many foener; the playne or fuperficiall number which ts produced of the multiplication of the tYpo numbers firft geuen the one into the other fhall be equall to the Superficial ni- bers Which are produced of the multiplication of the number not deuided into enery part of the number deuided. | Suppofethat there betwonumbers 4Band C:And °° 8 deuidethe number 4 B into certayne other numbers | how many foeuer,as into 4.D,D E,andE B, Then I fay that is Ywhich contayne one that the fuperficiall number which is produced of the pa multiplication of thenumber Cinto the number e¥ B is equall to the fuperficiall numbers which aré produ- HE ced ofthe multiplication of the number C into the ni- ber_4'D,and of Cinto D E,and of C into E-B. For Jet F be the fuperficiall number produced of the multiplica+ [)).: tion of the number C into the number .4.B,and.let@A | > be the fuperficiall number produced of the multipli- {4 cation oft Cinto AD : Andlet H / be produced of the | — multiplication of Cinto D E: and finally of the multi- 7 plication of Cinto F B let there be produced the = c UA | cr of Euchdes Elementes. Fol.63. ber / K.Now forafmuch as_4 B multiplying the numberC produced the number F: therefore the number C meafureth the number F by the yniries which are in the nnm« ber 4 B.And by the fame reafon may be proued that thenumber C doth alfo meafure the numbet G A, by the vnities which are in the numbere4 D,and thatlit doth mea- {ure the number AJ by the vnities which are in the number D F and finally that mea- fureth the number J K by the vnities which are in the number E B,Wherefore the na: ber C meafureth the whole number G K by the vnities which are in the number 42. But it before meafured the number F by tne ynities which are in the number AB wher fore either of thefe numbers F and G K is equem ultiplex to the number €. But num- bers which are equemultiplices to one of the felfe fame numbers are equall the one to the other (by the 6.definition. Wherefore the number Fis equa'lto the number G K, But the number Fis the fuperficiall number produced of them ultiplication of thenti- ber Cidto the number 4 B: and the number G Kis compofed of the fuperficiall num- bers produced of the multiplication of the naber C not deuided into euery one of the numbers 4 D,D E,and E B,/f therefore there be two numbers geuenand the one of them be deuided &c, Which was required to be proued. The2.Theoreme. The2.Propofition. If a right line be deuided by chaunce, the retlangles figures which are comprehended under the whole and euery one of the partes , areequall to the fquare whicheis made of the whole. ian / ppole that the right line AB be by chaunfe dee RSIS) nided-inta the point C. Then I fay that the reftaz YX) gle figure comprehended vnder AB and BC toe gether with the rectangle comprehendep vnder AB and AC isequall ynto the fquare made of A B, Dez Seribe (by the 4.6,of the first) vpon A Ba fquare AD EB: and (by the 31 of the fir/t) by the point Cdraw a line paral: lel vnto either of thefe lines ADaud B E and let the fame be CF. Now is the parallelogramme A E equallto the parallelogrammes AF and C E, by the firSt of this booke. But A Eis the )quare made of AB. And AF is the reclangle parallelogramme comprebended vnder the lines B A and AC: for it is compres bended vnder thelinesD Aand AC: but the line A D is equallv..tothe line A B, And like wife the parrallelogramme C Eis equall to that which iscontayned vuder. the lynes ABand B (- for the line BE is equal vntothe line ABVV her- forethat which is contayned vnder BA and A Ctogether with that which is contayned ynder the lines A B and BC is equall to the /quare made of the line AB. Tf therefore a right line be denided by chaunce the reétagle figures which arecomprebended ynder the whole , and euery one of the partes,are equal to the {quare which ismade of the whole: which was required co be demonstrated. Fe _ An otherdemonftration of Campane... . R iil. Sup- i Wn _* 0a Sin hale, - Sara uss ’ - a. , SL é tthe Some a ee a ne Barlaam. The fecond Booke Suppofe that the line 4 Bbe deuided into thelines 4C,C D,and DB.Then I fay that the {quare of the whole line 4 B,which let be 4 E BF, is equal to the re&angle figures which are contayned vnder the whole and euery one of the partes : fortake the line K,which let bee- qual to the line AB,Now then by the firft pro- | pofition the rectangle figure contained vnder tae lines 4 Band K,is equallto the re@angle figures contayned vnderthe line K and althe partes of the line 4 B. But that which is con- tayned vider thelines Kand AB isequall to the {quare of theline AB, and the reéangle figures contay ned vnder the line K and al the partes of 4B, are equall to the ae fi- gures contayned vnder the line 4B andallthe partes of theline AB: forthe lines 4 Band Kare equall: wherefore thatis manifeft whichwas required to be proued. i AO mn Me ee PR eee, mm en > 2 o ie) Anexample of this Propofitionin numbers. Take a number,as rt.and deuide it into two partes,namely, 7.and 4: and multiply rt,tnto 7,aadthen into 4,and there thalbe produced 77.and 44:both which numbers added together make 121. which is equallto the (quare number produced of the mul- tiplication of the number 11.into him(elfe,as you fee inthe example, Multiplication of the whole 1 bila cee 2 | ing his parces. } ees 77 7 44 4 | hy Sy the number produced of the | 3 | | ~ | wholeintohis partes. Iv ultiplication of thewhole = equal to into.hiaielfe, 7 7 = 2 ss TEE | Ir eke Dew tae , 121 the number produced of the whole into hinifelfe, “Thedémo aftration whereof followeth in Barlaam, ~The Jecond Propofition, Tfanumber geuen be denided into two other nu mbers: the fuperficiall numbers, which are pres . duced of the naultiplication of the whole inte either part,added together are equallto the Square nume — ber of the Whole number geuen, Von 4 ee “4 Suppofe that the number geuen be 4-B:and let it be denided into two other num: bers AC andCB. Then I fay that thetwofaperficiall numbers; which are produced, of the multiplication of 4B into 4 Cand of 4 Binto BC, thofe two fuperficiall num- bers (I fay) beyng added together, fhalbe equall to the fquare number produced of the multiplicatio ofthe number AB into it felfe.For let the numberef B multiplying it {elfe produce the number D, Let the number 4 C alfo multiplying the number 42 , . produce of Euchdes Elementes. Fol.6 4. produce the number £ F-agayne letthe namberC,B multipli- ing the felfe fame number -4 B’produce the nnmberF G . Now G forafmuchias the number e4#C multiplying the nimber ef Bi, produced'the number E F: thetefore.the numbere’A B meafu- | reth the numbet E F by the vnities which are in 4 C,Againe for- afmuch as thenumberCB multiplied the number 4 B,and pro: duced the number F G: therforethe number 4.2 meafureth the | number F G by the vnities which aresin the number CB.But the ) fame number 4B before meafuredthe numberEF by theyni- | 5 | ties which aréin the number AC. Wherefore the number 4B | meafureth the whole number &G by thevniities whcihvre in 4. | | B, Farther forafmuch asthe number 4.2 multiplying it felfe pro | | duced the number D: thereforethenumber 4% meafureththe + | number D bythe ynities which are in himfelfe.Wherfore it mea fureth either of thefe numbers,namely,the number D,and the number E G,by the ynities which are in himfelfe, Wherfore how multiplex the number D is tothe mumber 4B, fo multiplexis... | D the number EG tothe fame number, 48. But numbers which are equemiultiplices to one and the felfe famenumber,are equal the one to the other. Wherefore the number Ds equallto the number £ G.Andthe number Dis the fquare number made of. 4 thenumber_4B,andthe number EGiscompofedofthetwofu- — perficiall numbers produced of 4B intoB C, and of B Ainto A. C. Wherefore the fyuare number produced of the numberc4 B is equall tothe fuperficial numbers,produced of the number A Binto thenumber BC, and of A Binto AC, added together. If thereforea number be denided into two other numbers &c, which was required'to be proued. $e T he 3. Theoreme. ‘The3. Propofition. | Ff aright line be deuided by chaunce:the rectangle figure co- rehended under the whole and one of the partes,is equal to the rectangle fieuse comprebendedunder the partes,¢7 vnto the /quare which 1s made of the forefaid part. a amg) iV ppofe that the right line geuen A B be deutded by chaunce in the £4 point CT hen I ay that the reftangle figure comprebeded vnder the lee | lines. A B and B Cis equal vnto the rettangle figure comprehended ===—= ynder the lines AC andC Band alfo vnto the fquare which is made of the line BC. Deferibe( by the 46 of the first ypontheline BCa fquare CD Confiruttion, EB: and (by the fecond peticion extend ED vato F. And by the point Adraw (by the 31.0f the fir/t )a line parallel ynto either of thefe lines CD and BE and letthe fame be AF.Now the parallelos. = “ ‘4 grame A Eis equall vnto the para lleloz 7 grammes A DandC E,And A E is the reclangle figure comprehended vnder the lines AB and BC Foritis compres © D F bended vnder the lines A Band BE, Demonftratio ' 7 a iM } ine ” B) is if j fy ' ei , a) iD y ¥ : | B \/] ri ' ih} " i; ti u | | I | qi if ee SS Ses > a - = Se > 7. a See SRS eer - 2 are ee rena , — SSS SS SSS = ~ — eS Lea a TS ~ —— =~ Sars : epee a dss ———— ew cea toad TN x ree oe : : = -~ “ ; —- = — ——— ore « ~ mt a. Sey as a _ = ee nae m NR ST Ee — aS z = ey Bailaatte The fecond Booke*= © which line BE is equall vnto the line BC, And the paralelograme A Dis ee quall to that which is‘contayned vnder the lines-AC andC B: for the line D Cas equall ynto the line CB. And D Bis the fquare which is made of the lyneC D. VV herfore the rectangle figure comprehended yuder the lynes AB and BC is equall to the rectangle figure.comprehended vnder the lines AG and© Bw als Jorvnto the [quare which is made of the line BC, If therfore a right line be dee uided by chaunce,therectangle figure comprehended ynder the whole and one. of the partes,is equall-to the rectangle figure comprehended ynderthe' partes,: and ynto the {quare which ts made of the forefayd part: which was'requsred to be proued, ; iat Wot ots ds ' wWaeasd ¥% Anexampleofthis Propofition in: numbers; 21) v0 \ soc oun Suppofe a number,namely,14.to be denided into two'partes 8,and'6:The whole: number 14.multiplied into 8.one of his partes,producetit r1 2: thepartes 378 6.mul- tiplied the one into the other produce 48,which added to 64( whichis the {quare of 8. the former part of the number Jamounteth alfo to 112% whiche is equalltothe former: fumme.As you {ee in the example. OF 210.9 | oO: f ne { Bopthe partes: | the number produced of the ' d whole into oneof hispartes. Multiplication of thewhole }> {-’ into one of his partes, Multiplication ef. the oné 8 e sort | partintotheothers i ~ ee eq atin mA 64 ¢ , a e sede I I 2 ~ “Multiplication of the for-. 9 the number compofedofthe gv, One partinto the other, and ) ofthe former partinto him- . Paris ahvlelfe.: | ¥ Mer partinto it felfe, The demonftration hereof followeth ii Barlaam, The third propofition. +) If anumbergenen be denided into two numbers: the fuperficial xumber which és produced of the multiplication of the whole into one of che paresis equal to the (aperficiall number which is proe Aucedofthe partesthe one intothe other,andtothe (quare number produced of the forefayd part. » Suppofe that the numbergeuen be A B,which let be denidedinto two*numbers.4 CandC B.Then /fay that the fuper ficiall number whicheis produced of the multiplie cation of the number e4 ZB into the number BC, is equall to the fuperficiall number which is produced of the multiplication of the number 4 Cinto thenumbet C B,and. tothe {quare number produced of the numberC B-Forlet the numbere 4B multipli« eng the number C B produce the number D.And let the number _4C miultiplieng the number CB produce the number E Fzand finally let the number C 2 multiplieng him- felfe produce the number F G. Now forafmuchas the number 42 nail the aS WA number a of Euclides Elementes. Fol. 65. number C'B produced the number D.Therforethe number | Se B meafureth the humber D by the ynities whiche are in the | S number A B.Agayne forafmuch as the humber_A4 C multipli- | ed the numberC’ 2B,and produced the nuthber E F, therefore thenumber G8 meafureth the niber EF by the vnities which are 1n.4C.Agayne forafmuch as the number C B multiplied it | € felfe and produced the number FG: thetfore the number CB meafureth the number F G by thevnities which atein it {elfe, B But as we haue before proued the felfe fame nitber C B mea ] fureth alfo the number £ F by the vnitieswhichateinthenf- | ber 4 C,wherfore the number C B meafureth the whole tium- ber E G by the vnities which arein thé number AB, And it al- Ba fo meafureth the number D by thé vnities whiche are in the : number 4 5,Wherfore the number C # equally meafurethei- $ thernumber,namely,the number D,andthehimberEG.But , _ thofe numbers whome one and the felfe fame number meafu- 4 reth equally, are equallthe onetothe other, Wherfore the number Dis equalltothenumber EG.Butthenumber Disa fuperficiall number produced of the multiplication of the number 4 2 into the number @ C,and the number EGis the fuperficial number produced of the multiplication ofthe ni- A D 8B ber 4C into the number C2B,and of the fquare of the number C B.Wherfore the fuperficial number produced of the multi- plication of the number 4 2 into the number C Bis equal:to the fuperficiall number produced of the number 4 C into the number.C B, and to the {quare ofthe number C B,lftherfore anumber be deuided into two.numbers,the fuperficiall naber &cswhich was required to be proued. : | . Lhe 4.Theoreme. T. be 4. Propofition, Ifa right line be denided by chaunce, the fquare whiche is maae of the whole line is equal tothe [quares which are made | of the partes,¢x unto that rectangle gure which is compre- bended ynder the partes twife. | vg V ppofe that the right lyne AB be bychaunce denided in the pointe C, RD Then I fay that the fquare made of the line A Bis equall vnto 5 fquares “=s=iyhichare made of the lines A CandC B, and vntothe refban gle figure contained vnder thelines. AC and€ B twife.Deferibe: ‘ Pa9P og (by 9 46, 0f the first) Vponthe line AB afquere ADE ASC to B:and draw a line from B to D,and( by the 3 1,0f the +3 AGS we /* Gg K firft)by the point C draw a line parallel pnto either of | Sal : thefe lines A Dand B Ecutting thediameter BD in’ ey the pomt Gand let the fame be CF, And(by the point | “7 a | G (by the felfe fame) draw a line parallel ynto eyther pss iad ‘Su of thefe lines AB and DE, and let the fame be FI ® me a8 K, And forafmuch as-the line CF is a:parallel vnto Conftrathions. Demouftratis T he fecond Booke theline A D,and vpon them falleth a right line BD: ther fore(by the 29,0f the firft)the outward angle G B is equallynto the inward and oppofiteangle A D B,But the angle AD Bis (by the 5, of the firft) equall vnto theancle AB D: for the fide B Ass equall yuto the fide AD (by the definition ofa /quare). VV herfore the angle CG Bis equall ynto the angleG BC: wherfore( by the 6, of the firft )the fide B Cisequall vntothe fide CG. But C Bis equall yntoGK , and C Gis equallynto K B: wherfore G K is equall nto K, B.VV herfore the figure (GK B confifteth of foure equall fides, fay alfo that it isareétangle fie gure.For forafmuch as CGisa parallel ynto oe vpon the falleth a right line CB,therfore(by) 9,0f the 1, )the angles KB CandG CB are equal vnto two right angles,But the angle K BC is aright angle, wherfore ) angle BCGis alfo a right angle VV berfore( by the 3 4,0f the first the angles oppofite vnto them, namely,CG K ,andG K Bare right angles, VV berfore C G K Bisa reclans gle figure. And st was before proued that the fides are equall. VV her fore itis a [quare,and it ts defcribed vpon the line BG And by the fame reafon alfo Hi Fis afquare andis defcribed vpon the line 1 G,that is ,v- pon theline AC VV herfore the /quares HF and C K > aremade of the lines AC and C'B. Andforafmuch as 4; 6 the parallelograme AG is (by the 43, of the fir?) e- quall vnto the parallelogrammeG E,And AG is that which is contayned ynder AC and CB, for CG is equal yutoC By wherforeG E ts equallto that which is cons tained ynder AC andC B,VVherefore AGaudGE ” ss areequall vnto that which is comprehended vnder A Cand B twife. And the [quares H Fand CK are made of the lines AC and C BVV her fore thefe foure rectangle figures Fi F’.CK AG, and G Eare equall vito the [quares whiche are made cf the lines AC andC B,and to the retiangle figure which is comprehended vader the lines AC andC B twife.But the reéte angle figures F1F, CK, AG,and G Eare the whole rectangle figure ADEB which ts the (quare made of the line A BVV berfore the /quare which is made of the line AB is equall tothe {quares which are made of the lines A CandC B, andynto the rectangle figure which is comprehended vnder the lines A Cand C Bewifetf therfore aright line be denided bychaunce, the [quare whiche is made of the whole line,ss equall to the fquares which are made of the partes¢o vato the rectangle figure whichis comprébended vnder the partes twife:which Was required to be proued, | K S> wAn other demonftration. | I fay that the fquare of the line AB isequal vnto the /quares whiche are made of the lines A CandCB, ex vnto the relangle figure which is coniprebie ded vuder the lines ACandCB twife. For thefelfe fame difcription abiding,fore afmuch of Exuclides Elementes. Fol. 66. afmuch as the line ABisequall vnto} line AD, jangle ABD is(by the 5..of the firSt) equall ynto the angle AD B,And fora/much as the three angles of enery triangle are equal to two right angles (by the 22. 0f the first) therefore ¥ three angles of the triangle A BD,namely,the angles AD BD A,and BA D are equall to two right anoles.But the angle B AD is a right angle, wherefore the angles remayning AR D,and AD B, are equall vuto one right angle:and they are equally one to the other, wherfore either of thefe angles AB Dex AD B,is the halfe ofa right angle, And the angle BCG is aright angle, for itis equall ,., vito the oppofite angle at the point A (by the 29, of | the first) VV berefore the angle remayning CG Bis wet the balfe ofa pvt angle. VV herefore the anole CGB | 4) aS as equal ynto the angle CBG: wherefore alfo the fide (<1 B Cis equall ynto the fideC G.But BCis equall ynto ~ Ske GK, and C Gis equal vntoB KWV herefore the fiz | gureC K conjiSteth of equall fides:and in it isa right anole C B K .VV beres fore C Kisa fquare,and is made of the line BC. And by the fame reafon HF is a/quare,and is equall ynto that/quare which is made of the line A (VV beres fore C Kand Fare fquares,and are equall to thofe [quares which are made of the lines ACand CB, And forafmuch.as AG is equall vnto EG:and AGis that which is contayned ynder AC andC B,for GC is equal vnto C B: wheres fore EG alfo is equallto that which is coprebended vnder AC and CB: wheres fore AGand EG are equall vuto that reftangle figure whichis comprehended vader AC, and C Btwife.AndC K and HF ere equal ynto the [quares which are made of A(,and(B: wherefore C K,HE,AG, and GE are equal yne to yf [quares which are made of A CandC B sand vnto that rechangle figure Which is comprebended vnder A Cand C Bewife. But C K,HE,AG, andG EB are the whole [quare A E which is made of A B,VVherefore the /quare which is made of A Bis equall to the /quares which are made of AC andC B. and vne to the rectangle figure whichis comprehended vnder AC andC B t wife: which was required to be demonstrated, Teo Nee Flereby it is manifef? that the parallelogrames which confift about the diameter of 4 fquare muff needes be fquares, > This propofition is of infinite yfe chiefely in furde numbers,By helpe of it is made in thé additid & fubftration,alfo muiltiplicati6 in Binomials & refidu- als, And by helpe hereofalfo is demonftrated that kinde of equation, which is, when there are three denominations in naturall order , ot equa.ly diftant, and two ofthe greater denominations are equall to the thirde being leffe On this propofition is grounded the extraction of {quare roots,And many other things arealfo by it demonftrated, 3H Sy, An AL Corollarye ~ - — - - =—- ws <=> fae eres = a — — —--- wr — = —— : —— a ~—— = - CM « - * —+ ~—— ae wots - ———— “ = Se - — ——_ a —_ Sew A te Ante ll pny a 5 n a a = ine : - te — = ord — a _ ee * ee ee —— - - Sa - _. — 2 _ = — : ce NR BE a Se ee Fea ae —--. ee ; eee - a SF _ = ba. Ene ~—+- Ee ae _— =e = ~ ot ew = = + : j y * es. | ii i a ee maint) olin | bs Ae ei 2 as ee s a ig inca « 2 seal ¢ ne — ~* — a CR a epee ee " - >< =v. fo ae Bien. ~oy aan > < —— - i z => a y + - > o < ~ ook - on = *. ~ = = w : — ~—— - - S.. Sates < é = _ _ - ° ee x —_ ee 5 Say = A i ate tants Sp ee - ~ = Ta = 4 . —— =o —— - ‘ ste ~ PP. -) 4 = 2e . ~_ Qe = q ~ Pee - > F SRR ST SS Ca ee ea M = = Se eee = == == : E eek se ~—, : - os = ~~ z ~ - <4 aa x er ve ‘ ee oe "333 = —— are. ~~ — ———" ——— - sh ee ae 2 ann ptlain l * oa = 2 ‘- a Lo) aes as ae a eo ——~- - _ —- or “ . = —= RO = mA + a == ee Sa a 3 es > pe — Sa a ee : — Se — a - ———— —— < ~ “ wey —— —— — ———— ee — ——— = = =: : i ~ —~ _ = = - — 2 —- : _— S ee =-is —= = = ——- ~ ——- : - Pea as ii — es a. a -- oe - eng ee - AS: Sees « — = - = _- —— = oe = Pam tee rns) — = > ~ = a —_ — - ~~ _ — ——- ——— - ——— — — - ~-- +S ~ ee Sa tae a eee ie = —* ae ee 2 e ie, & rez Pn er any = "2 ve => = 3 a = : > 7 _——— — " Sarre = “ _— = = = bres <4ve3- é — = + --- ~--+ ~ SS cr SE. ~ ——s _ <= eas, —=. J . =7 Sas, 4. Te = ST _ ee “2 ae = —— == - Aan : : : : = - = => —=—" = —=- —— SL — = ———- -- eo — = = - $= os s an a8 et rns PR 2 - . = 5 is, = at = 3 a ry — = = bate yt.” = ee = is r cena, = ee — = pews = = ae S — — a PSA se lenacaceia = vaginas i i SS es - — . SSS = SS : SSS ea = ; > 5 = =F ee = Sas ~~ = -> =. £5 = = == ae ae = 5 Se oS ee eee a = == ¥ = — — —- = - Barlaam, The fecond Booke © Anexample of this Propofition in numbers, Suppofe a number namely,17.to be deuided intotwo partes 9. and 8. The whole number 17-multiplied into himfelfe,produceth 289.The eer numbers of 9, and 8. are 81. and 64: the numbers produced of the multiplication of the partes the one in- tothe other twife are 72, and 72: whichtwo numbers added to the fquare numbers of 9.and 8,namely,to 81.and 64. make alfo 289,which is equall to the {quare number of the whole number 17. As you feein the example, The multiplication of the af 9 whole into himfelfe. 17 4 g° hehe partes of the whole 119 17 . | 289. ~——— c¢thenumber produced of the. - | ( whole into himfelfe. The multiplication of eche 9. part into himfelfe, | 9 equall to the number compofed of eche art into himfelfe,and of the on The multiplication of the Sito thd biker bie : ‘ 9 _ | One partinto the other 3 8 93 The Sncstiratan wherof followeth in Barlaam, The fourth Propofition, If a number geuen be denided into two numbers: the fquare number of the whole,is equall to rhe Square numbers of the partes, and to the fuperficiall number which is produced of the multsplication of the partes the one into the other twife. Suppofe that the number geuen be-4 B: which let be denided intotwo numbers A Cand CB. Then I fay that the fquare number of the whole number 4, is equall to the {quares of the partes,thatis,to the fquares of the numberse 4 Cand CB,and to the fuperficiall number produced of the multiplication of the numbers 4C and CB the onéinto the other twife. Let the fquare number produced of the multiplication of the whole number 48 into himfelfe be D. And let C4 multiplied into himfelfe produce the number EF: And (‘B multiplyed into it felfe let it produce G A: and fi- nally of the multiplicatié ofthe numbers 4 Cand CB the one into the othertwife let there be produced either of thefe fuperficiall numbers F G and A K. Now forafmuche as the numbere#C multiplying it felf produced the number EF: therefore the num- ber e 4C meafureth the number EF by the vnities which are in it felfe. And forafmuch asthe number CB multiplyed the number C .4 and produced the number F G: there- fore of EuclideS Eleméntes. Fol.67. fore the number 4C meafureth the niberF Gbythe vnities |.\\ . whiche are inthe number CBZ, Butit before alfo meafured K the number & F by the vnities which ate in it felfe. Where- | fore the number 4B nuultiplying the numbet .4C produ- | ceth thenumber £ G.Andtheretore the numiber EG is the | 4 fuperficiall number produced of the multiplication of the | number 8 Ainto the numbere4C, And by the {ame rea- | fon may we proue that’the number GX is the fuperficiall number produced of the multiplication of the number A B into the number BC.Farther the number Dis the fquare of | the number .428.Butifanumber be deuidecintotwonum- 4! ¢ x] F bers, the {quare of the whole number‘ is equall to thetwo : | se 5 fuperficiall numbers. which are produced of the nuultipli- aie | cation of the whole into either the partes (by the 2, Theo- | | reme.) Wherefore the fquare number Dis equall to the fu- | | 34 perficiall number £ XK. But the number EX is compofed of 2 | | E the {quares of the numbers 4C and CB, and of the fuperfi- cial number whichis produced of the multiplication ofthe nuber 4 Cand CB the one into the other twife:& the num- ber Dis the {quare of the whole number 42. Wherfore the {quare number produced of the multiplication ofthe num-. __4 D ber 4 Binto himfelfe, is equall to the fyuare numbers of the partes, that 1s,to the f{quare numbers of the nabers 4C | and C B,and to the fuperficiall number produced of the multiplication of the num- bers 4 Cand CB, the one into the other twife, If therefore a number geuen be deui. ded into two numbers &¢.Which was required to be proned. The 5. heoreme. The 5.Propofition, Ifaright line be deuided into two equall partes, ¢s into two ynequall partes: the rectangle figures comprehended ynder the ynequall partes of the whole,together with the fquare of that which is betwene the fettios,is equal to the fquare which is made of the halfe. eid / ppofe that the right line A B be denidedinto to equall partes in the bee point C,and into to vnequall partes in the point D. Then I fay that the = reclangle figure comprebended vnder AD and D B together with the . [quare which ts made of C D, is equallto the fquare which is made of CB, Dea Cm/trattion, fcribe (by the 46. of the first) vppon CB afquare, and let the fame be CEF®B. And(hy the firft peticion )drawe a line from Eto BAnd by» point Ddrawe.. (by the x ,0f she firft) a line pas rallel yntoechof thefe lines CE and BF cutting the'diameter B EF in the point H, and let} fame be DG. Andagayne( by the felfe. EE aEnSENpae ence oe — ~ ore = iene =~ 7 ers —e a 52" _— —* ae - ———— ——— = = == a SS -—— SS SSS SS ee eee ee Demonftratio T he fecond Booke fame ) by the point Fidrawea line parallel vnto eche ofthefe lines A B and EFyand let the Jame be KO. andlet K.0 be equall vuto AB, And againe (by the felfe fame )by the point A draw a line parallel vnto either of thefe lines CL and B 0, and let the fame be AK. And forafinuch as (by the 43,0f the firft the fupplement Cis equall to the fupplemét FF. put the figure DO coe mon vuto them both, VV herefore the whole figureC 0 1s equall to the whole fre gure DF, But the figure CO isequall vnto the figure AL, for § line AC is equall vato the line CB. VV herefore the figure AL alfo is equal vnto the figure D F. Put the figure C FI common vnto them bothVV herfore the whole figure AHL ss equall vuto the figures DL and D F. But A His equall to that which is co tayned vuder the lines A D and D B,for D His equall vnto D B.And the fi- garesF Dt DL are the Gno- mon MNXVV herfore yGnow A mon MN X is equall to that whichis contayned vnder AD and D B.Put the figure LG coe mon vnto them both whichises .* qualto the/quare which is made of CD,VV berefore the Gnoma MN X and the figure L Gare equall tothe rectangle figurecoe prebended vader A Dand DB and ynto the fquare which is made of CD. But the Gnomon MN X, and the figure L G are the whole {quareC EF B which is made of BC, VV herefore the rettangle figure comprehended vnder AD and DB, together with the fquare which is made of C D,is equall to. the fquare which is made of CB, If therefore a right line be deuided into two equal parts, and into to vuequall partes the reétangle figure comprebended vnder the vn. equall partes of the whole,tovether with the (quare of that which is betwene the fections,is equall to the {quare whichis made of the balfe: which was requis red to be proned, This Propofition alfo is of greate yfe in Algebra. By it 1s demonftrated that equation wherein the greateft and lealt kare&tes or numbers are equall to the middle. | An example of this propofition in numbers, _ _ Takeany numberas 20: and deuideitinto two equall partes 10.and ro. and then into two vnequall partes as 13. and 7.And take the differéce of the halfe to one of the vnequall partes whichis 3. And multiply the vnequall partes,thatis,13 and 7,.cthe one into the other,which make 91.take alfo the {quare of 3.which is. 9. and adde itto the forefayde number g1:and {0 thall there be made 100. Then multiply the halfe of the whole number into himfelf, that is, take the {quare of to.which is 190. which is equal to the number before produced of the multiplication of the vnequal parts the one in- to the other,& of the difference into it felfe which is alfo 190,Asyon {ein the example: The of Euclides Elementes. Fol.68. The whole euen number, 20 IO ban dd ni pee equall partes 13 Lab > the ynequall pattes 3 The difference of one of the vnequal partes to the | alfe. 13 Multiplication of the vn- 7 equall partes the oneinto <9 the other, 9. ot 9 foo” { thenfibercom ofed of the mul- Multiplication of the dif- 3 tiplication of the vnequal partes ference into it felfe. nist: the one into the other, &of the 9 difference into it felfe equall to Multiplication of the half 10 into it {elfe. TO tens gma roo. | thenumbet produced of the halfeinto it felfe. The demonftration wheroffolloweth in Barlaam, The fifth propofition. If an enen number be denided into two equall partes and againe alfo into tWwe unequal partes: she fuperficiall number which is produced of the multiplication of the vnequall partes the one snte the other together with the [quare of the number fet betwene the parts, is equal tothe (gnare of haife she number: : | Suppofe that 4B be aneuen number: which let be denided into two. equall numbers 4 Cand CB,and into two vnequall numbers 4 Dand D 8,Then/ fay,that the fquare number which is produced ofthe multiplication of the halfe number CZ into it felfe,is equall to thefu- perficiall number produced of the multiplication of the vnequall numbers 4 Dand DB the one into the other, and to the fquare number produced of the number C D which is fet betwene the fayde vnequall partes. Let the fquare number produced of the multiplication of the halfe number C B into it felfe be E. And let the fuperfi- ciall number produced of the multiplication of the yne- qual nibers .4 Dand DB the oneinto the other,be the number FG:and let the {quare of the number DC which - is fet betwene the partes be GH, Now forafmuch asthe number BC is deuided intothe numbers BD and DC, therfore the {quare of the number 2 C,thatis, the num- ber E,is equall to the {quares of the numbers B Dand D C,and to the fuperficiallnumber whichis compofed of the multiplication ofthe numbers 2 Dand DC the one into theother twife,( by the 4.propofition of this boke) Let the fquare of the number B D be the number KL:& let N X be the fquare of the number D C: and finally of $id. the r x Dh 2 rey — RE en er rt nent nn a nnn a v\ rej erm rt we nen ee HX | | 4 4| “y ALN bo $2, % 4\ At by oe | i F ig a a de yee ee, I ie eK ak — — - —— - -~ Sines = - oe —- = — = =x - —e = - SSS : <= = ol — : eS ——— — ———— ae -— eee oe ee = a —<_ SS _— a ~~ in ‘a —— - » een Sens 2 : 9 > = —- So — a —— —S— = LS a eS ee - = = + mar < = > = . = ~_- . a a - — : =. = e oe [ose + — = —amairws = _—— —s~2 = SS E E = = — a _ = : = = ——— = << se =, = al 7 “; Sater =J * ~~ r =!" (> cle _ > ee t J —= y= = Sis + = > ——,, oe sewn a ———---— ee - - isan ee — - z a e a4 Ph Aa > te ‘2+ + be diears oe oe —_—-— + Fo: - a a — Syl ZT teyecond Booker \° the multiplication of the numbers BD and D C theone intorheothertwifsslerbe pro duced either of thefe numbers L Adand MN _. Wherefore the whole, number KX is ¢quall to the number £.And forafmuch as the number BD multipliyng it felfe produ- ced the number X L, therefor it meafureth it by the vnities which are init felfe, More ouer forafmuch as. the number C D multiplying the number BD produced the num- ber L 41, therefore alfo D B meatureth L AZ by the vnities which are in the number € Oxbut it before meafured the numbet K L by the ynities which are in it felfe, Where- fore the number D B meafureth the whole number K A4by the vnities which are in 8. But the number CB iseqial tothe number C_4. Wherefore the number‘DP mea- fureth thenumbetX ‘Ad bythe ynities which are in C_A. Agayne forafmuch as the na- berCD multipliyng the nutmber D B produced the number AZ W: therefore the num ber DB meafureth the number 4/4 2 by the vnities which are in the number C‘D:but it before meafured the number KM by the ynities which are iti the number 4C.Wher fore the number 2 D meafureth the whole number KW bythe vnities which are in the numbere4 D. Wherefore the number F Gig equall to the number K NV. Ft nunibers which are equenuultiplices to one and the felfe fame number,are equall the one to the other.ButthenumberG Aisequallto thenymber NX: foreither of them is fuppoe fed to.be the fquare ofthe number C D.'Wherefore the wholé nuiber K X is equall ‘to toe wholenumber F 4, But the number K X is equallto the number £;Whereforealfo the number F Hisequalhtothenumber E.And the number F Ais the fuperficial num+ ber produced of the multiplication ofthe numbers .4.D and DB the one into the o+ | ther together with the fquaré of the number DC.And thenumber Eisthe fquare of the number C B,Wherfore the fuperficiall number produced of the multiplication of the vnequal partes 4D and D2 the one into the other,together with the {quare of the nat ber. D C whichis fet betwene thofe vnequall partes, is equall to the {quaré of thetiumé bet C B which isthe halfeofthe whdle n umber 4B Iftherfore an euen number be de uided into two equallpartes;&c. which was required to be proued. Sap The 6.T heoreme: The 6 ‘Propofition. “ * Sf aright line be deuided into two equal partes,and ifuntoit be added an other right line direttly the rettangle figurecon- _ tayned onder the whole line with that which is added,¢g the line which is addedt ogether with the /quare whichis made of the balfe, is equall to the [quare which is made of the balfe line and of that which is added as of one li ne, § Bek: FA ppofe that the rigbte Z 4) line AB be denided ins La gelbad . ed SUSY si ofm 4.) aad gn : reo to twoequall partes in seater Secienyplerpmitiae isis ‘5 point C: @o'let there | added vnto it an other right line zi DB direétly,that is tofay, which ua being ioyned vnto A Bmakeboth ~-K.. Bo onevright line AD, Then Tay, 22 io 3 that the rectangle figure compre bended ynder A Dand DB, tos { gether b \ 7 : i f i > bn bhi . pie tint a) f \ of Euclides Elementes. Fol. 69. gether with the fquare whiche is made of BC is equal to the [quare whiche' is made of D.C. Defcribe( by the 4.6,0f thes, )yponC Da fquareC EF D,and (by the fir/t peticion)draw a line from D to.E:and (by-thes1,of the first) by} point ‘B draw a line parallel vnto either of thefe lines EC&X DF. cattine the diames ter D Ein the point Hyand let the fame be BG er by. the felf fame) by.5-_point FA draw to either of thefe lines A D and E Fa parallel line K,M:andmoreouer by the pointe A drawealine pae rallel to either of thefe lines CL andD M: andletthefame be A a ei viol 4 K.And forafmuch as AC is equal ; | vato CB, therfore (by the 36, of : % ew : the firft the figure A Lis equal vise re vuto Sheek C A. But( by the | Ee Set fie, ‘\ 43.0f the firft )C His equal ynz og fi | Mag }o. the figure HF, wherfore A Lis og bk equall ynto 1 F, Put the figure CM common tothem both, wher fore. the whole line. A Mis equal vnto the gnomon N X0;,But A aa M is that which is contayned ynder A D and D B: for D Mis equal vnto DEB: wherfore the gnomon N X 0 1s equall vuto the rectangle figure contained vna der AD and D B,Put the figure LG comiion to them both, which is equall to the [quare which is made of CB, VV berefore the rectangle figure which is cons tayned ynder_ADand D.B together with the [quare whichis made of C Bis es gualltothe gnomon N.X quall ynto BC VV herfore the gnomonK L M T he fecond Booke both: wherfore the whole figure A Fis equal to the whole figure CE, VV bers forethe figures. AF andC E are doubletothe figure AF. But thefigures A F and E-are the gnomon KL M,andithe fquare CF: wherfore th: gnomon K_ L M,and the fquare C F is double'to the figure A F, But the double to A Fis that whichis cone tayned vnder A Band BC twife, for BF is ex and the {quare C F is equall ynto the rectangle figurecontayned ynder A Band BE twife,Put the figure D G common vnto them both, which isthe [quare made of AC, VV herfore the enos mon KL Mand the fquaresBG andG D are eqhal yntothe rectangle fgurewhichis contats > = ned vader AB ey BC twife,cm vatothe fqnare | bas | Which is made of AC, But the giomon KEM, ex the fquares BG (o'DG are x whole [quare BAD E, er 9 part or fquare C F, which {quares aremade of the. lines AB eo of BC, therfore y /auares whichare made of A Bw BC are equall: vato the rectangle figure whichis contayned vader AB and B Crvife_and alfa’ vuto the fyuare of AC If therfore a right line be, deuided by chaunce:the fquare.. whichis made of the whole together wath the fquare which ts madeofone of the: partes,is equall to the rectangle figure which is contayned ynder th: whole and: the fayd part. twife,and to the [auare whichis made of the other parte.: whiche. wai veqnired-ta be demonstrated, ; : Fluflates addeth vnto this Propofition this Corollary. T he fquarés of, pwaeneynall lines do'¢xtéedethe rettangle fi igures contayned uncer the faid lines by the fquare of the exceffe wherby the greater lyne excederh the leffe. _U. Fonif the lineoB be the greater;and theline PC the feffe,it is marifelkchat the fquares of 4 Band BC are equall to thereGangle figure contayned ynde¢ the lynes Band B Crwife, and-moreouer to the {quare of the line 4 C,wherby the ine A Bexce- deth the line BC, ei | | 2 By this propeficion mot wonderfully was found out the extradion of toore {quares in irrationall numbers,befide many other ftraungethinge:. ~.. ,. Anexample of this propéfition in numbers, ~(Lakeany number as43.and deuideitinto two partes as into 4.8.9. Take the {quire of 13. whichis 169, takealfo the (quate of 4. whichis 16.andadde thet: two fquares togethe? Which make't 8 5, Then multiply the whole number 13-into'4 ihe forefayde parttwife,and you fhall produce'§2.and 5 2: take alfothe{quare of the other part,that 1s,0f 9. whichis 81.Andaddeitto the produces of 13 .into 4.twife,that isynto 52.and 52. anid chofethreé numbers added together thallmake 185. whiche is equall to the uniber compoled of the (oaares ofthe whole and of one of the partes, vhich is ale e2 . The’ of Euchdes Elementes. Fol. 71. fs { Se \ pareesof the whole, 13 . ‘i | ‘The whole Multiplication of the wholeinto it felfe, Multiplication of one of the partes into it felfe. » The numbercompofediof the {quares of the whole, and of one of the partes Multiplicationof the - equall to whole into the forefayde parttwife. ‘thenumber compofed of the whole into the fore- {aid part twife,and of the 8p. »» ofquare of the other part, Multiplication of the o- ther part into it felfe. The demonftration wheroffolloweth in Barlaamy, “ . The feuenth propo/ition. ~ Tfanumber be dinided into reonumbers: the fynare of the whole manber together With the Square of one of the partes, ss equallto the fuperficiall number produced of the multiplication of the whole unmber inte be forefad pars twefe together With the fquare of the other part a3 Suppofe thet the number. 7 Bbe deuided into the numiberse ¥ Cand CB, Then / ‘ay that the {quare numbers of the numbers B 4 and AC are equall.cothe fupetficiall number produced of the multiplicatié of the number B 4 into the numbet AC twile, together wath the f{quare.of the number B C. For forafmuch as( by the 4.0f this books) the {quare of the number 4B isequall tothe (quares of thenumbers B Cand C 4,and. tothe fuperficiall number produced of the multiplication of thenumbers BC and C Athe onc into the other twife:adde the {quare at the numberef C’common to then. both. Wherfore the fquare ofthe nunibere 4 B together with the fquare of the num- ber 4 Cis equall to two fquares of the number .4 C and to one fquare of the number. Band alfo tothe {uperficiallnumber produced of the. multiplication of the numbers BCandC A the one into the other twife. And fora{ntueh as the fuperficial number pro: duced ofthe multiplication ofthe numbers B AandC A the onc into the other once, iS equallto the fuperficiall nuber produced of thé multiplication of B Cinto C A once, and tothe fquateofthe number C A(by the thirdof this booke ):therfore the number produced of the niultiplication of B Ainto 4 C twifeis equallto the number proda- ced of the multiplication of B Cinto C A twife,and alfo to two {quares of the number A. Adde the{quarénumberof BC common to them both. Wherfore two fquares of the number ef C and one fquare of the number C B together with the fuperficiall number Til. pro- ae a Se tm De ten een ws Bsr eee . ND Se — T he fecond Booke > duced of the multiplication of B Cinto’C 4 twife are equall to the fuperficiall, number, producedicf the multiplication of thenumaberB 4 into the number 4 twile toge-; ther with the {quare of the number C B:Wherforethe fquare of the number 4.2 tage-| ther with the fquare of the niiber 4 C is equal to the {uperficial niber, produced of the; multiplication ofthe number B 4 into the number.4-Citwife,together with the fquare’ of the number CB, If therfore a number be deuidedinto two numbers &c. which was! tequired tobe demonttrated. c+ 4 | = =] ———— rw j i} ad | 4 ee eM all | it tn ae { ie ik rll iM ' Hy q | a ‘ t Loe ‘ if ’ anf ¥ t Lf 7 yt HM , ihe } uri? walt Hid | mT ait tre it hy | WEE a si ‘al Wish ih ; Pei bra ; VE vr Wt ae a ah i : ' ay}. a a ie mh | ) ; i i his bei | Hide rit : ff ial. tl | | 64 the {quare of the whole 4B Thy: -* salt) 1 ety 25 «6thefquareokthepart zc. | Tothe fquare of thewhole ; Guare Ch Slab Ohaiesst oils | if WW a Hts Hy 1 | i} 4 80 the {uperficiall number 9° thefquare of the other part # @, D 8 9 : s ; 4 ~ ‘x - " % : e 2 atata oo » } o>, . atl , adie 7 _. : _ > * , . s > . ; - fa « the fquare of the part LA et ee noon Lee 40 ) 80 the fuperficialnuniber produced of the multiplication of the’ , whole inte the part twife : : ex 3 3 a % ii t? ay ang ~ ; ; : % ~~ Ss, J Bas wf i 3 eeag es ; : 4 te ble . : y Pte ~Pthefquareofthe otherpari® | | | The8-T heoreme.”° ThE8Propopition. Ifaright line be deuided By chaticesthe rectangle figure com- sh (pretended ynder the wholeand one ofthe partes fourestimes, — togetherspith the fquarewhichissuade of the other parte, is’: Ss eeguell tothe lquare which is made ofthe whole and the fore- a a 2 ‘Maid parts of one lings. * ae Ree 1 V ppofe thatithere be acertayneright.line AB, and let. sl INE NN | tt be denided by chaunce methe point C: "Then L fay that: | Zeon the relanglesignre comprébended ynder AB and BC- | Bl TOR fie foure ityutes,together with the (quare, whichis made of. AR Cis equall to the fquare madeof ABand BC as of one. So line.Extend the line AB (by the fecond petition). And Se Lae ag At (by thethird of thefirft) vato CB put anequalllyne B: “phe any lgpenglapny Do And (by theseufthe firft,) defcribe vppon AD a fanare.A ED, And defcribe a double figure. And forafmuch as ( Bis equal ynto BDbue€ Bisequall ato GK, (by the 34,0f the firft) and likewife BD me | of Euclides Elementes. Fol.72.- is equall ynto K N, wherefore GK alfoisequallynto KN: and by the fame reafonalfo'P R , is equal ynto O.And forafmuch as BCis equall nto BD, and GK vuto K_N, therfore (by the 3 6.of the first) the figureC Kis equail vnto the figure K D;and the Gun figure G'Ris equall ynto' the figareRN. But (byy 43,0fy i, )the fieure C K is equal yn to the figure RN: for they are the fupplementes of the parallelograme CO. VV hers forethe figure KD alfois es quall vnto the figure NR. VV berefore the/e figures ‘D KCK GR RN, areequall the one to the other. VV heres fore thofe foure are: quadrus pleto the figureC K, Agayne forafmuchasC Bisequal vn« toBD, but BD is equall ne to B K that is,ynto C G.And | | | . C B is equall vnto G K that is nto G P: therforeC G is equallyntoGP, And forafmuch as CG is equall ynto G P and PR is equall vnto RO, therefore the figure AG is equall ynto the figure MP and the figure P Lis equali nto the figure R F.But the figure MP is(by the a3, of the first) equal ynto the fieure PL, for they are the fupplementes of the parallelogramme ML: wherfore the figure alfo AG ts equall ynto the figure RE. VV herfore thefe foure fioures A G,MP,? L,and RF are equall the one tothe other: wherfore thofe foure are quadruple to the figure A G.And itis proued,that thefe foure fieuresCK, KD GRi RN, are quadruple to the figure KV WV berfore the eight figures which contaynethe gnomonS\T'V jare quadruple to the figure A K. And fora/much as the figure A K ,isthat which is contayned ynder the lines A BandB D, for the line B K is equall vnto the line BD: therfore thacwyhicheiscontayned yns der the lines AB and BD foure tymes is quadruple vuto the figuée AK. And it 1s proued that the gnomon S$ TV is quadruple to AK.VV ber fore that which iscontayned vuder the lines A Band BDO foure tymesis equall ynto the enomo STV Put the figure X which is equall to the /quare made of A-C common nto them both.VV herfore the rectangle figure comprebended vnder the lines AB and BD foure tymes together with the fynare which is made of the line A C,is equallto the enomon S'T Vand yntothe figure XH, But the oromon S TV: and the figure X Hare the whole fquare AEE D, which is made of A D : wherfore that which is contayned vnder the lines A B and BD foare times together With the Jqnare which is made of AC; isequall to the fquare which is Bit so oe i a Ns ay, made —. = tre — 3 — my — —* SSS See eee: SSS Se eS es ee = =—s = ———- —— (—- ~ a eeeee ey : Sn ees "% = : <= r . a ae —_——-- Sa a - Sere =O: tet a = a et te 5 ae Se Se: prone ~ : = = : - = 5 " - 2 > a - - > aie = ol ed a ~~ 66S So < ——- x — —$—— ~ —— ~ — a i be Se - gM Kd 7 ——— —— as - 3 “ o eee Ss a - —— - - ~ ~ en ths « ta Se : 2 ve = ~~ ie a ~~ nme. =. ete VEN. “ester a wee ares Saco = z ~ ethic MME oa a . =e . te sate a noe = - = = - = ——- oe Ne a ee 3 a 5 Verb. Gis —— Sens Ee CR => =< PE Es ei en oe ee es — — > ee Te ' = ‘ . ae + x ; x? eet = SS = 3 =F Se =e = ee S26 ee SX > - A ee “S ~ FN - NS a. ‘ x —~ > es a 7e — —- = = ast = - => 3 > - ne i : > = = F eee = ~— = = ro ; re Se a A - ee Se + ot eso =. SoS SS =< i ——S rs 7s 5 = = —— -=-- = —< aan ne — -—-- — =: ree = — — —— z ae ae << Soe “= Pigs Sai: A The fecond Bookes' : madeofAD.But BD is equall vnto BC.VV herfore theretfangle figure cone tayned foure tymes ynder.A B and BC together with the [quare which is.made of AC, is equall vuto the /quare whichis made of A D,that is, ynto that whiche is made of A Band B Cas of one line, If therefore. a right lyne be-denided by chaunce, the reftangle figure comprehended vnder the whole.and one.of the partes foure tymes, together with the [quare whichis made on the otherpartsis equall to the fquare which is made of the whole and theforefatd part, as of one | tine, which was required to be demonstrated. “ ¥ | An example of this Propofition in numbers, ar Pere Takeany number as 17.and deuide it into two partés,asinto 6,and 11. And muiti- ply 17.1nto 6,namely one of the partes foure tymes, atid you fhall produce 102, 102,, 102.and 102,Take alfo the {quare of 11.the other part,whichis 121: andaddeitwnto the fourenumbers produced of the whole 17.into the part 6.foure tymes,& yousthall make 5 29.Then addethe whole number 1 7.to the forefaid part 6. which make'2 2° Os take the {quare of 23, whichis 529. whichis equallto the number compofed*of the whole into the fayd part foure tymes, aud of the fquare ofthe other part, which num- ber compofed isalfo 529.As you fee in the example. The whole, 17 6) : a f partes of the whole Multiplication ‘of the _.wholeinto one of the partes foure times, the nuniber compofed ofthe whole into one of theipartes II | fouretymes, & of the fquare Multiplication of'the o- | of the other part ther part into it felfe, | Additioit of the whole in- | equal to ~ } tothe part, q | Multiplication of the nii- ber copofed of the whole and the forefaid part into it felfe,. © wees the fquare of the number cb. pofed of the whole & the fore {aid part, The Cee ee eee Se of Euchides Elementes. Fol. 73. The demonftration wheroffollowerbin Barlaam. The eight propofition: If anumber be deuided into two numbers the fuperficiall number produced of the wultiplicarion of the whole into one of the partes foure tymes together With the {quare of the other parte,ts equall te the [qnare of the unmber compofed of the whole number and the forefayd part, ; Suppofe that the number 4 B be deuided into two numbers 4CandC B. Then fay that the fuperficiall number produced of the multiplication of thenumber 4 Bin to the number C B foure tymes together with the {quare of the number 4 C,is equall to the fquare of the number compofed of the numbers 4 B & CB. For vnto the nume ber 8 Clet the nuinber B.D be equall. Now forafmuch asthe fquare of the number 4 Dis equal to the fquares of the numbers 4 Zand 5.D,& tothe ? fuperficiall number produced of the multiplication of thenumbers 4B & 2 | B Dthe oneinto the other twife( by the 4.0f this booke) And the numb er + . B Disequall tothe number.B.C;thereforethe fquare ofthe number 4D. 3 | is equall to the {quares of the numbers 4B and BC, and to thé fiperficiall > + C number produced of the multiplication of the numberse 74 B and BC the | one into the other twife,But the fquares of the numbers -4 B and B Care e- | quall vnto the fuperficiall number produced of the multiplication of the’ 6 | numbers 4 Band BC theoneinto the other twife, and to the f{quare of 4 C(by the former propofition) Wherfore the {quare of the number 4 Dis : equail to the fuperficial number produced of the multiplicationof the ni- | bers 4 Band BC theone into the other foure tymes,and to thefquare of [ the number e4C,But the {quare of the number ef Dis the {quare ofthe... 4 number compofed ofthe numbers 4 Band BC: forthenumberBDis e-. qual to thenumber B C.Wherfore the fquare of the number compofed of thenumbers ABandB Cis equall to the fuperficiall number produced ofthe multiplication of the numbers 4 B and B Cthe one into the other foure tymes, & to. the {quare of the num> ber 4 C,/f therfore a number be deuided into two numbers,&c. 8 8 8 $ 2 2 2 2 16 I 16 16 SEESEETEE 64, thefuperficialltnumber produced of the multipli- . cation ofthe numbers 42 and BC the oneinta. j 6 the other foure tymes. 6 36 thefquareof AC, 10 10 100 thefquareof thenumber compofed of 4 Band BC. 64 the fuperficial naber produced ofthe multiplicatié made 4,times ~ 36 thefquare numberof AC. Soe eR X 100 Vi. T he rm —— — fn — Eee weeperenstege eee eee - —_—— : . Pade = SSS a > - _ _ ee Saas — —— > ;: — = = —— : -———— — — —_=— _ =- a . = . a ee eee _.- = —__ = = = = ——— - — = = SESE AOS Se reagee = r: ~ =. SS == ~ LSS = SSS ee tei Se a eS 6 eee —= ——SS— = _ = a SSS ina a Eine a < = ’ —2 Re =. tot — = — — — eine - rar ~ = = = : — ae a +, es " a - _ = — .— ee SS Sr eed == =r 2 - = SSS SS ee ee - eS ea See —— a The fest Baste The 9:F beoreme. The 9.Propofition. $a Ufaright line be denided into two equall partes, and into twornequall partes, the /quares witch are matte of thecne- quall partes of the whole,are. double tothe Jquares,which are made of the balfe lynesana of that lyne which is betwene the Jettions. | aa ee ee oa. os sts. > as -- . - _ 3 ae ah + Tae et - — - Ss yp ee eee 2 -& - ++ _ - . —— — —_ — = = ~ = 15 ~ = z : Ses me : F : : . . ees a ee ee ee — = mena ved ~ _ = oe Se a. ee ai ik Ltinca tem Sealant — = setendlnn. wantollliasaa Fs one =e aa « . ee - —* aay - —t a, eee 4 : : —— ee : er i ee —-— — :— ~— ———— ——— — ar v SS - “ > : 5 Pe ae eae ee = = = - — es ES a Foret - - — a —— : = =. = —— : = ——- or = i= —: ~ —- Bn ee ee a a es a 2 —— - = aS = = SS == SS = 23 - ; oe 7 — ene SS . — = = = - — — ————— . = ren ee en ae au a as Se = = RES gener ; ————— So Se — SS ae a 2 (SS ae —_ he oe ee PS a ee ee: i a ee = > : SS = rar = = eer. = 2 Ee = a - eas nae = “Sy 7 : ; == — ~ ——— - a ~ —- SSS = ———== = => += - o-=- 2 ae 7 7 = > an = ~~ = = Sa —s . oP ig _ as — we -~ — —e ote reed pata eee No Scosersrey: a : tr ——— == ———————— = = —— — a =— = SA inion 3 ‘ ‘ om . . fuse initial = : — =: so ou oe — = a = a pe — SS eS ry az = eer = = =v = — a ——— 7 - ae ene ee “t+ tee re el ee Er perpendiculer lineC EF, AndletC Ecby the 2 of the first) be put.equall vnto esther of thefe lines”: AC eC B: and(by the - Sarst peticto ) draw-lines from Ato-E, and from E toB And (by the 31. of the first) ky the point ) | D draw vnt6 the line EC a parallel lyneand let the ‘fame be DF sand (by the Selfe fame) by the point F draw vnto A Ba line parallel and let the Jame be FG, And ( by the first peticion draw a linefrom.A to F, And forafmuch as ACis e quall ynto CE, therfore( by the kof the first) theangle EAC is equal nto the angle CB A And forafmuch as the angle at the point Cisa right angle: therfore the angles remayning E AC,and AE Cyare equall ynto one right angle, wheree fore eche of these angles EAC and AEC isthe halfe of aright anole, And by the fame reafon alfo eche of thefe angles EB Cand C E Bis the halfe ofa right angle VV herfore the whole angle AE Bis a right angle And forafmuch as the sangle G EF is the balfe of a right angle,but EG Fisa right angle. For( by the _29.0f the fir ft it is equall vnto the inward and oppofiteangle,thatés.pnto EC - B: wherfore the angle remaynin is EF Gis the balfe ofa rightangle VV heree fore( by the 6 common fentence)the angle G E Fis equall vato the angle EFG, VV ierfore alfo (by the 6.of the Tirft)the fide EG is equallynto the fide F G.Ae gaine fora/much as the angle at the point Bis the balfe of a right angle, but the angle FD Bis aright ancle,for it alfo( by the 29.0f the firft) ts equally nto:the : giteg Be OSA TE Ss a 8 ae a inward A of Euchdestlémentes. Fol.74. inwarde and oppofite angleE CB. VV hereforethe angleremayninig BF Dts the halfe of a right an ole. VV herfore the ai ole at the point Bis equall “pnto the angle DF B. VV herfore.( by the. 6. of owes fide DE-isequall tnto. the fideD B. And forafmuch as ACis equall ynto€ By therfore the fqnare which is made of AC is equall ynto the {quare which ts made of C E. V. Vherefore the fquares which ave made of € Aand.C E are double to the {quare Which is made of AC. But (by the 47- of the first ) th ot which is made of E Ais equallto the (quares which aremadeof AC andC E (For the angle AC Eis aright ane gle) wherefore the fquare of AE ts double to the fquare of AC. Agayne forafe much as'iE G jsequall ynto-G.F, the fquare therfore which is made of E G is ee qual to the [quare which is made of G F.VV herfore the fquares which are made of G EandG F are double to the fquare which ts made of G Fo But ( by the 47. of the first) the [quare which is made of EF is equall to the '[quares which are made of EG andG F. VV herfore the {quare which is made of E Fis double to the [quare which is made of G F. But GF is,equall ynto CD. VVherefore the fquare which is made of = m4 | EF is double to the ; ES fquare which is madeof'©"°> ~ CD. And the {quare wwhiche is made of AE is double to the [quare ‘which is: made of. ACeso15 VV herefore the [quares which are made of AE and EF are double toy fquares which aremade “4 CC eee of AC and CD. But( by the 47: of the first the fquare which is made of AF is equal to the [quares which “are made of AF-and E P( For yangle AB Fisa right angle) VV herfore the ~ sunt which is made of AF ts double to the {quares which are made of AC eFC ‘But (by the 47~of the firft yfquares which are made of A D and 'D F areee nall toy [quare which is made of A F.For,j-angle ot.» point Dis a right. angle. ae herfore Hie aaa: which are made of AD and D F are donble'toy {quares which are made of AC andC D. But D Fis equall pnto DB. VV herfore the fquares which are made of AD and D B, are double tothe {quares ~which-are made of A CandC D. If therfore aright line be denided mto two equall partes ‘and into two. ynequall partes sthe quares which are made of the pnequall partes ‘of the “whole are double to the fqauares which aremade of theshalfe ne, and. of that lyne which is betwene the feEtions: which was required to be proved. Sonn lige An example of this propofition in numbers. Take any elem number as 12.And deuide it firftequally as into 6,and 6, & then vh- S equally asinto 8 a 4.And take the difference of the halfe to one of the randd pane V.ije whic : —_—— ee —— = = = ss — =~ —__- —_ Ses = ~—— baw. TRA 00) Sabo i a I Sai ee = - R238 -o—-* ~~ — oe —< = » Sa WUD ~ y + + = ‘ . "7 - - +e — ue - 7 tlie at al 8 - es r —— “«~ SN ere ee x ey © ee ae é a Ea ie Pyar s ul — i oe e< by a Cg at < ~ ~ ies ~ ape Pome > = —— ~ ~ —_——— - - —— « — —- ~~ ee et Sat = a _ a oeereen = poh i + oe aii " . E - a AS pag ‘ PY ibndeeat : = - 2 =~ ~ —< naderewe : = —— _ - ; - = ee - , . - 7 : "uot —— — - -- - - ——s : ~ ~ rs a - ——_ < - 2 een eee ——— So — ———— ——_-—— ra = — - ‘: ; - - = ad EEE Se ree —— : : == Yt <_ Rava Ie Se a= = 2 : < Baie \etetecca stones ers ae ae eo = neta ; Se hag 025 EN Fien> vee ee =<, oe = — —- > 4h. ~ ST > aed : = - ao. . = - rx? : eT One ew - a ss a ter mc Samedios ce Oona —- re grr = ’ = : = ey I _- - ne a a a ee ee = Ene SS a . —- ee OE ae ae pr _or me . A or — ae a J . ~ pee ‘ 4 . os ? = ee 5 = 3 z 7 ar ~ ———— Coens : —~ = es wae, ST Ae ~~ lee - ane * a er =. sow a o —— es a a ~* ee x ahh mas aa < a ae — Pees. : = 3 ee <= = ee 4, : > ek Seem re — ee repeeeer = rs * > eeraiees - — : —— —— ae es he rc ae oe Se eS =e ee = ———- = - ——— ——S ——— = —— SS eS = : —— = = rn SS ee : : => Sa —_—- Se ee SESS —— SS eee SSS = = —————— : ee se a 4+ = = ~ - ae Si ~ SS Sg Bo ee 3 < < @ e = 3 Sake See ° aoa a pe ne * a a : : Ss = == Ee a= = =x? = eee ee = —— =< : “= = : -- = z > 4 r ets ‘ = , 7 > - ne =< : Tn ig = = — - - — -_ = _—T — Sueael -- a pe => “a — ae . - = re Ss Seed are 5a ee 5 “ “ —— swe = : — ~ a —— ______ : : oo = a os = saints Fa i = " — —— . ie — —— ~ _ _ — = .—- = =e = = —— : — - —~—— 4 = ie 2 ate > = porrenmnrnnr eras oat os 2 se x Se: _ —s SSS ee sk oe — Soe te ee = —s- a = =—— — = spewed = 2 —— = - Sren, ol cael . . 2 -- ——— —— a . — ~ ~ — ~~ -—- — =e - ie oe = — =~ = ~ — = = = - — — — ~ . — = = = a = = a Ss _ = 2 SS ea = 3 SS SSS ———————— — —— — rs ye #- which is 2.And;take the {quare numbersiof the vnequall: partes 8,andi4,which are 64, and.16:and adde them together, which make 80. Thenmtake\the {quarés of the halfe-6, and of the differéce.2:which are 35,and 4:which added. togéthet make 46.Vnto which number,the number compofed of the fquates of the: Whequull partes ;whiche.is 80;:is double.As you fee in sheexanipltssrtit inns 2 Re auhtone ae. | The whole. ts la } sig equal partes > theynequall partes: wol yg!) “ithediffrence ofthe halfa.td ee sone Ofthepartes \ Multiplication ofeche .. | vynequal part into himlelf, © the number compoled of the | ~{quares of the vnequal partes Poe ag pee doubleto Multiplication of the half and of the difference eche ‘| anto himf{elfe,. the number. compoled of-the. {quares ofthe halfe;and ofthe’ ‘ ‘.., differenges..\ 544 The demonftration-wherof followeth in Barlaam, ~~ Theninth Propofition. If anumber be deuided into tWo.equall numbers and acaine be denided into twe inequall partes: the Square numbers of the vnequall numbers are double ro the Square which is made of the miusltiplicat i- on of the halfe number snto it /eife, tegerber with the {quare whiche is made of the number fet bee tWwene them, i | ie | i For let the number ¢# B being an even number be denided intotwo equall numbers ACKCB: & into twovnequall nfibers:A D and DB,Then J fay that the {quarenum- bers of 4 D. and D B,are double to the fquares which are made of the multiplication ofthe numbers _4CandC Dinto peaiieee Forfora{much as the number.4 Bis an euen number,and is deuided alfointo two = aa numbers: 4 Cand C Band afterward “Into two vnequal niibers .4 Dand DB: therefore the faperficial niber produced of the “mulciplicatia-of the nitbers AD.& DB, WW oneinto.the other together with the fquare of the number D C,is equal to the {quate of the number ACT by the fift propofition ) .. Wherfore the fuperficiall number produéed of the multiplication\of thé numbers AD wand D Brthe otie into the othertwifestogetherwitht wo{quares of cs ea CDy is doubleto the fquare of the number ; : : e4C. Forafmach as alfo the number 4 Bis deui- ‘ded intotwo equal numberse 7 CandCB stherforethe (quate nnttber of ABis qua- druple tothe fquare number produced of the multiplication.of the number e-4 C into it felfe( by the 4,prdpofition) . Morecderforafmbch asthe fupeificiall number produ. ced of the multiplication.of thenumbers A ‘D& D.B the one into the other. twife-so- gether, with two (quares of the nunaber.D Gis double tothe {quare Ee of CA:& ora{mach ~~ of Euchdes Elementes. Fol.75. forafmuch-as there are two numbers, of whiche the one is quadruple te -- one and the felfefame number,and the other is double'to the fame num \* ~~ p ber: therefore that number whiche is quadruple. fhall bedoubleto that number whiche is ‘double. Wherefore the fquare of the number 4 B is\ double to thenumber produced of the multiplicatié ofthe numbers €7 t.| D and D B the one into the other twife together with the two {quares of | the number D: C;Wherfore the number which is' produced of the maiko): tiplication of the numbers -4 D and DB the one into the other twife, is. | leffe thé halfe of the {quare ofthenumber 4 B by the two {quares of the” ; numbers D C.And forafmuch.as the niber produced of the multiplica- tion.of thenubers.4D & D&B the oneintothe other twife,together.with : the niber cépofed of the {quares of the numbers 4 DandD BZ is(by the ke 4:propofition Jequall ro the fquare ofthe number 4 # ; therfore the ntt- | ber compofed of the{quares of the numbers 4 D&D Bis greater then the halfeof the fquarentberof-/ #8, by the two {quares of the number | D.C.And the {quare of the numbere¥ B isquadruple to the fquare of $ thenumber-AC.Wherfore the ntiniber compofed of the fquares of the. numbers 4:D.andD Bis greaterthenthe.double of the fquare.of the number.4.C bytwo{quares of the number.D C,Wherfore the {aid num- ber is double to the fquares ofthe numbers dC and CD, Jf therefore a j number be denided &c.which was required to be demonftrated, A g Ure 3 > gaat iat 64 the (quare ofthe viequall part-4.D 4 ‘the (quare of the vhequallpart 8 2) kbs ' | 25 thefquare of the halfe 4c, 3 St 9 the fquare of CD,namely,of the number fet betwene, 25 9 ee CO 34 the {quares of the halfe,and of the number fet betwene, 64 = 68 a + RR 68 thef{quares of the ynequall partes. | Theto.Theoreme, Thero.Pr opofition. “Ifa right line be deuidedintotwo equal partes,¢> unto it be “< s added another right line direttly:the qnare which is madeof the whole (> that which is added as of one line, together with a Be Stes Eek . j as oh " ow a rer f Rens - sh wees Kaen & Constrnéttion. Densonstrate tion. The fecond Booke the fquare whicheis made of thelyne whiche is added 5 thee: two fquares( ifay are double tothefe fquares namely;to the. Square which s made of the balfe line, <> to the fquare which is made of theother balfe lyne and that whiche is added ,. as of one lyne. | JV ppofe'that acertayne right line A Bbe deuided into two equall partes 4 lin the point C.And vntost let there be.added an other right line directe: Ny namely;B 2, The I fay, thatthe fquares which are made of the lines A Dand DB are dorble to the (guares which are made of thelines ACandC D. Rayfe vp.( by the Lie Of the first)_from the point yato the right line ACD 4 perpenaicater lyneyaid let the fame be C End let C E (by the 3,0f che fir/t) be made equall vnto either of thefelines AC aid CB. And (by the firft petitia on) draw right lines from Eto. and from E.to:B..And{ by the 3s, of the first) by the pot E,draw aline parallel vnto C D,and let the fame be E F And( by 3 Jelf fame )by the pot D draw a line parallel vnto C E and let the fame be DF. And fora/much as por thefe parallel lines CE¢x DF lighteth acertainright line EF, therfore by ihe 2 9.0f the firft,) the angles CEF and EF D are equal vato two right anglesVVherfore the angles FE B,and EF D are le/se then to right angles, But knes produced from angles leffe then to right angles( by the fifth peticton )at tie length meete together.VV herfore the lines EB andF D beyng produced on that fide that the line BDis, will at thelength meete toe gether.Produce them ind let them meete together in the point G, And (by the firft peticion draw alne from Ato G, And forafimuch as the line AC is equall vntotheline CE, theangle | | | alfod ECis (bythesofthe = = nee .. firft equal vnto the agle E AC, And the angle at point Cis aright angle VVbirfore -- eche of thefeangles EAC ep and AEC is the: balje of right angle. And by thi fame reafoneche of thefeansles C EB, and EBCis thebalfe. A ; of aright ancleVVheefore ~~ ~ * — the angle AE Bis arisht angle. And forafmuch as the angle E B Cis the halfe of 4 right anglestherfore(by the 1s. of the firkt)the anple D BG is the half of a G nightangle. But) angh BD Gis ee equal ynto the anole D CEafar they are alternite angles)V Vherfore theangleremainine DG B is the halfe of a right angle VV herfore (by the 6 common fentence of the fir/t\the ane gle DG Bis equal toibe angle D BG,VV herfore (by the 6 .of the firft) the fide of Euchiles Blementes. Fol. 76. BD is equall yntothe fide GD) Aeayne forafninch as the angk EG Fis the halfe of aright angle: and the angle at the powmtek 1s aright angle: for (by the 34-0f the first) it 15 equall pnto the oppofite angle EC D. VV herefore the anole remayning F EG ts the halfe of aright angle. ¥V-herfore the angle EG Fis e- qualltothe angle FE GVV herfore (by the 6..of the first theyide bE 1s equall ‘nto the fide FG. And forafmuch as EC 1s equal pnte C Aypthe ‘fquare alo which is made of E C is equall to the [quare which is made of C A, VV herefore the.{quares which are made of Cand CA ave double tothe fquare which ts made of AC. But the [quare whichis made of Fea is( by the 47: of the first) ee qualloontothe fquares which are made of E Cand CA. VV herefore the [quare whichis made of EA is double to thefquare wlichismade of AC. Againe-fors afmuch as G F 1s equall vnto BE Fthe fquare alforwhich ts made of G Fis equall to the [quareswhich is made of E-E. VV herfore the /quares which are made of G F and E Fare double to the [quare which ts madeopE F.But(bythe47: of the first) the {quare which 1s made of E.G is equal to the fquares which are made - of GFand E F. VV berefore | the fquare which 1s made of E Gis double to the fquare ‘which is made of E F.But E Fisequall pntoC D, where forey [quare which is made of EG is double to the {quare phichis made of CD. And it is proued the [quare which ismade of E. A1s. double to the [quare which is made of | A CVV herfore.the {quares whith are made of AE and EG are double to the fquares which are made of AC and C D.But( by the 47-of ths first ) the fquare which is made of AG ts equall to the [quares which ave made of AE and EG. VV herefore the {quare which is made of AG 1s double tothe fquares which. are made of AC andC D. But vnto the fquare whicke ts made of AG are equall the {quares which are made of A D and D G.VV herfore the [quares which are made of A Dand DG are double to the [quares-which aremade of AC and D C. But DG 1s equall ynto D BVVherfore the [quares which are made of A D and‘D B aré double to the {quares which are made of A Cand DC. If therfore avight line be deuided into two equall partes, and ynto st be added an other lyne directly, the [quare which is made of the whole and that which is added, as of one line together with the [quare which is made of the line “which is added ,thefe two [quares( fay.) ave double to thefe [quares, namely, tothe [quare. which 1s made of the halfe lyne, and to the [quare which is made-of the: other halfe hue and that ‘which 15 added.as of one lyne : which was required to be proued. | ' \) . f} { i | L | { i 4 j | , ! ' ly A iD i i ' : 7 1} 4 r] ‘5 Ny iit ey iv : ai) ) Bu) ¢ 7) 5 a. a A wy : a fi 4B Ni the . f ‘oa ‘ i : "+ A H - \ ie a ih!) | ee 4¢ : . 4 ‘ ’ H ae ‘ OH ell ya ie | | Whe! “tf : ‘ > | : ; . s | .7 j hf yy iZ if 1 uJ an > 7 7 as a a mi i mu A j ; : SH } a i : : ; iw ii | it } 4 | | ‘ hi BA ee g An other demonStration after Pelitarius. V.uu. Suppofe SS aS eG a T he fecond Booke Suppofe that the lyne AB be deuided into. two equall partes in the poynreC® And ynto it letthere be added an other right lyne dire@ly, namely, BD. Then ¥ fay that the fquare of A D together with the fquare of B Dis double to the fyuares Of A CandCD, C3 ) Vpon the whole line A D defcribe a fquareA D EF. And vpon the halfe lyne.A C de- {cribe the {quare AC G H.And produce the fides G Hand CU till they cut the fides F P & D F,wherby thalbe defcribed the figure H L K F,which thalbethe {quare of the line C.D :as (by the Corollary of the 4, of this boke,& by the 34.Propofition of the 1. itis manifeftif we draw the diameter C D, For the lyne K Fis equallto the line CD. And. making alfo the lines HM andH N equall to either of thefe lynes AC and CB, drawe the lynes M O and N P cutting the one the other right angled wifein the point Q. Ei- ther of whichlynes let cut the fides of the {quare A.D: ; E Fin the pointes O and P. Now itnedeth not to prove that the figure H Qis the fquare of the lyne A C,feyng thatitis the {quare of the line CB: as the figure Q Fis the {quare of the line B D : neither alfo needethit to proue that the parallelograme H P is equall to either of the fupplementesE HandHD : nor that the fupple- mentes N'O and QU are equall, Forall thisis manifeft eu€ by the formeof the figure,for thatall the angles a- g bout the diameter are half right angles, & the fides are equall. Wherfore if we diligently marke of what partes the {quare HF which is the {quare of CD, is compo. fed,we may thus reafo. Forafmuch as the whole fyuare E Dis compofed of the two fquares A Hand H EF and of the two fupplementes EH and H D,we mutt proue that thefe fupplementes with the fquare Q F(which is the {quare of the line B D) are equall to the two fquares A H andH F, For then thal! we proue that thefe two fquares AH && Hi F taken twile are equall to the whole {quare D E together with the fquare of QF; which thing we tooke firft in hand to proue,And thus do I proueit. | The Supplement E His equall to the paralleloprame HP, And the {quare AH to- gether with the lefferfupplemét, N O,is equall to the other fupplemét H D (by the firft common fentence fo oftentymes répeted as is neede ) wherfore the two fapplementes E HandH Dare equall to the {quare AH and to the Gnomon KHLP Q O. If therfore vnto either of them be added the fquare QF: the two fupplementes E Hand H D to- gether with the {quare of Q F thal be equal to the {quare AH, & to the Gnomon KHL PQ Oand to the fquare QF. But thefe three figures do make the two fquares AH and H-F.Wherfore the two fupplementes EH andHD together with the {quare Q Fare e- quall to the two fquares AH and:H F, which was the fecond thing to be proued. Wher- fore the two fquares A H and H F beyng taken twife are equallto the whole fquare D E together with the fquare of Q F. Wherfore the {quare D Etogether with the {quare Q Fis double to the {quares AH and H F: which was required to be proved, q An example of this Propofition in numbers. Take any euen number as 18: and take the halfe of it which is g.and ynto 18. the whole,adde any other number as 3,which maketh 27, Take the {quare number of 21. (the whole number and the number added) which maketh 441. Take alfo the {quare of 3 (the number added) which is 9. which two {quares added together make 450, Thenadde the halfenumber 9, to the numberadded 3. which maketh 12, And take the {quare of 9.the halfe number and of 12, the halfenumberand the number added which fquares are 81, and 144,and which two {quares alfo added together make 235: yato which fumme the forefayd number 450, is double, As you fee in the example. The 5 : of Euclides Elementes. Fol, 77. Théewholé, | 13 SCs Thenumberaddéed. 3 9 } the halfe 21 21 21 Multiplication of the whole and 42 the number addedinto himfelf, rT NS aes 44° [ 44! | Multiplication of the number 3 9 added,into himfelfe. Bede 450 The number compofed of 9 the fquare ofthe whole & ' ~ £ thenumber added and of Multiplication of the halfe in- 9 the {quare of the number to himfelfe. 9 added, : 8:1 .-: '‘¢doubleto Multiplication of the halfe, & 12 1 the number addedinto it felfe, 144 iy 325 the number compofed of the {quare of the halfe and 144 of the {quare of the halfe | and the number added. si The demonfiration wheroffolloweth in Barlaam, ‘The tenth Propofition. Tf az.enen nomber be denided into tivo equall nombers, and unto it be added any other nomsber: the Square nomber of the whole somber compofed of the nober and of that whichis added , and the fquare nomber of the nober added:thefé two fquarenobers( I fay )added together , are double to thefé [quare nombers namely,to the {quare of the halfe nomber, and to the (quare of the nomber compofed of the halfe nomber and of the nowber added, Suppofe that the nomberc B being an eueh nomber be denided into two equall nombers AC andC@ : and vntoitlet be added an other nomber BD . Then I fay, that the fquare nombers of the nombers AD and DB are double to the {quare nombers of efC and CD. For forafmuch as the nomber AD is deuided into thé nombers 4B and 8D: therefore the fquare nombers of the nombers 4D and DB are equall to the fu- perficiall nomber produced of the multiplication of the nombers 4D and DB the on into the other twife,together with the {quare of the nomber AB( by the 7 propofitis) But the {quare of the nomber AB is equal: tofower fquares of eitherof the nombers 4CorC8 (for ACis cquallto the nomber CB): wherforealfo the fquares of the noni- bets 4D and DBZ are equall to the {uperficiall nomber produced of the multiplication of thenombers 4D and DB the one into the othertwife, and to fower fquares of the _nomber BC or CA, And forafinuch as the fu perficitall nomber produced of the multi- plication of the nombers AD.and DZ the oneinto the other,together with the fquare of the nomber CB, is equalto fquare of the nomberCD(by the 6 propofitid): therfore thenomber produced of the multiplication of the nombers AD and DB the one into the other twife together with two {quares of thenomber.CB,is equall to two fquates ofthe nomberCD , Wherefore the fquares ofthe nombers_4D and DB are equall to ae aks two T he fecond Booke two {quares of the nomber CD, andto two fyuares of the nomber AC. Where fore they are double to the fquares of the numbers 4C andCD.And the {quare of the nomber ¢4 D is the {quare of the whole and of the nomber added : And . . 3s CD Pp the {quare of DZ is the fquare ofthe nombe r added: the fquare alfo of the nomber : T is che {quare of the noi compofed of the halfe and of the nomber added : If there+ fore an euen nomber be deuided.&c, Which was required to be proued, ‘ g ’ 8 3 +8 —— | 64 thefquareofAD 4 thefquareof DB 3 5 | ‘ | | | e 25 the {quare of C D,namely,of the number compofed of the halfe and Yip _ of the number added, | - 7 | ae oi 35 64 5 | 3 3 @ CEM 4 hss | 9 the fquare ofthe halfe 4c, 8. ae 68 §341°° A ; SaT het. Probleme, T be 11.Propofition. To deuide a right line geuen in fuch fort, that the reitangle ‘figure comprebended ynder the whole,and one of the partes, | Teal be equall vnto the {quare made of the other part. ay’ ppofe that the right linegeuen be A B. Now it is required to denide / Se the line AB in fuch fort, that the rectangle figure contayned ‘ynder Sf) 0) the whole and one of the partes, Jhall be equall vnto the {quare which LNCY\ === 1s made of the other part.Defcribe (by the 46. of the firft) >pon AB Construction. afquare AB CD. And ( by the 10.0f the fe Fe firft )-denide the line AC into two equall partes in the point E,and draw a line from BtoE. And (by the fecond petition )extend CA vnto the point F . And (by the 3. of the Jirft )put the ine E F equal vnto j line BE, And{ by the 46. of the Jirst ypon the line A F* defcribe a fquareF GAH. And (by the 2. petition )extend G HZ nto the point K. Lhen L fay that the line AB is deuidedin the point Hin fuch fort, that the rectangle _ Sigure which is compreheded ynder AB and BF 1s equall to the /quare which is made of Demonfiratio AAT. For forafinuch as the rightlneAC — | 15 deuided into two equall partes.in the poynt E,and nto it is added an other right line of Euclides Elementes. Fol. 78. AF. Therefore (by.the 6.of the fecond) the reftangle figure contayned ‘ynder C Fand F A together with the [quare which is made of A Eis equall toy {quare whichis made of EF. But E Fis equall pnto EB . VV herefore the rectangle fagure contayned ynder C F and F A together -with the fquare which is made of E Ais equall tothe fquare ‘which is made of EB. But (by 47. of the fir/t ) ‘vato the {quare which is made of EB are equall the [qnares which are made of BA and.A BE. For the angle at the poynt A is aright angle. VV herefore that which is contayned bnder C F and F A, together with the [quare which is made of AE, is equall to the [quares which are made of B.A and AE. Take away the {quare whichis made of AE which is common , to them both: V, V herfore the rectangle figure remayning contayned ‘bnder C F and F A is equall vnto the [quare which is made of AB. And that which is contained ‘ynder the lines C Fand FA 1s the figure F K.. For the line F A is equall vnto the line FG. And the {quare which 1s made of AB is the figure AD. VV herefore the jie gure F Kis equall vnto the figure AD . [ake away the figure AK which 1s common , to them both. berefore the refidue, namely ,the figure F His equall ynto the refidue namely,‘vnto the figure HD . But the figure H Dis that which is contayned vnder the lines AB and BH, for AB is equall yne to BD... And the figure F His the fquare which ts made of AH VV, herfore therettangle figure comprehended ‘bnder the lines AB and BH is equall to the [quare which is made of the line H A.VV: herefore the right line genen AB is deuided in the point Hn fuch fort that the reftangle figure contayned bnder AB and B His equall to the {quare which ts made of A E1:-which was required to be done. ss Thys propofition hath many fingular vfes. Vpon it dependeth the demonftration Many avd of that worthy Probleme the 10-Propofition of the 4.booke : which teacheth to de- /inguler vfes {cribe an Ifofceles triangle,in which cyther of the angles at the bafe thallbedoubleto ofthis propa the angle at the toppe . Many'and diuers vies ofaline fo deuided fhall you findeinthe tion. 13.booke of &uchae. err hs Thys is to be noted that thys Propofition can hot as the former Propofitions of thys fecond booke be reduced vnto numbers . For the line EB hath vnto the ening Rae line AE no proportion that can be named, and therefore it can not be exprefied fe reduced vn by numbers. For forafmuch as the fquare of EB is equall to the two {quares of so sumbers. AB and AE (by the 47.0f the firlt) and AE isthe halfe of A B, therefore the line BE isirrationall. For euenas two equall {quare numbers ioyned together can not make a {quare number : fo alfo two fquare numbers, of which the one is the {quare of the halfe roote of the other, can not make a, {quare number. As by anexample . Take the {quare of 8. whichis 64. which doubled, that is, 128. ma- keth nota fquate number. So take the halfe of 8. whichis 4. And the {quares of 8.and 4. whith are 64.and 16. added together likewy{e make not a {quare num- ber. Forthey make 80. who hath no rootefquare . Which thyng mutt of necefii- tie be.if thys. Probleme fhould haue place in numbers. But in Irationall numbers it is true,and may by thys example be declared. This propofi- X.il. Let ~ Aero ee - Pa eer wpa — - - 7 =: Cit oct ' ance — ee. + ene nr - le ail ee * mangapnseeonteananmnremars “ es oto a — Sa en ale S he -—- se a pe aaa ~ -_——~. pe A be 5 Ped - — ni ee = - ~ ——_ nae oe ee T he fecond Booke Let 8,be fo denided,that that which is produced of the whole into one of his partes fhall be equall to the {quare number produced of the other part. Multiply 8.into him felfe and there fhall be produced 64. thatis,the{quare -4 BCD. Deuide 8, into two. equall partes,thatis,into 4, and 4.asthelineed EorEC,. And multiply 4.into hym felfe;and there is produced 16, which addevnto 64, and there fhall be produced 80: whofe rooteis 35 80: whichis theline £ Bor the line £ F by the 47, ofthe firft. And — forafmuch as the line E Fis % 80, & the lyne E Ais 4, therfore the lyne ef F is /F* 80—4,And fo much fhall the line AH be. And the line BH thall be 8—/ 4 80—4, that is,12—v/ 3 80.Now thé 12— 7% 80 multiplied into 8 fhal be. as much as V3 80—4. multiplied into it felfe.For of either of them is produced 9 6—#/5120. SeTheu.Theoreme. The 12.D’ropofition. Ln obtufeangle triangles, the fquare which is made of the fide Jubtending the obtufe angles greater then the [quares which are made of the fides which comprebend the obtufe angle, by the retlangle figure, which 1s comprebended twife under one of thofe fides which are about the obtufe angle , yponwhich being produced falleth a perpendicular line 5 and that which is outwardly taken betwene the perpendicular line and the obtufe angle. a at OT DE Vppofe that ABC be an obtufeangle triangle haning are — PD But (by the 4-7.0f the fir i ehe res which are made of the lines CD of Euclides Eleméntes. Fol. 77- €D and 1B. For the angle atthe point Disa right angle..And vnto the Squares which ave made of AD. and DB by the felfe fame ) is equall the Jquare whichis made of AB.VVherfore the {quare which is made of C B,is ¢- quall to the [quares which aremade of CA and AB. and vnto the rectangle fie gure contayned ynder the lines C Aand AD twife. VV herfore § fquare which is made of CB, is greater then the fquares which are made of CA and AB by the reftangle figure contayned ynder the lines CA and AD twife. In obtufe- angle triangles therefore the fquare which ts made of the fide fubtending the obe tufe angle,is greater then the {quares which are made of the fides vwhich come prebend the obtufe angle,by the rectangle figure vvbich is comprehended twife Ynder one of thofe fideswhich are about the obtufe angle pon which being proz duced falleth a perpendiculer lyne,and that which is outwardly taken betwene the perpendiculer Lyne and the obtufe angle: which was required to be demone jirated. Of what force thys Propofition, and the Propofition following, touching the meafuring of the obtufeangle triangle and the acuteangle triangle, with the ayde of the 47.Propofition of the firlt booke touching the rightangle triangle, he fhall well perceaue,which fhall at any time neede the arte of triangles in which by thre thinges knowen is euer fearched out three other thinges vnknowen,by helpe of the table of arkes and cordes. The12z.Theoreme. The 13. Propofition. $a» /n acuteangle triangles,the fquare whichis made of the fide that fubtendeth the acute angle,ts leffe then the [quares which are made of the fides which comprehend the acute an- gle,by the retlangle figure which is coprebended tmife under one of thofe fides which are about the acuteangle, vpo which falleth aperpendiculerlyne, and that whichis inwardly ta- ken betwene the perpenaiculer lyne and the acute angle, Pig id “ppofe that ABC -be anacuteangle triangle hae Ip) 77g the angle aty point B acute eo( by the t2.0f = the first from the point A draw 'ynto the lyne B Ca perpendiculer lyne AD. ThenI ay that the [quare which is made of the lyne AC is if, then. the |quares ‘which are made of the Lyne C Band B A by therettan ole figure conteyned vnder the lines CBand BD twife. For fora{much as the right lyne BC is by chaunce denided in the point D therfore ( by the 7. of the fecond) the [quares X.iy, whith 5 _ o Demonfirasié The fecond Booke which are made of the lines Band BD are equall tothe reftangle figure con- tained ynder the lines C Band D Btwife and vnto the [quare whiche is made of ineCD. Put the {quare-whiclhis made of the line D A common ‘ynto them both. VVherfore the fquares which are madeof the lines CB, 2D and D. Aare equall vnto the rectangle figure contayned ynder the lines C Band BD twife, and ynto the fquares which A aremade of A Dand D C.But to the [quares whiche are made of the lines BD and ‘D A is equaly [quare which is made of the line AB: for th'angle at y point Dis a right angle. And pnto the [quares whiche are made of the lines AD and D Cis equall the fquare-whiche is made of the line A C( by the 47.0f 9 firft):-wherfore the fquares which aremade of the lines Band B Aare equal to the fquare which is made of the line A C,and to that swhich is contai ; ned ‘bnder the linesC Band BD twife . VVherforethe 3 a e, {quare which is made of the line AC beyne taken alone is leffe then the [quares which are made of the lines C B and B A by the rectangle figure , whichis con: tained bnder the lines C Band BD twife. In rectangle triangles therfore the Square which is made of the fide that [ubtendeth the acute an gles leffe then the. Squares which are made of the fides ‘which comprehend the acute angle, by the rectangk figure ‘which is comprehended twife ynder one of thofe fides which are about the acute angle, vpon whichfalleth a perpendicular line , and that which is inwardly taken betwene the perpendicular line and the acute an 1ole:-which was required to be proued. heures | : q A Cor ollary added by Orontius. Hereby is eafily gathered,that fich a perpendicular line in reétangle triangles falleth of neceffitie vpon the fide of the triangle, that is, neyther within the trian- gle,nor without. Burin obtufeangle triangles it falleth without,and in acuteangle triangles within. For the perpendicular line in obtufeangle triangles, and acute-. angle triangles can not exactly agree with the fide of the triangle : forthen an ob- tufe &¢ an acuteangle fhould be equal to a right angle,contrary to the eleuenth and twelfth definitions of the firft booke . Likewife in obtufeangle triangles it can not fall within,nor in acuteangle triangles without: for then the outward angle of 2 triangle fhould be leffe then the inward and Oppolite angle,which is contrary to the 16.of the firtt. we : (Saas _ And this is to be noted, that although properly an acuteangle triangle, by the definition therof geué in the firlt booke,be that triarigle,whofe angles be all acute: yet forafmuch as thereis no trianele, bur that it hath an see angle this propofiti- on is to be vnderftanded;&is true generally in alll kindes of tangles wharfocuer, and may be declared by them;as you may eafily protie. ) 7 | pshinslvSaatgneeea vy) & sete 3A The _. a : — = =e of Euclhides Elementes. Fol.80. The2,Probleme. The 14.Propofition. Vato a rettiline figure geuen,to make a/quare equal. Wane V ppofe that the rectiline figure geuen be A. Itis required to make a h ef I (quare equall vnto the rectiline figure A. Make( by the 45- of 5 firft) Conftenttion. WZ | Lo Z *y- G , i Nica| Vato the rectiline figure A an equall rectangle parallelogramme BC "DE. Now ify line BE be equall ynto the line E D,thenisythyng done whiche was required : for puto the rectiline figure A 1s made an equal [quare - , BD.But if not one of thefe lines BE is ED the creater. Let BE be the greae ter and let tt. be produced bnto y point F. And( by the 2 .of the firft) put bnto ED an equall line EF . And( by the to. of the firft ) denide the line B F into. two equall partes in the point G . And making the centre the pot G , and the Space G B or GF deftribe a femicirceB HF. And | (by the 2. peticion ) extend the line DE | ous pntoy point 1. And( by the -peticion ) A | | draw a line from G to £4. And foraf{much | Demonfiratiz as the right line FBis denided into two | | equall partes in the point Gand into two bnequal partes in the point E therfore (by the 5. of the fecond )the rectangle figure comprehended ’ynder the lines BE and E F together with the [quare which is made of the line EG,is equall to the square which ts made of the line GF. But the line G F is equall ynto the lineG H.VV herfore the rectangle figure comprehended ynder the lines B E and E F together with the [quare which ts made of the line G Ets equall to [quare which is made of the line G H.But nto the fquare which is made of the line GH are equall the fquares whiche are made of the lines HE andG E ( by the 47. of the first. WV herfore YY whichis contained ynder ‘y lines BE and EF together with J [quare which is made of G Eis equall to 9 quares which are made of Fd Eand CE. Take away the [quare of the line EG common to them both . VV herfore the rectangle figure contained bnder the lines BE eo E Fis equall to the fquare which is made of the line EH. But that whiche is contained dnder the lines B Eand E Fis the parallelogrammeB D, for the line E F is equall ynto the line EDV)Vherfore the parallelogramme BD is equall to5 [quare whiche is made. of the line HE . But the parallelograme B D is equall ynto the -rectiline fi eUre A.VVherfore 5 rectiline figure A is equall to the [quare-which tsmade of ¥ line FLE XV herfore ynto the rectiline figure geuen A, is made an equall fquare. defcribed of the line E H:-which ‘was required to be done. ie | ¢ The ende of the fecond Booke of Buclides Elementes. »* o ‘ m0} SC; ~~ t of this booke. a — Shae - > 3 2 x = niin t = — —— a 7 atiey: x : ~ . ne oe ~ —_—— : —_— -—- s een —— mm a tase xs —— aes — ———— = ——— ——— = I ee — es _ pee isk te ee SS ae = = = ————sS— - —— = = — ——————————— ; Ss - riggs, inlg Aah — > = Pa ~ dal ot i a i pe ” Oe re earns = - “ = rome - iy RE ET Ae ees 2 0 ee ee ee ae es NE ; — = — —— — — . — _— z ~_ - aw WASSr tse. wee — - “ - ia = 7" = ; - 7 - — —— 3 — = — = -- ——- _- — Be de EAS sme + _ = = 7 = ee nat . _ i —— a i . = = se Ses = — _—+ ~ — sad? os = a os q Thethird booke of ig clides Elementes.’ 6 es or » en oe Theargument C22 HESS His third booke of Euclide entreateth ey = f\\ of thie molt perfect figure, which is.a circle. Where- sa W°-rasS), fore itis much more to, be eftemed then. the two : tug; bookes goyng before , in which he did fet forth the Fe mott fimple proprieties of rightlined fisures . For

. er i bere, E circumference andat the centre of a circle, and of the varietie and differences of: them. Wherfore the readyng of this booke , is very: profitable to the attayning to the knowledge of chordes and arkes.It teacheth moreouer which are circles con-. tingét,and which are cutting the one the other :.and alfo that the angle of contin- gence is the leaft ofall acute rightlined angles:and that the diameter ina circle is the longeft line that can be-drawen ina circle. Fattherin it may wellearne how, three pointes beynggeuen how foeuer(fo that they be'inot fet ina right line),may. be drawen a circle paffing by them all three, Agayne,how in afolide body , asin 2. Sphere, Cube,orfuchtyke,may be found thetwo oppofite pointes . Whicheis.a thyng very: neceflary and. commodious : chiefly for thofe that shall make inftru- mentes fernyng to Aftronomysand other artes. Definitions. > ' Equal circles are fuch,whofe diameters are equall, or whofe bynes drawen from thecentresareequall.~ | The circles A andB are equal,if theyr diameters namely, E FandC D be equallsor if their femidiameters , whiche are lynes drawen fromthe center to the circumference, Why circles Se se ae Ne pte take their ~*~ Ahe. reafon. why. circles: equality of take theyr equalitie, of the e- vers or femi= — {6midiametets is , for that a digmeberse | of his endes fixed .\ As ifyou namely AF andBD beequall, © *” thetrdiame- qualitie of their diameters or - circle is defcribed by'one re-*” : uolution or turyng about of > “the femidiameter, hauing one \ + _ imagine the lyne AE to haue. \ Sco ricceea ee ema ~ . his one pointnamely A faftened,and the other end namely Eto mouerou ener > o nd tilt of Euclides Elementes. Fol.81. it come to the place where itbee4 to moue, itfhal fully deferibe the whole circle. Wherefore if the femidiameters bee equall , the circles’ of neceffitye mutt alfo be equall: and alfo the diameters, By thys alfo is knowen the definition of vnequall circles. Definition of | vnequall cure Circles whofe diameters or femidiameters are unequal, ave alfo unequal , eAnd that circle chess which hath the srearer diameter or femidiameter; is the greater circle: andslat circle Which kath é

eA right line is fayd to touch a circle, which touching the cir- Seto it cle and being produced cutteth it not. — As the right lyne E F drawen from the point E , and paffyng by a poiatof the circle, namely by the point G to the point F ons Jy toucheth thecircleG H,and cuttethi¢ 6 not,nor entreth within it.For a right line E entryng withina circle,cutteth anddeut- : Ly deth the circle. As the right lyne K Lde- videth and cutteth the circle KLM, and entreth within it: and therfore toucheth it in two places . But a right lyne. tou- chynga circle,which is commonly called a cotingent lyne,toucheth the circle one- H LA A contipent ly in one point. Cine a Circles are fayd to touch the one the other, which touching the Third defini one the other,cut not the one the other. ae As the two circles AB and BC touch the oneithe. other , For theyr circumferences | touch together in the poynt B., But neither of them cutteth or deuideth, the other . Neither doth any part of the one enter within theo» ther.And fuchatouch of circles is euer in one The touth of poynt onely: which poynt onely is. common circles 38 ener to them both , Asthe poynt B isin the confe. en * posneé S renceof the citcle AB, and alfoisin the circite ference of the circle BGs =>. : T be third Booke Circles may Circles inay touch together two maner of wayes, either outwardly the one =, ft . Wholy without the other : or els the one being contayned within the other. wer of Wayess = Asthe circles D Eand DF : of which the one D E contay- néth the other, namely DF: and touch the one the other in the poynt D:and that onely poyntis common to them both: neither doth the one enterinto the other . Ifany part, ofthe . Oné enter into any part of the other,then the one cutteth and deuideth the other, and-toucheth the one the other notin one poynt onely as in the other before, but in two pointes, and haue alfo a fuperficies common to them both. As the cir- cles GH Kand HLK cut the one the other in two poyates Hand K:and the oneentrethinto the other: Al- | fo the fuperficies H K is commonto them both: oy sae EE For itis 3 part of thecircleG H K, andalfo it isa ve Def isla-00d) pattofthe circle WL XK. 5 Fourth def- ‘Right lines in acircle are fayd to Piso sa be equally distant from the cen- Oar Bear | | tre, when perpendicular lines drawen From the centre ‘nto thofe lines are equall. eAnd that line 1s fayd to be more di- Stantsypon whom falleth the greater perpendicular line. : om — © ey the circle ef BC Dwhofe centreis E,thetwo lynes | A Band C Dhaue equall diftanee from thecentre E : bycaufe thatthe lyne £ F drawen from the centre E perpendicularly ypon the lyne 4 B,and the lyne EG drawen likewife perpendi- jarly from the centre £ vpon the lyneC D are equall the one to B theother . Butinthe circle K L 4 whof€ centreis N the lyne 4 KX hath greater diftance from the centre ®_ then hath thelyne ZL A4: forthat the lyneO NX drawen from the centre W_perpendicularly vppon the lyne H K is greater then thelyne N_P whichis drawen fro the centre AL perpendicularly vpon the lyne LeAl, 3 » Solikewife inthe other figure thelynes 4B and DCin the - circlee7d BC D are equidiftant fronithe centre G,bycanfe the lynes O G.andgG P perpendicularly drawen from the centre G vppon the fayd lynes 4B and DCare equall . And thelyne A Bhath greater diftance fromthe centre G then hath the thelyne EF , bycaufethelyne OG perpendicularly drawen * from the centre G to the lynee 4 B is greater then'thelyne H G whiche ts perpendicularly drawen from the centre G to the lyne EF. ? A. dow i « te SS ee ee ——— sae SS ee —————— = —— =< = === = = ———. = Se Fae — : ~~ ; etc = = ae ne ae a = = =—= — = = = _s : = Sour tae ee Se > — 5 = = —e So = = — = = - = = = - -- — ~— — - _—— z : =a z SS Po z= - . — - . A — =~ ——- —= = as - -s ayes mas teed a See ae aE S SS Se “a $ cS eee a ee ~ —— : = = 2 , aw — — - —— SS 7 Sa Se ae ee a Fi = a _— = = = So oe = eee at See s2 IE eg dagen pe Oley a eS = > oc eS = = —— = —— 2 3 — * a ame ee rn Ea wa See = iS RS ame ; — —— — = = = = 3 ->— —.—- - ate ee ee = ee ee ee . -. mee : = ———— > ==. = ——— == ~- —- —. —— — = = = — = =~ —— = = SS = 2 Semen; acta é een er FES <= = = - = = ~ = -- ~-———- ee oem : -— —— => te Afetlionor fegment ofa circle, is 4 figure coprehended wader aright line and a portion of the circumference of a circle. e a3 | oS -As the a Oo ne spticenet ys —— = a of Enchides Elementes. Fol, 82, Asthe figare AB Cis. feSion of acircle * ° ‘3 {ueS bycaufe itis comprehended: vnderthe right >> B J lyne 4C and the circumferenceofa circle 4 aN BC. Likewile the hgure DE PisafeGion df acircles“forthatitis comptehendéd vader im FE therightlyne DF, and theeiréiference DE F. And the figure 4 2B C fer‘ehat it eotaineth within itthe centre of thecirclediscalled: the . oreater fection'of a circlenand thefigure DBF isthe lefle fection of a circle ; Bycanfe itis wholy withoutthe centreofthe circle as it was hoted inthe 16 ; Definition of the firft booke, : 4} | re An angle ofa [eClon or fegment, if that angle which ts con« Sit def tayned vender aright line and the circuference of thecircle. “°° » Astheangle ABCin the feGionAB Cis.ay angleofafece tidn , bycaufe itis contained of the circumference B'A.G.and. ; therightlyneBC. Likewife the angle C B D isan angle of the feGion BDC bycaufe itis contayned ynderthe circumference BDC,and the right lyneB€ . And thefe angles are commonly called mixte anglesybycau{e they are contayned vadera right lyne and a‘crooked3 And thefe:portions of circiimferences are: commronlycalied arkes, and theright lynes:are calledchordes, orrightlyneés fubtended, And the greater feGtion hath ener the greater angle,and the leffe feGion the lefleangle, Mixt angles. Arkes. Chordes. “An anoles layd tobe in a Jettion, whe in the circumference 18 gepeirn ies taken any poynt,and from that poynt are drawen right lines friviom tothe endes of the right line which is the bale of the Jeoment, “the angle whichis contayned onder the right lines drawen f from che poy, it C1 fay )fayd to bean anglein'a fettion, ; ~ Astheangle AB Cis'ananglein the fedion ABC ,bycaufe,. B fromthe-poynt B beynga poyntin the circumference AB Gare. ... : drawentworighrlynes B Cand BA to the endes of the lyne AC | whichis thebafeof the feion ABC. Likewife the angle AD G is aifangteinthe {eGion A D Cybycaufe from the poynt D beyng 5 fy - in the circufereice A D.Carédrawentwo right lynes,namely,D “| C & DA totheendes ofthe right line A C. whichisalfo the bafe ; to the fay Ffecion A D G:So-you fee, itis not all one tofay,an ans...) gle of afection,and aftanglein afegtion,An angle of afecion co- e's fifteth-of the touch of a tightlyne anda crooked .,Andan angle... 7 in afe@ion is placcd on the circumference ,vand is contayned of tworight lynes . Alfe the greater {c&ion hath imit thellefleangle, and the Jeffe feGion hathinit the greater abaie., \ . | dh pe ——— 7 D Difference off an angle of & Section and of an angle in a Setti0tbe “* ao A But when the right lines which comprehend the angle dove» riots dcp. -ceateany circumference of a circle,thenthat angle 1s fayd to" .. betirvelponidentsand vo'pertaine to that circumferences — Bes Aa.ij. Asthe ~ Thethird Booke io Asthe rightlynes B Aand B C which containe theangle AB ©,and receaue the circumference A D C therforethe angle AB Cas fayd to fubtend and to pertaine to the circiference ADC, And ifthe right lynes whiche caufe the angle, concurrein the centre of acircle : then the angle is fayd to bein the centre of 2 circle, As the angle EF-D is fayd to be in the centre of a circle, for that it is comprehended of two right lynes F E and F D: whiche concurre and touchinthe centre F, And this angle likewife fubtendeth the circumference EG D: whiche circumference alfo, F is the meafure of the greatnes of the angle E F D, a els 2a ARS HEGSL tO D A Setlor of a circleisCan angle being fet at the ~—z N ent h defi e- 6 s sition, centre of'a circle) a figure contayned ynder the right hines which make that angle,and the part of the circu mference ree ceaued of them, As the figure A B Cis a feGor ofa circle, forthatit hath an angle at the centre,namely the angle B.A C,& is cotained of thetwo right | » dynes A Band A C( whiche contayne thatangle and thecircumfe. | rence receaued by them, ? 33 3 t 3 Like fegmentes or fettions of a circle are thofe, which baue ..,. equal angles,or in whom are equal angles. : Two defini- Here are fet two definitions of like fe&tions of sions. . acircle, The one pertaineth to the angles whiche Firp - are fetin the centre of the circle and. receaue the : circumferéce of the fayd feGions:the other per- taineth to the angle in the feGion;whiche as be- fore was fayd is oot: in the circumference . a if theangle BAC, beyngin the centre A and re- BO seal of the circumference BLC be equallto, =F F G. the angleF EG beyng alfointhe centre E and | oF receauied of the circumference FK G, then are the two fe@ionsB C Land F GK lyke by the firft definition. By the fame definition alfo are the other two feGions like,name« lyBC D,and FGH, for thatthe angle B A Cis equall tothe ancle FEG, ape Tenth defint tiok. 8s Pp Alfo by thefecond definition if B AC beyng an angle placed in the cit >” cumference of the fe@ion BC A bee.” . angle E D F beyng an angle in the fe- », Gion EF D placedin the circumfe. | . Fence, there are the two fections BC A, and EF Dlyke the one to the o- ther. Likewife alfo if the angleBGC beyng in the fectionBGG be equall, TT a, to the angle EH F beyng inthe feaia ~~ i EHF the twofe@ionsB CG andEE™: H fone. =< —— Sebancae =a ne “Se ose > — . a SS-— =r ——— _——— ———. a -> ya ——— ——— = a ee : =i ee = === — = . = ib ioea eno am . ss =" = Oe a he ~ . i ee oF Ss ore 4 -—— —_ = -= . SS = ‘ — — a) ae —— . naeear ra Se ants ie - os 4 .-. =F ma < = SSS - e “ 2 v Mies | +e LT Sa See Sees a — ; REE SET? ae meneEe : © : vane : cad. ~~ = » See a ——— nt a nel “ _— Se no - —pa a = — : - - a _ . Hare lyke. And fo is it of angles beyng equall in any poyntof the circumference. ee 4 : P - Gee or ¥ é oF Ba . ait aa of Euchides Elementes. Fol.83. Euclide defineth not equall Sections: for they may infinite wayes be defcribed. For there may vppon vnequall rightlynes be fet equall Sections ( butyetin vyne- quall circles) For from any circle beyng the greater,may be cut ofa portion equall to a portion ofan other circle beyng the lefle’. But when the Sections are equall, and are fet vpon equallright lynes , theyr circumferences alfo fhalbe equall. And rightlynes beyng deuided into two equall partes , perpendicular lynes drawen from the poyntes ofthe diuifion to the cir- cumferéces fhalbe equall. Asifthe two fe@i- ons ABCand DP E F, beyng fet vppon equall B E ryghtlynes AC& DF, beequall : then if echof the twolynes 4 C & DF be deuided into two e- quall.partes in the poyntes.G and H, & from the {ayd poyntes be drawen to the circumferences | two perpendicular lynes B Gand E\H, the fayd Pet\occ) perpendicular lynes thalbe equall, | A G C 2 H F ST he Probleme.” The P ropofition. To finde out the centre of a circle geuen. NO) ee that there bea circle geuen ABC . It “is requie \ WS) Key || red to finde out the centre of the circle ABC. Draw init : \\ gs SY LLY Ne, @right line at all aduentures, and let the fame be AB. HEN| And ( by the 10. of the firft) deuide the line A B into two RNG! eguall partes in the poynt D. And( by the 11.0f the fame ) CF oat fi the poynt D raife vp vnto AB a perpendicular line Hick Say | D Ceo (by the fecond petition Jextend D C bntey point E. And (by the 10. of the firft ) deuide the line C E into two equall partes in the poynt F . I’'hen I fay that the point F is-the centre of the circle ABC. For if it be not,let fome other point namely G ,be the centre. And ( by the firft pette tion draw thefe right lines G.A,G D, and GB. And for afmuch as AD is equall ynto DB, and DG is common vatothe Loth, therefore thefe two lines AD. and DG are equall to thefe two lines GD and D B,the one to the other and ( by the 15. definition of the firft ) the baleG A is equall to the bafe G B. For they are both drawen from the cee tre G tothe circumference : therefore ( by the 8. of the firft) the angle ADG is equall to the angle BDG . But when a right line ftanding pon a right line maketh the angles on eche fide equall the one to the other , eyther of thofé angles ‘(by the 10. definition of the firft is a right angle. VV herefore the angle BDG i ia a... is # Why Euclide defineth not equall Settt- ONS. Conftrulitos. Demonfire- ston leading #0 4n impof~ Abilstie. The third Booke is aright angle: but y angle FD Bis alfo aright angle by conftruction. VV her: fore(by the 4.. petition )the angle F D Bis equall to the angle BD G,the grea ter to the leffe , which is impofsible .VV herefore the poynt Gis not the centre of the circle A BC. In like wife may we proue that no other poynt befides F is the centre of the circle ABC.VVherefore the poynt F is: the centre of the arcle ABC: which was required to be done. ; | Correlary. Correlary. FLereby it is manifest thatifin a circle aright line do demde aright line into two equal partes,and make right angles oneche fide : in thatright line which deuideth the other line into two e- quall partes is the centre of the circle. SeThe1.Theoreme, The 2.Propofition. Ltn the circiiference of a circle be také two poyntes at all ad- uentures : aright line drawen from the one poynt to the otber Shall fall within the circle. . V ppofe that there be a circle ABC. Andin the circumference thera SAR of le there be take at all aduentures thefe two poyntes A¢x B. T hen Demonftra- Kren vA Say that aright line drawen from A toB fhall fall-within the circle #i0 sopofti. SSE 4B C.For if it do not Jet it fall without the circle as the line AEB a4 “9 Litie. doth which ef it be pofsible imagine to be aright line. And( by thePropofition going before )take the centre ache circle and let the fame be ‘D. And{ by the firft petition )draw lines from D to A,and from D toB. And extend D F to E. And for afmuch as (by the 15. definition of § firft) - ; D Ais equall vnto DB. Therefore the ane gle D AE is equall to the angle DBE. And for afmuch as one of the fides of the triangle D AE, namely the fide AEB is produced, D therefore ( by the 16. of the firft ) the an gle D EB is greater then the angleD AE But \4 B the anglk DAE is equall nto the an gle P26 DBE.VVherfore the angle DE Bis rede ws ter then the angle D BE. But (by the 18.0f neg the firft.) bnto the greater angle ts [ubtended _ “A __ the greater fide. VV herefore the fide D Bis ee greater then the fide D E, But( by the 15 . definition of the fir/t )the line D Bis equall ee of Euchides Ellementes. Fol. 84. equall ynto the line DF.VVherfore the line D F 1s greater then the line D E, namely, the leffe greater then the greater: which is impofsible VV herfore aright line drawen from Ato B falleth not “without the circle. In hike fort alfomay we prone that it falleth not in the circumference : VV hereforeit falleth within the circle If therefore in the circumference of a circle be taken two poyntes at all ad- sentures: aright line drawen from the one poynt to the other [hall fall within the circle: “which was required to be proued. SepT he2.Theoreme. ~The 3.Propofition. Ifinacirclea right line pafsing by the centre do deuide ano- ther right line not pajsing by the cetre into two equall partes: it (hall denide it by right angles. And if it deuide the line by right angles,it hall alfo deuide the fame line into twoequall partes. iy V ppofe that there be a circle AB C ,and let there be init drawen The firft pars Ve aright line pafsing by the.centre, and let the fame be CD, deutding of this Propo- Hal other right line AB not pafsing by 5 centre into two equall partes #1" === in the poynt F. Then I fay that the angles at the poynt of the deutfion are right angles. I ake (by) firft of the third) Confirnttion. the centre ofthe circle ABC, and let the fame be E. And ( by the firft petition ) drawe lines rom Eto A ex from Eto B. And for afmuch Deémouthra- as the line AF is equall vnto the line FB,and — the line F E is common to them both, therfore thefetwo lines E F and FA are equall ynto ber of Ne thefe two lines E F er FB. And the bafeE A is equall puto the bafe E B( by the 15. defini« tion of the firft).VV herefore( by the 8. of the firft) the angle AF E is equall to the angle BE E, But when a right line Standing vpon a. right line doth make the angles on eche fide equall the one to the other eyther of thofe angles ss ( by the 10.definie tion of the firft a right angle. VV herfore either of | thefe angles AF Ee BFE is aright angle. VV herefore the line CD pafsing by the centre, and deniding the line AB not pafSing by the centre into two equall partes maketh at the point of the denifion right angles. But now Juppofe that the line CD do denide the line AB in fuch fort that it The fecond maketh right angles. Then fay that it denideth it into two equall partes, that eet 15,9 the line A Fis equall ‘ynto the line F B. For the fame order of COnfEUCLION Tem aS remayning for afmuch as the line E Ais equal ynto the line E B (by the 35. des tion. | Aaa.iilf finition _ - a a - — ~— - “ ch i i - - A ey Rett ig Se ae eso ail Rann . Rivmaeiitiinns dei Signa at + ; SS " 2 we ne as : are NS “= Bisons . yoo aoe — -— - - s 2 - i aad % ==: : > = 7 = = * - Lz ——— — as bs —— Se = = = a —— = re — > ——— - ‘ = see ee hese : ~ == : aw = =~: a : =_—— = oe = = = — = <== —————— === = - —= ——~ —— SS eee — —S— <= —— = oe = mt meet a - = " : — = -— Saas = 2 === ~—+>- —_ — - fn — = oe weed . wr a spe Se =: oflgirten ace Las - - a. <= : oes Se . =— — —— : : : : > = we 4 r * 3 gs a — > = SS ee a ee 5S ees ee = =s = S ba ‘Fe ~* sr —— —-— ~-—-~ —_—— ==> > : fe * - is > . = ee = See - - ~" =.= Sn - a _ —— —_ — —_— — = , ate + ei —_— ee —— = ao Byes ln so = ~ , —t mee = en —— = — a —— = Ps —— cn —- =—— = = : == a = — oe : = a = F ‘ * C Rates _ — ———— = —————— —— — = : =~ = - ~=s > ad ——_—_- --—----- — ------— — - = _ Sass FR = <= a Sst Se r =— = = : — a = — - _ = = : eee <==-= — ~ . ae ~-- ~ =e os —- ‘ ~—- _ - 3 : ae Seat = —S—e SS - - SS a - = - E va ae AER SNE Pt TET 2 oe Demon ra- tion leading toan mpof~ fibilitie. . it be pofsible let them deuide eche the one the equall ynto EC,<¢ BE vnto ED. And take from F to E. Now for afmuch as a certaine right line FE pafsing by the centre deuideth an other line A Cnot pafSing by the centre into % two equall partes it maketh where the deuifie bat on is right angles (by the3.of the third ). VV, herfore the angleF E Ais aright. angle. Againe for afmuch as the right line FE, pafsing by the centre , deuideth the right line BD not pafsin ¢ by the centre into two equall partes stherefore( by the fame )it maketh phere j deuifion 1s right angles VVI isaright an cle. And it is proued that the angle FE Ais aright angle. VV here fore, by the 4. petition ) the an gle F EA is equall vnto the angle FE B namely the leffe angle dnto the greater : which ss impofsible . VV herefore the right lines AC andB D denide not eche one the other into two equall partes. If therforein a circle tworght lines not pafsin ig by the centre, deuide the one the other , they ' T he third Booke. inition of the firft.). Therefore the angle E.AF is equall ynto the.angle E BE (by the § «of the firjt). And the right angle A FE is (bythe 4. petition )equall totherightangle BF EB. VV herefore there are two triangles E AF, 27 E BF hauing twoangles equall to two angles, one fide equall to one fide, namely the Side EF which is common to them both and fubtendeth one of the equallangles, wherefore (by the 26: of the firft,) the fides rema yning of the one are equall dn- to the fides temayning of the other .VV herefore.the.line. A. F 4s. equall ‘nto the line FB. If therefore in a circlearight line pafSing by the centre do deuide an other right ine not pafsing by the centre into.two equall partes,, it [hall denide it by right angles. And if it deuide the line by right angles it {hall alfo deuide the _ feame.line in'o two equall partes :-which-~was required to be demon/ftrated. SpT he 3. Theoreme. The 4... Propofition. Lfina circletworight lines not pafing by the centre, denide the one the other : they /hall not deuide eche one the other into, two equall partes. Bee oe tex, V ppofe that there bea circle ABCD, and let there bein it drawen two as ; right lines not pafsing by the centre and deniding the one the other, and we let the fame be AC and BD, which let deuide the one the other in the poynt E. Then L fay that they denide not eche Se the one the other into two equall partes . For if other into tuo equall partes, fo that let AE be the centre of the circle ABCD, which let bel. And( bythe firft petition.) draw a line erfore theangle FEB so MGS foal of Euclides Elementes. Fol.8s. prall not deuide eche one the other into two equall partes: ‘which wa required to be demonjftrated. In this Propofition are two cafes.For the lines cutting the one theother,do ey- ther,neyther of them pafle by the centre,or the one of them doth pafle by the cen- tre,& the other not. The firlt is declared by the author.The fecond is thus proued, Suppofe that in the circle ef BC D theline BD pafling by the centre doe cut the line e#C 6 not pafling by the centre. Then I fay that the lines ' eC and B&D donot deuide the one the other in- to two equall partes, For by the former Propofi- tion the line B D-paffing by the centre and deui- ding the line e fC into twoequall partes, it fhall alfo deuide it perpendicularly.And for afmuch as the line 4 C deuideth the line B D into two equall partes & right angled wife:therfore by the Correls lary of the firftofthys booke,the linee4 Cpaffleth 4 by the centre of the circle : whichis cOtrary to the fuppofition. Wherfore the lines e4 Cand B Ddo not deuide the one the other into two equall D partes : which was required to be proued. ST he 4 Theoreme. The5.Propofition, If two circles cut the one the other,they haue not one and the fame centre. ING ppofe that thefe. two. circles PVIABC; and CBG.do cut the YF" one the.other in the poyntes C and B. Then I fay that they haue not one ¢> the fame centre.For if tt be pofsie ble let E be centre'to them both. And( by the fir[t petition draw a line from E to C. And draw another right line EFG at all aduentures. And for afmuch as the poynt Eis the centre of the circle ABC, therefore ( by the 15+ definition of the first the line E Cis equall vnto the line EF. Agayne for afmuch as the poynt Eis the centre of the circle C BG » theres fore by the fame definition )the line EC ts equall ‘pnto the line EC. And itis proued that the line E-C is equall vnto the line E F : “wherefore the tine E F alfo is equall ynto the line E G namely the lefJe ynto the greater : whica is impofSie ble. VV berfare the poynt Eis. not the centre of both the circles ABC,eo-C BG, In like fort alfo may-we prone that no other poynt is the centre of bith the fos | circles. Two tafes in this Propo= fittons Confiruttion for the fecond 6afen Demonfira- $10. Conflenttion. Demonfira- 120 leading to an impoffibs~ lities Demonfirae avin leading to an 1m pof= poilitie. ; line F Bywherefore the ine FE: alfo is es , qual ynto the line FB, namely the leffe _» Sore the pont F isnot the centre of both the circles: ABC and C OF. Yn like °°? fort alfomay wwe prone that.no other poynt isthe centre of both the fayd circles, Z wocafes én thys Proe pofitione Thethird B ooke 23 circles Lf therefore tino circles cut the one theother y they haue not.one and the famecentre: which was required to be proued.. SapThe5.Theoremes’ > The. Propofition, Iftwo circles touch the onethe other , they baue not oneand the famecentre, | A Jd V ppofe that thefe two circles AB Cee CD E do touch the one the other | | Boy an the poynt C. I’hen I fay that they haue not one and the fame centre. | == For if it be pofsible let the point F be centre nto them both.And (by the | uft petition )dvaw a line from Fto@sand pod 3% Sesic ioe draive the line FE B at-all aduentures. And for afmuch as the poynt F isthe cene tre of the circle. A B C,therfore( bythe 15. definition of the first jthe line F Cis equall ‘bnto the line FB. Agayne forafmuch as the poynt Fis the centre.ofy circle CDE, therefore ( bythe Jame definition) 'the line . FC is equall ynto theline FE. Anditis prowed,that the line F Cis equall nto the | BS: bnto y greater: which is impofs thle VV her centre : which was required to be demonftrated, ene ad ee « —— co If therefore two circles touch theone the other: they haue not one and the fame. In thys Propofition are two cafes ¢ for the circles - touchyng the one the other,may touch eyther within of without. Ifthey touch the onethe other withingthenis it by the formerdemonftration manifett, thatthey haue not both one-and the felfe famecentre. It is alfo thanifettif.4. they touch the one the other without: for that cery cen- | °\ treisinthe middeft ofhyscircle, ae ai $4. CC HINtee aaa of ace betaken an pont wbichisaam ey ae peeks ‘ oe &S SY3 } . a - Las ew . 2 ” ~ ” <~ * al — of Euchides Elementes. Fol.86. thecentre of the circle, andfrom that poynt be drawen ynto the circumference certaine right lines: the greatest of thofe ~ Hines hall be that line wherein is the centre , and the left fhall be the refidue of the fame line. And of ail the other lines, that which 1s nigher to the line which paffeth by the centre ws greater then that which ts more distant. And from that point can fall within the circle on ech fide of the least line onely ting equall right lines. . V be AD. And take init any poynt befides the centre of the circle jand let the fame be F. And let the centre of the circle( by the 1. of y third) be the poynt E. And from the poynt F let there be drawen ‘pnto the circumference ABCD thefe right lines FD, FC, and FG. Then I fay that the line F Ais the greate/t: and the line FO is the left. And of the other lines, the line F Bis greater then the lineFC, and the line F C 1s greater then the line FG. Drawe (by the firjt petition )thefe right lines BE,CE, andG E. And for afmuch as( by the 20. of the firft jin ene ry triangle two fides are greater then the third therefore» lines EF Band EF ave greater then the refidue , namely then the line FB, ‘But the line AE 1s e quall puto the line BE (by the 15.defte. , nition of the fir{t). VV herefore the ines BE and EF are equal ‘nto the line AF .¥V berefore the line AF is greater then then the line BF. Agayne for afmuch as.the line BE is equall ynto C E (by the 15, definition of the firft and the line F E 1s common vnto them both therefore thefe two lines BE and EF are equall vnto thefetwoC Eand E F . But the angle BE F 1s greater then the angle C EF .VV herefore (by the 24.0f the fir/t the bafe BF ss greater then the bafe CF: and by the fame reafon the line C Fis greater then the line FG. Agayne for afmuch as the lines G Fand F E are greater then the line. EG( by the 20.0f the firft). But ( by the 15.definition of the fir) eh line E Gis equall ‘ynto the line ED: VV herefore the lines G F and F E are greater then the line ED take away EF svhich is comon to thé both ,wherfore 9 refidue G F is greae ‘ter thentherefidue FD? VV berefore the line F A ts the greatest and the line FD 1s eb the line FB is greater then the line FC, and the line FC greater ROW V ppofe that there be acircle ABCD: and let the diameter thereof S Conftrattion. The firft pare of this Prope fition. Demon/flra- FLO%e Second part. Pa — ae. aa =e ee - ao ae eee = —— —— — ~~ _ > = = Spee F< an ae eres, — ~ * ae - = — ae! = : or: 2: ~ £>>74 = =A a — —- = > —~ er ? se “ . = = = = a —— = = _ —— . — _ - — = > > - m 4 — ee oe a —— é _ =, $ —s ae irs . aes > = a re ——— as = Ses == —— Fes << -s 5 re x = = = = = = ~ SS Toa a — ¥ ==, = = = x = = ~ = — +e Ses =e SETAE Be ae 3 i = t ss =£ “= 2° 2Se08 me => a = aes = ~—— as = >. — ————— = a re. mae =: Ae Sah = SS < rer p == _< = ~—: ae 2 we 4 Fa pall _ — ~ L < > SSS es ee eres oe See ASS Caw Soe : _— == eee Ses - =~ ao ——— —~- : _—— = =~ ~- -_———s _ —- _ —< — —_ a — —— = = - = > — — ~ - Ps SSS B28 a o>». : * . 2 bird part. This demon Frated by an argument lea- ding to an im-~ po/ssbelte. on other deo pnon|iration of the latter part of the Propofition beading alfo toan smpofst- bslitie. vt Cor ollsrye Thethird Booke as greater then the line FG. Now alfo I fay that fromthe poynt F there tan be drawen onely two equall right lines ito the circle ABCD oneche fide of the leaft line namely F D.. For (by the 23. of the firft) vpon the right line genen EF and to the poynt mit namely E,make'ynto the angle GEF an equall ane gle FE FH: and{ by the firft petition )draw a line from F to H. Now for afmuch as (by the 15. definition of the firft) the line EG is equall pnto the line E H, and the line E F 1s.common bnto them both, therefore thefe two lines GE and EF are equall nto thefe two lines HE and EF, and (by conftruéfion) the angle GE F is equall vnto the angle H EF : VV herefore( by the 4.0f5 firft) the bafe FG is equall ynto the bafe FH. I fay moreoner that from the poynt. F can be drawen into the circle no other right line equall bnto the line FG . For if it pofsible let the line F K be equall vnto the line FG. And for afmuch as F K as equall nto FG. But the line F H is equall pnto the line F G, therefore the line F K_ 1s equall vnto the line FH. VV, herfore the line which ts nigher to the line which paffeth by the centre is equall to that ‘which 1s farther of , which we bane before proued to be impofsible. Or els it may thus be demonjtrated. Draw ( by the fir/t petition) a ine from E to K_: and for afmuch as ( by > 15. dee jinitio of y firft) j line GE is equall nto J line EK, and the line F Eis common to them both ,and the bafe GF is equall ‘wnt the bafe F K., therefore (by the 8. of the firft ) the angle G E Fis equall to the angle K EF. But the angkeG EF is equall tothe anole HEF .VVheree fore (by the firft common fentence ) the | angle AE F 1s equall to the angle K EF the leffeonto the vreater : which ig impofsible. VV herefore from the poynt F there can be drawen into the circle no other right line equall ynto the line G F. cate, but one onely . If therefore in the diameter of a circle be taken any poynt, whichis not the centre of the cirs cle,and from that poynt be drawen nto the circumference certaine right lines: the greateft of thofe right lines Shall be that-wherein is the centre: and the leaft hall be the refidue. And of all the other lines sthat which is nigher tothe line which paffeth by the centre is greater then that which is more diftant. And from that poynt can fall within the circle on ech fide of the leaft line onely two equall right lines : which was required to be proued. q. A Corollary. Hereby it is manifeft, that two tight lines being drawen frd any one poyntof the FR pos of oe ra: the other of the other fidevif with hi diamer ter they make equall angles,the fayd two right lin - Asi AC are the two iis FG a FH, i a aa 3 Deal g0be of Euclides Elemenites. Fol.87. $@The yz. Theoreme. « Lbe8.Propofition. Tf withoutacircle be taken any poynt and from'that poynt be dravven into the circle unto the circumference certayne fight lines,of which let.one be drawen by the centre and letthe refit be drawen at all aduentures: the greate/t of thofe lines which fall mn the concauitie cr hollownes of the circumference of the circle,ts that which paffeth by the centre : and of all the other lines that line which ws nigher tothe line which paffeth by the centre1s greater then that which xs more diftant.But of thofe right lines which end in the conuexe part of the circum/e- rence,that is the mn which 2. drawen from the poynt tothe diameter : and of the other lines that which isnigher tothe leaft is alwates leffe then that which is more diftant. And from that poynt can be drawen unto the circumference on ech fide of the leaft onely two equall right lines. Poul & ~~ 3 a, PP ofe 9 the circle genen be ABC, 33 wha withouty circle ABC, takethe ==|point D : and fro 9 fame point draw certain right lines into y circle bnta the'cire. cumference,<¢7 let thébe DA, DE ria ee ir D C:e7 lety line D A paffe by > centre. Then E fay, of right lines which fall inthe concauitie of y circumference AE FC ,y 1s, withing circle ,j ereateft is y which paffeth by y centre that is, DA. And of thofe lines wich fall ypon5 conuex part of y circumfes rence ,y leftis 9 whichs drawen fra § point D yntoy end of ¥. diameter AG.And of the right lines falling Win the circumferece, the dine D E 1s greater thenJ line D Fes the. ; line ‘DF is greater theny line DC. And of. ~ the right lines which end inj conuex part of. : the circumference, is , without § circle that | which ts nigher mtoD GJ left is alwayes leffe theny which is more diftat that 1s;the le Kis ef then theline D Land the line DL is leffe then theline ‘f) EF. Take( by the fi rf of i the third }the centre of the arcle ABC ,andlet the Confirstlion. agi: ee SS ee | Bb. Jame : iif an | im ME | i ' hit by if Hil} Whit " if 4 ae i | | ate i : by Hi i aH AB | i} an wim: | ; ’ i 1 Hy i { teh I I SLUM So nEE a Wa aT 4 anu J + ? HH : 4 : \ , nie ie it [ ’ iT ' ha ‘ } ; vi ; whi ii T - : ‘oh tae a i Tt i B v Tbe fir fl part of this Propo- fist0ne Second part. Third part. T he third Booke fame be M: and (by the firft petition ) drawe thefe right lines ME,MF,MC, MH,M Land M K.. And for afmuch as ( by the 15. definition of the firft) theline AM is equall vnto the line EM, put the line MD common to them both .VV herefore the line AD is equall ynto the lines EM and MD. But the lines E Mand MD are (by the 20. of the firft) greater then the line ED. VV herefore the line AD alfois greater then the line ED. Agayne for afmuch as (by the 15. definition of the fir/t)the line M E 1s equall ynto the line MF, put the line MD common to them both: VV herefore the lines EM andM D are equall to the lines F M and MD ,and the angle EM Dis greater then the angle FM 'D: VV herefore ( by the 19. of the firft.) the bafe E D is greater then the bafe FD. In like fort alfo may we proue that the line FD 1s greater then the line C D VV herefore the line D Ais thé greateft and the lineD Eis grea ter then the line D F, and the line D F 1s greater then the line DC. And for afmuch as ( by the 20. of the x firft) the lines M K and K D are greater A thentheline MD .But( by the 15+definis tionof the firft) the line M G is equall’yns | tothe line M K_. VV herefore the refidue K Disvreatertheny refidueG DVV here fore the line G Dis bs then the line KD. And for afmuch as from the endes of one of the fides of the triangle MLD , namely, Af D are drawen two right lines M K and K_D meeting within the triangle therfore (by the 21. of the firft) the lines MK and K Dare leffe then the lines MLe LD, of which the line M K_is equall vnto the line ML . VVherefore the refidue DK is leffethen therefidue DL . In dike fort alfo may we prone that the line D Lis wie : then the lne DH . VVherefore the line | D Gis the left, and the line DK_ is leffe then the line D L, and the line DL is leffe then the line D H. Now alfol fay that from the poynt D can be drawen ‘pnto the circumference on eche fide of ‘DG the least onely two equall right lines. Vpon the right line MD, and bnto the poynt init M make (by the 23. of the firft) vnto the an: gle K MD anequall ancle DMB. And (by the firft petition) drawe a line from D toB. And for afmuch as ( by the 15. definition of the first ) the line M B 1s equall ynto the line M K put the line: M'D common to the both where fore thefe two lines MK and MD ave , aan to thefe two lines'B Mand MD the one to the other, and the angle KM Ds (by the 23. of the fir/t) equall to the angle BM 'D:VVherefore( bythe &. of the firit )the bale D K is equall fo the of Cuchdes Eleiientes. Fol.88. 4 to the bale DB. ore ape See ea And for afmuch as the lineD K° , : | ts equall puto the tine“D N © But “wnt the | line D K is equalbthelnie DBT herfore\ 1 (by the firft common fentence the line DB . Ml wsequall ynto the line. DN“ VV herefore - 4 that which ts nisherynto OG the least ts 4 eqnall toy Which is more diftants VV hith™ 4 weihaue before proued tobe impofsible: ~~ °° ae i | Or it may thus be demonstrated. ‘Driv ** An other de>. bl) (by the firft petition ) a line from MtoN. heme And for afmuch as ( by the 15. definition’ of pert,leading alfotoan ime the firft)the line. K_M is equall vnto the ee | oe line MN ,and theline M D is common to: them Both. And the bafe KD ise pasintiey guall to the bale D.N (by fuppofitian ) thepe ore( by the 8°. of the firft) the-ane gle KMD isequall to the angle DMN. But the angle KMD 1s equall to the angle BMD. Wherfore the angle BMD is equall to the angle 7 Ar D. the lefJe vnto the greater: which is impofsible. Wherefore from the poynt D can not be drawwen pnto the circumference ABC on eche fide of DG the left, more then two equallright-lines. If therefore without a circle be taken any poynt and from that poynt be drawen into the circle “nto the circumference certaineright lines,of “which let one be drawen by the centre and le? the rest be. drawen at all aduentures the oreateft of thofericht lines which fall inp concauitie or hollows nes of the crreumferenice of the circle is that phich pafseth bythe centre .. And of all the other lines that line *phith ys nigher to the line which pafseth by the cene tre ts greater then that which is more distant. But of thoferight hnes whichend bi the connex€ part of the circumference , that Ime is the lest which is drawen From the poynt tothe dimetient®: and of the other lines that which is nighey to the leaft'is alpayes leffe then that-which is more diftant . And from that poynt can be drawenynto the circimference on ech fide of the left only two equall riche - dines which was required to be proued, © wae eee Thys Propofition is called commonly in old bookes amongeft the barbarous, 4 . a Cauda Paxonis, tharis, the Peacockes taile. s - galled Cauda | | : | Panonis. CURSART S090 ' q A Corollary. Hereby it is manifeft,that the right lines,which bein g drawen from the poynt 4 Coralary. Bb.ij. geucn Confiru€tion. Demonfira- tight. | onl 3 The third Bobke a geuen without the circle, and fall within the circle, are equally diitant, from the jeaft,or from the greateft (which is drawen by the centre).are equall.the. one to the other : but contrarywyfe if they be vnequally diftant, whether they light vpon the concaue or conuexe circumferenve of the circle,they are vnequall. SapT he 8.Theoremes ‘The 9:Propofition. If within acircle be taken x poynt, and from that poynt be drawen vnta the circumference moe then. tro equall right linessthe poynt taken 1s the centre of the circle, | sfx iA ppofethat the circle be ABC, and within it let there-betaken the b | noynt D. And from D let there. be drawen.wnto the circumference Meta) ABC moe then two equallright lines, that is, D.A,D Band D.C, ~_ LbenTfay that the poynt D is the centre of the circle ABC. Draw (by the firft petition) thefe right lines cm AB and BC: and( by the 10. of the : sir tToisie firft )deuide thé intotwo equall partes Z \ Rid ¥ in the poyntesE and F: namely,the line AB in. the poynt, E,.and theline BC i the poynt. F..And draw} lines ED and FD, and ( by the fecond pes tition extend thelines ED andF'D on eche fide to the poyntes K.,G and FL ..And for almuch as the line AE 45 qual dnto the line E'B and the line ED is common to them both, there: A | fore thefe two fides AE and ED are equall nto thefe two fides BE , and ED: and ( by [uppofition } the bale D A is equall to the bafe DB. Wwherfore (by the 8 . of the firjt) the angle AE Dis equall tothe angle BE D. Wherfore eyther of thefe angles AED and BED is-a right angle . Wherefore the line G K deuidethy line AB into two equall partes:and maketh right angles.And or afmuch as ,if in a circle aright line deuidean other right line into two.equall partes in [uch fort that it maketh alfo right angles ,inj.line that. deuideth is the centre of the circle( by the Correllary of the firft of the third )..T herfore ( by the Jame Correllary) in the line G K ts the centre of the circle ABC. And ( by the Jame reajon )may we proue that in $ line H Lis the centre of the circle ABC, and the right lines GK ,and HL hane no other poynt common to them both befides:the poynt-D. Wherefore the poynt Diis the centre of the circle AB G. Tf therefore within a circle be taken a poynt and from that point be drawen pnto th e circumference more thentwoequall right lines, the poynt taken is the centre of the circle: which was required to be proued. ~ ps | | q An of Euclides Elementes. Fo].8 9. q An other demonstration. | «ce 0 Let there beitaken within the circle_A BC the poynt D. And from the poynt. 4p 0. = D let there d&deawen. nto the arcumferencé more then two equall right lines, monfiration namely, 0) A,D B, and DC. Lhen I [ay that-the poynt ‘Dis the centre of the a P= circle. For if not, then if it be pofsible Papi ae let the point E be the centre : and draw Sis a line from Dto E,and extend D Eto : the poyntes F and G. Wherefore the \ line F G ts the diameter of the circle ' ABC. And forafmuchasin FG the | ian'h | diameter of the circle ABC 1s taken | Kee a poynt, namely ‘D, which is not the | : centre of that circle , therefore (by the \ \ / ee of the third ) the line DGis y gree Bans | se . | My / B\ teft, and the line D.C ts greater then fo se te the line D Byand the lmeD B 15 ereae tea then the line D A. But the lines DC,D B,D A, are.alfo equall ( by fuppofee tion ): which ts impofsible. Wherefore the poynt Eis not the centre of the circle ABC. And in like fort maywe prone that noother poynt befides D. W herefore the poynt ‘D is the centre of the circle A BC : which was required to be proued. if > Ep Fa ial The 10. Propofition. Acircle cutteth not a circle in moe pointes then two. y RG Or if it Le pofSible let the circle ABC ent the circle D EF in mo pointes Deseeiire: NG Ee : in Ate. > ae 6 : ; i ss ZL then two that is in BiG 5H,e7 E, And drawe lines fro B to Gand from tion leading LOS B to H. And( by j 10. of the firft deuide either of the lines BG ¢ BAL ibilisie : into two equall partes in y pointes ees so = K and L. And ( by the 11. of thee * | forft) from the poynt K_ raife bp ae dnto_y line BH a perpendicular line K.C, and-likewife from the poynt L ra:fe vp Unto 7 line BG x: Tag = gincrrerdicalar- line L My and CPEs Mite oo OL “Ct Aey ¥ wed ye . a \ mete 3 extend the line C K ‘to the Dopnt <— | A,and LNM to the poyntes X and E, nd for afmuch astrs . aii) Lmur sit. wer & the arcle ABE ihe right line AG denidenethe right le BE) yg ye Into 120 ConA patos’ andmaketh right angles ,therforet by the3 of thethird) QUO 7 Bb.iy. in the ee = —— oe os Ne kore ey ee mm > ess = _ 7 Aa 5. : 3 -“ ns , : “ + (acta , . ee eee ee ae, ~~ oh a derndgerbesclees ~ eneGpas ele eos een aa = ~— a ow 3S ne _ . eae i I OR te | = ss a be = - * s. ae —e2. ~— = = ~ a ee ~— en ee - ——_— ae =e LE — eS | RE eee = = oe BR ee a ep two tameliig —s = y 7 ———— —-- —~ sae ee t--2o: SSS om — St oe 7 ‘ —- Se ee a er a oe NS as nee c . 6 ne ee ‘ - > —s D “ rt - — _ | ry ~ * , * an — " J ’. $ ¥ T: as eben Seees Se wTP a fone + etl cal a OL a & ae ee &4 7 ew" a Mi : - s = C2. Sat Pea : ~ oe * : er ae ae en = Seis - a -— = ee So ——- —— i ae geomrrsataooerres ebm mee - R = . - =p i ——— __ _ -_ ———— - - — s = % - = tat ae = —"s Sat}: = = = = ———— : —* Soa Se — ps - = = ——— — ae — ae " a ee Sree, - = x So ete ~ =. a ; = ~ += — ae a z z —- — - —= pas “s —_. Soe teee 7 Sy . 2). Ke a <= Pe pera ~ a ~ ar one ee a, z Se. a _ ~ ~~ Ae ee a aes a : = = a = ages * “ - ~ c— = wd x - a4 e > - nd eS wy ™ a Yaa x ~ a 5 ecore + = -_ " wars _ aan aae a ~ ~ — _ os — - ron i =. % - ~ . = rs - =—- ~ i 2 siege = = Fe — <+ --- + age a — — tia Se aS Sa — — a — > ys Ss % Se : ~~ . : ~ — = — tla - . r-: ——-— = -: Pans — = = = Ae <2 = =. ne eS S55 : — > oer a ee . - : < — = J narrow = > a ps r= = - nein oe a - -- = = ——— —. = ee eee rs = == tS —— : = rs = : 7 : hE = = SS 2S _ ———— ——S—— —<———SS = z a ee et ee Se a ST x EE 3. — ——= = as - = — a To Dis Ss Ss, — —— > - Sar pe ne na ag — SS ——— A at = we a : = == = — = = = - . an a Pe SDSS Sa ES an = == The 10. Theoreme. The ti .Propofition. Ifo cireles touch theone the other inwardly , their centres : =e Ct Be «being oo of Euchdes Elementes. Fol.go. being geuen: aright line ioyning together vheir centres and _ produced, will fall vpon the touch of the circles. MESS ppofe that thefe two circles ABC, and ADE: do touch the one the | 4 OA. | | . Confira&ion. SY other inthe poynt A. And ( by the firft of the third ) take the centre of ~e. -the circle AB Cand tet the fame be F: and likewife.y centre of the circle ADE and let the fame beG. I’hen I fay that.aright line drawen from Eto G se : : : i ay monftra~ and being produced, will fall ypon the poynt A. For if not, then if st be posible: sion leading let it fallas theline FG.D kL doth, And draw theferightlines AF 7A, Go, to ;.therefore the tne G Ais equall to the lineG'D.And ison & | apt # | And it is proued that the line F Ais equall to the line FC. Wherefore the lines PYfand AG are equall vnto the'lines F CandG D:W herefore the-whole line . POYs'gréater then the lines F Hand AG: But it is alfo leffe (by the 20. of the. pref eswhich ison pofsible.. Wherfore-a'ri eht line drawen from the poynt F to the poyat'G hall paffe by the poynt of the touch namely sby the poynt A . If theres fore two circles touch the one the other outwardly, a right line drawen by their centres ,fhall paffe by the touch : which wwas required to be demonftrated. | * q An other demonftration after Pelitarius. — pherdes’ \. Suppole that the twaicircles <8 C.and D EF. do itouch the one. the, other outa monfiracion wardly in the poynt.ed@: And let & be the centre ofthe circle e# BC: From which ey after'Pein= poyncrproduce by thetouch ofthe circles theline.G4 to the poynt F ofthe circuni- pincleading®’ ference DER x Wihich forafmnchasit. paffeth not bythe centre of the circle DEF aljoto awa (as the adverfary aftirmeth ) draw 4%). nae \ eek = ehott Gntihge the'eitannference pe Boog from the fanie centre G another. rehcline G Ki which if itbepofi. — * ble let pafle by the centre of the cir- dle DEF ,namely, bythe poynt H: sow h suing MPEHE poynt Bysaithecircaference. “\. sintiagyaie —- j 4 oy sof sel + ¢ . + tk ex 4 . < = eG, a ats sare ax C% wo A \otet\ DUAPan the popaeD yerlerthedp2 sai NV sort Oo: avhonwun POheepeynt therof bé ia che ‘point loop N\ wetiw\ G taken without the circle D E F ts ed, Ke And forafmachasfré the peyrit’? 6904 72. > } i328. : 23 CETIQTION i has Line ; 7 2% ri a ° 8 o 5 drawenthe line GX paffing by the centre H, and fro the fame poynt is'drawenalfoan | : : ‘Other ¥ = = Se ewe a ee Sr » —_ of Eachdes Elementes. Fol.ote ther line not paffing by the centre,namely,the line GF . Therefore (by the 8. of thys booke) the outward part GD oftheline GK fhall beleffe then the outward part G_A of thelineG F.Butthe line G is equall to the line GB. Wherfore the line G Dis lefle then theline G 8,namely,thewholeleffe then the part : which isabfurde. SeThe 12.Theoreme. The 13. Propofition, Acircle can not touch another circle in moe poyntes then one, whether they touch within or without. | Or if it be pofSible,let the circle A BC D touch § circle EB FD hel Ry oey-y| fir/t inwardly in moe poyntes then one,that isin D and B.T ake pat: (by the firft of the third the centre of the circle ABCD and let Sy dad the fame bey point G: and hkewife> centre of the circle RBF D, maaan and let the fame be 5 poynt F.Wherefore( by the 11. of the fame) aright line.drawen from the poynt G. to the poynt Hand produced will fall ype on the poyntes'B.and D : Jett fo fall as the line BG HD doth.And for afmuch as the pyynt Gis the centre of the circle ABC D,therefore ( by the 15. definitix on of the firft) the line BG 1s equall to KK the line DG. 1Wherfore the line BGs greater then then the line F1 WD: Where fore the line B EH1ts much greater then the line HD. Agayne fox afmuch as the poynt 1 is the centre of the circle EBFD,therefore (by the fame definis tion ) the line BA is equall to thedine FLD: andit ts proned that it is much greater then it : which is impofsible. A circle therefore can not touch a circle ine wardlyin moe. poyntes then one. Now. Fay that neither outwardly alfo a circle toncheth a circle in moe poyntes then one.For if it be pofsible,let the circle AC K. touch > circle ABCD outwardly in moe poyntes the one, that | isin Aand C: And (by the firft petition ) draw ‘a line from the poynt A to the poynt C. Now for a/much as in 9 circumference of either of the circles ABC D, and AC K_, are taken two poyntes at all aduentures namely,A and C therefore ( by the fecond of the third ) a right line ioyning together thofe poyntes ‘halt fall within both the circles . ‘But it falleth within the circle ABC D <> without the circle ACK :-which is abfurde.. W herefore a circle fhall not touch a circle onte wardly im moe pointes then one and it is proned } neither alfo inwardly. Wheres fore a.circle cannot touch an other circle-mmoe poyntes then one, whether they touch Of circles which touch the one the other inwarde ly. Of circles Which touch _ phe one the other ont warily ‘Lhe third Botke tonch within or without s-which wis required tobe demonftrated. qa i aia 5 Pare ey ~ a scree a - . = ee _ > _ = - -s+-6 ~ ee ee _ a el a RR nl inate ae — = na = - ES = = iin = - sue = ne - ee — ~=- < ~ ae ™ = = = - —s [reo >< _—— =<— =— ——s — ~ . am a —— - ; — = = ~ — qn other demonstration after Pelitarius and Finffates. ) An other de- Suppofe thatthere-be two circles ASG and A DG, which ifit be poffible let rouch monstration the one the otheroutwardly in moe poyntes then one, namely, in A and G.Letthe after Pehta~ centreof the.circle AB,G bethepoynt I, and letthe centre of the circle AD G be the rius @ Fluf- poynt K, And draw aright line from the poynt I tothe poynt K, which (by the 12, fates,of carcles of thys booke ) fhall‘pafle both by the Which tooch poyht A and bysthepeynt.G: whichis od ~ . ‘ok ‘ - she one the not poffible :-for then two right lires other out fhould include a fuperficies,contrary.to | 5 Yardly. the laft common fentence. It may alfobe < dius demonfirated ‘Draw a line trem ‘the. | wy centre, Lto the centre K which fhall patie by One of thetouches, as for example by Sseee38s the poynt\A > And draw thefe right lines Smasitdss Gk and.G I, and foxthall.be made atria |. oe © dngle, whofe two fides\G K and-GI fhallnot be greater’ then the fide IK : whichis contrary to the? 2 def the firlt, ; ; : “Buthow ifit be pofible tet the forefiyd circle AD G touch thecircle ABCinward. Of care fes Jy walinoe poyntes then'one namelysin the pointes A and’G : and lerthe centre of the’ which iooch circle AB G be. the poynt I, as before? and let she one the the centre of the circle ADG be the peynt K, as these alfo before. And extend aline fromthe poynt I eo aral 2 to the poynt K', which fhall fall vypon the touch (bythe 11.0fthys booke). Draw alfo thefe lines KG,and IG. And forafmuch as theline K G is equall to the line K A(by the 15.definition of the firit) adde the line KI conimon to them both. Wherefore the wholeline AI is equallto thetwo lines K Gand KI: but vntotheline AL is equall the line IG (by thedefinition ofa circ'e). Wher- fore in the triangle IK G the fide IG '!s notdeffe i then the‘two fides IK and KG: whichiscon- trary tothe 20. of the firit.” \ Sap be iz. Theorem. : T he 14. Propofition. Ina circle, equall right lines,are equally diftant from the cé= tre. And lines equally distant from thecentre,are equallthe one tothe other. tear “ D 77 j ; ° A Cc The fie part a peg ppofe that there be a circle. a. of this Theo- (CoGi CD; and let there beinit he reme. is Os ie \ “x drawen. thefeequall right lines / AB and CD. Lhent fay that they are | | | Confiruttion. Gy tant from the centre. LT he by. Ava ee ¢ —— the firft of the third )the centre of thecire .\ te ABCD andlet the fame bey poynt BE. And ( by the 12.0f the first) from the point E draw nto the lines ABerCD D sma) —— perpendscnlar — —% ane 1 of Euclides Elementes. Fol.92. perpendicalar lines E F and EG. And (bythe firft petition) draw thefe right lines A Eand CE. Now for afmuch as acetameright line EF drawen by the centre cutteth a certaine other right line AS not drawen by the centre , in {uch fort that it maketh right angles stherefore ( ly the third of the third ) it deuidetl> it into two equall partes. Wherefore tive line A F is equall to the line FB.Wher- fore the liné AB is double to the line AF : and by the fame reafon-alfo the line CD. is double to the line CG. But the lune A Bis equall to the line CD. Wher- fore the line AF 4s alfo.equallta the line CC. And for afmuch as ( by the 15.dez finition of the firft) the line A E is equall-tvthe line EC, therefore the [quare of the line EC is equall to the [quare of the ine AE. But bnto the [quare of the line AE, are equall( by the 4.7. of the fir[t jibe [quares of the lines A F e7 F E: for the angle at the poynt F is aright angle. And ( by» felfe fame )to the {quare of the line EC are ee the {quares of the ines EG and OC: for the angle at the poynt G is aright angle. Wherefore: the fquares of the lines AF and FE are equall to the fauares of the ines CG ana G E: of which the /quare of the line AF is equall to the {quare of the line CG: for the line AF is equall to the line CG . Wherefore ( by the third common — fentence )the [quare remayning, namely, thé [quare of theline F £:, wsequallto the quare remayning namely, to the [quare of the line EG. Wherefore the line E F is equall to.thehine.E. G. But right lines are fayd to be equally diftant from y cens tre’ when perpendicular lines drawen fro B D the centre tothofe lines, are equall (by the 4. definition of the third ). Wherfore the lines A'B and C'D are equally diftant jrom the centre. But now fuppofe that the right lines: AB and CD beequally diftant from the centre ,that ts let the perpendicular line E F be equall to the perpendicular line EG. Then I fay that the line AB isequall tothe line C D. For the fame order of conftruction remayning , we may snltke fort prone that the line AB ts double to. the line AF, and that the line CD-1s double to the line CG. And for afinuch as the line AE 1s equall to the lne.C E., for they are drawen front y cene tre to the circumference, therfore the [quareof the line AE 1s equall toy {quare of the line CE. But (by the 4-7. of the firit) to the [quare of the line AE are equall the fqguares of the lines EF and F A, And (by the felfe fame ) toy [quare of the line C E are equall the fquares of tht lines. EG and GC.Wherfore the fquares of the lines EF and F A are equaltothe fquares of the lines EG and GC,Of which the fquare of the line EG i equall to thefquare of the line EF, for the line EF is equall to the line EG. Wherefore by the third common fene tence ) the {quare remayning namely the [quare. of the line A Fis equall to the [quare of the line CG .Wherefore the line AC is equal dato the line CG. - — the Demonfra- ti0n. Demon fra- $108. The fecond. pare which is she conuerfe of the jirfte a | The third Booke the line’ AB is double to the line AF and the line C.D is double. to the line CG . Wherefore the line: A B is equalt. to theline CD... Wherefore. in a circle equallright tines are equally diftant fromthecentre. dnd lines equally diftant, . from the centre are equall the one toy other :-which was required to be proued. a ee ae < = > ~ » = a = see =— _ — _ ~ = ee Pe a a ———— on GE a —— os ~ eee ee <6 anne ae O38 ee, a ee a en atemes5 ieysnn = ae os eee et _ - _ q An other demonstration for the Sirf part after C ampane. SS ee See win other des ~ Suppofe that there bea circle ef BD C,whofe centre let be the poynté. Anddraw monfiration in ittwoequalllines 4B and CD. “Fhen1 fay that they ate equally diftant from the of the first centre.Draw fron.the centrevnto thelines 4B yy MW oe ain | part ajter and CD, thefe perpendicular lines EFand&G.....\ .' Campane And(by the 2. part of the’3.of this booke Jthe ."Y\" line ‘4 B thal! be equally deuidedin\the-poynt F; } \ and the line CD dhall beequally denided in the\ov; poynt G. And draw thefe rightlines E AEB, © | EC, and ED. Andfor afmuch asin the triangle‘ ¥ \ ef EB thetwo fides ef Bands & are dquall(o% » to, the two fides C D,and CE of thé.triangle , CED, &thebafe E B isequalltothe bafeED, , ~ therefore (by the8. of thefirft) the angle at'the point 4 fhall be equall to the angle at the point C.Andforafmuch as inthe triangle 4E & the... two fides AE and AF ate equail\ro the two fides CE andCG ofthe triangle CBG, andthe <2 8 Os aka Bt angl¢ EA F istequall to the angle C E\G, therefore(by:the'4. ofthe fir) the bafe E F is equall to the bafe‘E G.: which for afmuch as they ar¢ perpendicular lines, therefore the lines 4 B & C D are equally diftant fro the.centre;by the 4 definition of this booke.. \SaThe 14. Theoteme.” The 15. Propofition: ey al —— — ~ —— + = =s _—- + —- Sioa 5 ee =~ eee ew were ee wate ET = a = Rae eo = ~ - > ad SS - 3 Ns me = _ ok = ——e = _ » dna arele, the greateft line ts the diameter, and of all otber lines that line whichis nigher tothe centreis alwayes greater then that lne whichis more diftant. uid Vippofe that there bea eircle MR A BCD and let the diameter =e thereof be the line AD and let the centrethereof be the poynt. Edna: pntothe diameter: AD let theline BC benigher then theline FG. Then T fay that the lite\AD is ‘the oreateft, and ‘the line BC isoreater then $line FG, Draw (bythe 12. of the first) from the? AS Confiruition. centre E to the-lines BC and FG pers. °°\} pendtcular lines B HT and EK. And) > for afmuch as the lineBC is nigher"yne’ to the centre then the line FG stherfore & .@ a a tf s 4 : 1 * ; : It 5 ‘ im Me i tee i" Py fa ; h¢ | oie - : mae lt me. . : | ae "| : ibis ’ Wi} Tha Mie i ined) ; apy tial Ba mS Jae aclh ae ‘ye i oy Thiet WAS i YO TP | ri} 45 welt rae aL ey Lib} i WAR Me ORS ae te rhe ail i yh W = Wei } 4 : : - - } + + Le pt ! jared - : ; ‘42 : i bi (fie af A ae a! f >| - > ; ( i ; rh wey ii) Al i} {! ih . at Abia a ~— —— — Sty > ee + ene ieee * ss - -- = a ee — ~~ 7 . . ; of Euchides Elementes. (by the 4.. definition of the third) the line EK is greater then the line E Hi. And (by the third of the firft ) put vnto the line EH an equallline EL. And (2y the 11. of the firft) from the point L raife op Unto the line EK a perpene dicular line LL.M: and extend the line L.M tothe poynt N.. And( by the first petition ) draw thefe right lines, EM,EN, EF, and EG. And for Pp ds the line E Ff 1s equall tothe line EL , therefore (by the 14. of the third, and Demonftra- by the 4-- definition of the fame ) thelineB Cis equall to the line MN.A igaine "°% or afmuch as the line A E is equall to the line EM, and the line4ED to the line EN, therefore the line A'D is ez guall to thelines ME and EN. But the lines M-Eand-EN are( by thezo. of the first) greater then the line MN. Wherefore the line A Dis greater then the line MN. And for a/much as thefe two lines ME and EN are equall to thefe two lines FE and EG ( by the 15.definition. of the firft ) for they are drawen from the centre to the circumfes rence and the angle ME N is greater then the angle F EG, therefore( by the \ " Guz ~ 8, Fi Sunt” 8 ee E Kis Ven Fol.93. G 24. of the firft) the Lafe MN 1s greater then the bafe FG. But it is proned that the line MN. 1s equallto the ine BC : Wherefore the line BC alfo is greae ter then the line FG . Wherefore the diameter AD is the greateft and the line BC is greater then the line FG’, Wherefore in acircle,the greatest line is the diameter and of all the other lines that line which is nigher toy centre is alwaies greater then that line which is more diftant : which was required to be proued. — q An other demonftration after C, ampane. In the circle e-f BC D,whofe centre let be the poynt E, draw thefelines,~4B,AC, e4 ‘DF G,and 7 K, of which let the line e4 D be the diameter of the circle. Then I fay An other den that the line 4 D isthe greateft of all the lines. And the other lines eche of the oneis fo much greater then ech ofthe other,how much nigher itis vnto the centre. Toyne together the endes of all thefe lines with the centre , by drawing theferight lines ie KLE H,and EF, And ( by the 20. of the firft) the two fides E F and EG ofthe triangle E F G, thall be greater then the third fide FG . And forafmuchasthe faydfides EF & EG areequalltotheline AD (by the definition ofa circle) thercfore theline ef D isgreater then the line FG.. And bythe fame realon itis greater then euery one of the reft of the lines ,if they be put to be bafes of tri- angles ; for that enery two fides drawen fré the centre monfiration after Cam- pane. ! y ; : +} i " it A Ti U i ; it / Ht he ’ | f = os eee ki ih) aR ‘ iia The irik pare wa. of thas Theo- veme. Demon ftra- gion leading to an abfardi~ sie, Fhe third Booke x centre ate equall to the line ef D.. Whichis the Arft part of the Propofition. Agayne,for af- ra much as.the two fides EF and EG of the tri- augle EFG,are equall to the two fides EZ and'\E.K of the triangle Ef K, and the angle FEG is greater thenthe angle HEX therfore (by the 24:0f the firft) the bafe FG is greater thenthe bafe H K.And by thefame reafon may it be proued, thatthe line AC is greater then theline 4B. Andfois manifelt the whole Pro- pofition. 2 SaT he 15. Theoreme. — The 16. Propo ition. Tf from the end of the diameter of a circle be drawenaright line making right angles ; it /hall fall without the circle sand betwene that right line and the circumference can not be drawen an other right line: and the angle of the femicircle is oréater then any acute angle made of right lines but the o- oO : ye eee ther an ele is leffe then any acute angle made of right lines. V ppofe that there beacircle ABC: whofe centre let be the point D, : YK a by.theyir{t petition’), from the: poynt, Fitothepoynt C drawe aright preg, ofit- s = line EC. Uhen Lfay,that C Fisa perpendicular line to D-E.Forif Jor pawl By the t2. of the fr ft) from the poynt, Frorbednie Qk w parnendiculark. A SEs line FG. And for afinuth, asthe angle > ac FGC usar ight angle, ther efor e the angle : Ne * angle PG Cis greater Hoey angle ‘bGG. but bnte the greater angie i subtended : the ereater fid sde( by the 19. of the firft-).. where vefore the line Cis greater then.the. \ ' line F G. Bubthe line FC is equall to theo =e line FB, fi for they are drawen from the... centre to th ech cuty erence: Wherfor the: a BS line\F B alfo is greater then thedine ee namely,the leffe then the greater: whichis BS, a ‘ble Wher sire the line F Gis not ap ver pendi lar line: wnto the line D E.And im like fort'may we proue that “no other line is a per -pendicular line vutoj.line DE befides the.line F C: Where det ethe tne E {ha per penckcular, linea DE. Sf therefore a, right line touch ° om ta Chie ee circle, = ] f ) it Hi nf din oxberde- wsonfiration afier Orot- 6345. Dewonfira- tion leading $o a: impef~ fies ‘stse. — Scivcle,do from 9 centre to F-touch be drawen a right line yp right line fo drewen' fall be aperpendscular line'toy touch line : “which was required to be proued: ap ’ y Another demonstration after Orontins. Suppofe that the circle geuen beABC, which letthe right lyne DE touchinthe point C, And let the centre of the circle be the : point. FiAnd draw aright linefromFtoC,Then. .. (is ll J {ay that the line F Cts perpendicular ynto the lin¢ DE. For if the line F C be nota perpediculer ynto the line D E, then, by the Geahede ofthe X- defi nition of the firft boke,the angles D CF &F CE {hal be vnequall-é& therfore the one is grea- ter then aright angle,and the otherisleife then arightangle. (Fortheangles DC FandFCE are by the 13.0fthe firftequall totwo right an- gles) Letthe angle FCE,ifit be poflible, be grea- terthen aright angle, thatis, letit beanobrufe angle.Wherfore the angle DC F thal be an acute augle, And forafmuch as by fuppofitié the right AS line D Etoucheth the circle ABC, therefore it ; Gis & cutteth not the circle, Wherefore the citcumfe.' rence BC falleth betwene the right linesD C & C F: & therfore the a¢nteand re@iline angle DC F thall be greater then the angle of the femiaircle B CF which is contayned wnder the circumferéce B C & the rightline C F,And fo thall there be gené a re@iline & acute angle sreater then the angle of a femicircle: whichis contrary tothe 16, propo- fitiorrof this booke.Wherfore the angle D C F isnot leflethen a rightangle,In like fort allo may we proue that itis not greater then aright angle.Wherfore itis a rightangle, and therfore alfo the angle F C Bisa right angle, Wherefore the right line F Cis a per- peadicular ynto the right line DE by the 10. definition of the firft: which was required to be proued, @ os SeeThe 17-Theoreme. — Ther 9. Propofition, “\Pf aright lyne doo touche acircle, and rom the point of the \tuuch be rayfed vp unto the touch lyne a perpendtcular lyne, inthat lyne/o rayfed vp is the centre of the circle. . Waid ppofe that the right line D Edo 4 SS we i ae ABCin we point —'C. Ana from C raife bp( by > 11.0 the firft Yonto the line DE we Nc lar line CA, Then I fay,that in the line s . CA is the centre of the circle. For ifnot, | A is then ge be pofsible, let > centre be -withe out the line C A,as in ¥ poynt F. And (by meh petition ) draw aright line from C tof. Andforalmuch as acertaine right tine DE toucheth the circle ABC jand From the centre to the touch isdrawena- "= =. ~* E right line CE, therefore (by the 18. of the third) FC ise perpendicular line to 4 ale ‘4S =, of Eucliles Eleméntes. Fol.96. DE. Wherefore the angle F C Essa right angle... But theaugle ACE is alfa. aright angle: Wherefore the angle FCB ts equall tothe angle AC E,namely, the leffe wntathe greater : whichis impo[Sible Wherefore the poynt F isnot the centre of the arcle ABC. Andinltke fort may Wwe prone , 9 it is no other where but in the line AC. If therefore aright line do touch a circle and from the point of the touch be raifed vp vnto the touch line a perpendicular line , mn that line fo wifed wp ts the centre of the circle: phich-was required to be proued, | SeT he 13. Lhearene, .. The20. Propofition. In acircle an angle Jet at the céntre,ts double to an angle fet at the circumference , fo that both the angles haue to thew bafe one and the fame circumference, = +5 Vppofe that there be a circle ABC, and at the centre thereof namely, C5 the poynt E,let 9 angle B EC be fet ex at the circumference let there | be fet the angle B.A C, and let them both haue one and the [ane bafe, = namely,the circumference BC. Then I fay, that the ange BEC is double to the angle B AC. Draw y right A | . line A E,and ( by the fecond petition Jexe ; tend it tothepoynt F. Now for afmuch as the line AE is equall tothe line EB, forthey are drawen from the centre pnto the circumference, theangle EAB isee ¢ quall to the angle EB A( by the 5 .of the firft ). Wherefore the angles E A Band EBA are double to-theangle E AB. But ( by the 32. of the fame ) the angle BEF is equall to the angles E AB and EBA: Wherefore the angle BEF 1s | double to the angle E AB. And bythe ) fame reafon the angle F EC is double to the angle E AC.W berefore the Tebole angle BEC is double tothe whole angle BAC. | | Avaine fuppofe that there be fet an other angle at the circumference ,and let the fame be BD C. And{ by the firft petition draw a line from D to E. And (by the fecond petition extend the line DE nto the poynt G. And in lke fort may “we proue,that the angle G EC is double tothe angle ED C- Of which the an- gle'G E B is double to the angle E DB. wherfore the angle remayning ih 5 a is dowble to the angle remayning BDC. Wherfore in acircle an angle fet at the centre is double to an angle fet at the cireumferencefo that both the angles have to their bafe one and the fame circumference ; which was required to be demon Strated. . = LAS (a Bo) Dp. B BE Conti. ZT wo cafes in thys fro0 pofition the one when the angle fet at the circumfes FENCE INCH deth the cen ber. Demonftra- * $80%« The other whe she fame angle fet at the circumfe- rence incla~ deth not the center. — Pee ~ —— Se arn " . = a. _ = ~ Stee — SET Eo = — — = ——— =. —. = — ——- Se = SSS = — EE ee ena ene a ee = pine = = : = | - we ee ee er Ty Bs Latrae ee a ete Sa BAD is equall tothe ancle BAD. Wherefore ina circle the angles-ohich. ie con/ifteimoneand the felfe fame fegment are equall the one to the other: “which Wasxequired to be proued. ee ES ge es a b Bag “A ; \ } , | ; | : : , + a0 Ibi propofition are three cafes,For the angles confifting in one and thefelffame fegment, the fegincnt may either be greatet théa fenjicircle, or lefle thena {emicircle, or elsittafemicircle,Eorthe firkt cafe thedénionftration before put ferueth. nd But now fuppofe that the angles BLTD and BE D doconfift in the {e@id B_AD, which fol fay thatthe angeles’ Bi4 D and B ED are’e- vall.For draw a right linefrom .4 to Ei Andlet greater then a {emicircle, And therfore by the akeinut, namely, the anglese4 B Laud EDA are equall the one to the other. Andforafmuch — the inward andoppo- ‘33in the trianole er BG MWatec angie 6G D, and by thefame.reafon the two-angles E DG and G ED of the triangle BGD.Wherfore the two angles 4B Gand G A a of Euclides Elementes. Fol.97 by the firft commé6 fentence.From which if there be taken equall angles, namely, 4BG, and ED G, the angleremainyng B 4G hall be equall to the angle remayning DZ G,that is,theangle 8 A Dto the angle D E B(by the third common fen- tence) which was required to be proued, \ The felfefameconftru@ion and demonftrati- 5 cape daca a a | D on willalfo.ferue,ifthe angles werefet ina femi- 2 | circle'asitis playne to fee,in the figure here fet. / LaT he 20.T heoreme. The 22. Propofition. Hfvithin a circle be defcribed a figure of fower fides, the an- gles therofwhich are oppojite the one to the other, are equall totwo right angles. ConfruGion. eal two right angles: therfore y three angles of the triangle ABC, namely, Demenfira- AB,ABC, and BC A,are equall to le AC Bisequall to the angle ADB, Foy they confiftin one and the fame feg: ment A DCB . Wherefore the whole angle AD Cis equalltoy angles BAC and ACB: put the angle ABC come monto them both. Wherefore the angles ABC,BAC,and AC B,are equall ta D A! the angles ABC and ADC. But the. | angles ABC, BAC, and ACB, are equall to two right angles. wherefore the angles ABC and ADC are equall to two right angles . And in like fort alfo may we proue,that the angles BAD and DCB are. equall.totworight angles. — = a ae megs ete. ee ee The third Boke. . oa angles. If therefore within a circle be defcribed a figure of fower fides, the ané gles thereof which are oppofite the one to.5 other, are equall to two right angles: which was required to be proued. $7 be 21. Theoreme. The 23. Propofition. Upon one and the felfe Jame right line can not be defcribed two like and vnequall fegmentes of circles falling both on oné and the felfe Jame fide of the line. Detionitra- | Or if it be pofsible,let there be defcribed ypon the right line AB sion leading SA! two like ex bnequall fettions of circles namely, ACBéz ADB, fibilitien f Qi Nt falling both on one and the Selfe fame fide of the lne AB. And ie & aie (by the firft petition) drawe the right line ACD, and( by the LP Say £3 yy te , ' pean —=S" third petition ) drawe right lines from Cto 8, and from D to B. - | fegmetes of circles are they which haue ; | equall angles ( by the 10. definition of thethird )-1i berefore the angleACB | | | isequallto. the angle AD B, namely, | ; the outward angle of ¥ triangle CD B. _ tothe ward angle : which( by the 16. oe ane enone and the felf fame right line can not be defcribed two like ex vnequall [ege Ha ediicion Here Campane addeth that vpon oneand the felfe fame right lyne cannot be of Cam pane acicribed two like and vnequall fections neither on one and the felfe fame fide of demoanfirated fite fide,Pelitarius thus demonftrateth. Let the fection AB C befet vppon the lyne’A C,and vpon the other fide let be fer the fection AD Cyppon the felfe fame lyneAC, ve and let the fetion ADC belyke vnto the fe@i-* on ABC, Then I fay that the feGions ABC andA D C being thus fet are not vnequal.For ifit be pof- fible lerthe fection AD C be the greater, And ‘de- hide theline A C into two equal partes in the point E.And draw the right lyne BE D deuiding thelyne A Crightangled wife,And draw thefe rightlynes A Demonfirs 8%, CB,ADandCD, And forafmuchas the le@ion tien leading &D-GCisgreater then thefeQionA B C;the perpen toan impofG- dicularlynealfo ED thall be greater then the per. bslttie. pendicularlyne EB : as is before declared in the eude-of the definitions of this third booke, Wher- ‘enw, aby . Ly ¢ of Euchidet Elemente. Fol. fore from thelyne E Dcutofalyneequalltothelyne E Bawhichlet beEF. And draw theferightlynes A F and C F. Now then(by the.4,ofthe firftthe triangle A EB thall be equall to the triangle A E F,and the angle EB A thall be equall to the angle BF A, And “by the fame reafon the anele EB C fhall'be equall tothe angle EFC. Wherttore the whole anglé A BC is equall to the whole angle A F C.But by the 21.ofthefirit,the ane gle A F Cis greater then the angle AD C.Wherfore alfo the angle A B Cis greater then the angle A DC. Wherefore by-the definition the fections ABC and AD Care not lyke,whichis contrary to the fuppofition, Wherefore they are not lyke and ynequall: whichwas.requited to be proued. Ss The 22.Lheorme. The 24«Propofition Like fegmentes of circles deferiled uppon equail right lines, are equal the one.to the other. N AN ppofe that pon thefe equall right lines AB and C D be deferibed Py yathe/e like fegmentes of circles namely, AEB and C FD. Then l fay, that the feoment AEB is equall to the fegment CF D_. For putting the fegment AEB vpon the fegment C F D,and the poynt Avpon_y poynt C, and the right line AB ‘pon the right lneC D, the poynt B alfo fhall fall vpon the poynt D, fory line A Bis equall to the line CD. And the right line A B exe attly agreing with the right line CDy | = . Pe segment alfo AEB phall exactly agree with the fegment Peale CFD , For if the | : tion leading right line AB do. \ se : - Piteas exactly agree “with * theright line CD, andthe fegment AEB do not exaltly agree with the fegment C FD, but dif: fereth as the fegment C G D doth: Now( by the 13. of the third) a circle cutteth not a circle in more pointes then two,but the circle CG Dcutteth y carcleC F D in more pointes thé two ,that is in the points C,G,and Drwhich 1s( by the fame) impofsible.. Wherefore the right line AB exaétly agreing with the right line CD, the fegment AEB fhall not but exactly agree with the fegment C FD: Wherefore it exattly agreeth with it, and is equall ynto it.. Wherefore like feg mentes.of circles defcribed vpon equall right lines are equall the one to the other: twhich-was required to be proued. ) Se eee ——— See - es *F = _ 5 _ Oat eee .- ———" ~ This Propofition may alfo be demonftrated by the former propofition. For ifthe fe- 4, she den ions. A EBandC ED beingtike and fet ypon equall right linese 4B and CD, thould | be vnequall thenthe oie beyng put vpon the other,the great: rthall exceede the Jefle: but the lined Bis one line with the line CD; {9 that therby thal follow the contrary of the former Propofition, : monfiration. oe ee eee Pelitarins ~ He SSE Se an ae = a - ~ oo Sas ea eee sn other de- morftratio af éer Pelitaritse Confiradtion. Three cafes in thes Propofi- g70n . The pitt cafe. Demonstra- Sion. Lhe third Booke» -Pelitarins demonStrateth this Propo(ition an other way. Suppofethat there be two right lines eB & C D which let be equall : and ypon thé fet there be fet thefe like {eQions 4B K,and CD E. Then I{ay that the {aid {e@ions are equall.For if not,then let C ED be the greater {feGion. And deuide the two linese 4B and C Dinto two equail partes,the line 4 Bin the pointe F, and the line CD inthe EK pointG.And ere&two perpendicular lines £ Kand G £,And draw thefe right lines .4 K& KB; EC,ED, And forafmuchas the fection C E Disthe greater, therefore the ; perpendicular line G Eis greater then the perpendicular F K: From the lyneG E cut of aline equall to the line FX,which let be. 4 F oa C G G Hand draw thefe right lines C Hand H D.And fora{much asin the triangle ef K F the two fides 4 F and F Kare equall to the two fidesCG & GH ofthe triangle C H- Gand the angles at the pointes F and G are equal (for that they are right angles) there fore( by the 4.0f the firft) the bafe 4 Kis equall to the bafe C H,and the angle 4 KF to theangle C 4’ G,And by the fame reafon the angle B K Fis equallto the angle DH G, Whertfore the whole angle 4 K Bis équall tothe whole angleC # D.But the angle C H Dis greater then the angle CE D by the 21. 0f the firft, Wherfore alfo the angle 4 KD is greater then theangle CE D,Wherfore the fections are not lyke, which is contrary to the fuppofition, Lhe3.Probleme. The2s. Propofition. eA Segment of a circle beyng geuen to defcribe the whole sin. cle of the fame feoment. MASS ppofe that } ferment geuen be 8 ec 4B a t is ania to deferibe MS the whole circle of the fame fege ment ABC. Deuide( by the 10. of the firft the line A C into two equall partes in the poynt 'D.. And (by the 11. of the fame) from the poynt D raife Dp bnto the line AC a perpendicular lineB D. And ( by the firSt petition ) draw a right tine from A to B. Now then the angle .4'BD being compared to$ an ile BAD, as esther greater then tt or equall pnto it,or lefJe then it. | FirSt let it be greater. And (by the 23. of the [ame Yopon the right line BA, and Ynto the poynt init A, make nto the angle ABD an equall angle'B A E. And by the fecond petition extend the line B D vnto the pont E. And by the firft petition ) draw a line from Eto C. Now for afmuch as the angle AB E is ee quall tothe angle B AE, therefore ( by the 6 . of the firft the right line E Bis equall to the right line AE. And for afmuch as the line AD is equall toy line C,and the line D E is common to them both; therefore thefe two lines A | si an £ 2? E af Euchdes Elementes. Fol.o9. andlD E, aree gua tot hefe tivo tines CD and DE theone tothe other . Aad each AD EG (by the a: petition )equall to the angle C DE, for either of Ahem isa right angle, Wherefore (by the 4.. of the firft.) the bafe AE is equall to the bafe. CE . But it is proned that the line AE is equall to'y line B EW hers fore the line BE al 0us equall tothe line C EW herefare thefe three lines A FE A E Band EC are equall the one to the other. Wherefore making the centre E, and the Space either-AE, or EB; or EC, defcribe ( by the third petition }.a cir cle,and it [hall paffe by the poyates’ A, B,C. Wherefore there is defcribed the hole circle ofthe fegment cenen. And itis manifest, that the fegment.ABE is leffe then-a fenucircle, for thecentre-E falleth without it. | _. Lhe like demonftration ayo. will patsy oe ferue if the angle A‘BD. be. equal ta : the angle PAD? For thetine AD being equal to either of thefe lines, BD, and. DG; there are three lines s:. DAD Band DC equall theone'to’ theothey. Sathat the paint D frallke, the céntreof-thecrcle being complete; A D dnd ABE pha bea femicircle: ows But if the angle ABD be leffe-then-the angle BAD, then bythe: 2300f the fin) vpon the right line "BA, and ynto the point in it A-nakeynto the angled 'BD.an equallangle-withing segment ABC. anid fothecentréof the\crcle: fhaltfailin$ line D: B " did tt ‘fhall be the pointe E- and the Jegment ABE .. Jeali begreaterthenafemicircle JV Lerefore a fegment et D C = bein g venen there 1s de/ cribed the hole circle of the fame Jegment sswhich “was required to be done: 2h 4 ere | ¢ A Corollary: Flereby tt is ma nifeft that ina semicircle the angle B eA D is equal tothe angle DB eAibut ina fection lefSe then a femicir- Ce, itt leffe :snafettion greater then afemicircle,stis greater. There is aloarether generallwayrto:finde out the | forefaidcentre,\which will ferne indifferently for any 2: 201! 7: fection whatfoeher: And that is thus. Bakean the cir ner cumMerence geuen or {eGion.4 ACthree pointes atall' auentures which let be 4,B,C. Arid-draw-thefe lines 24 Uap. ey re Bahd BC(by-the firftpeticion) And(by-the rolofthed: + six Nu Vio ss: % of fir(t) deuideinto two equall partes either of thefayder}n eA 3 bo NSS soni! Wid . j 1 : a . > ; * a ae, Ne lines,the linee#\Bin the point Py &the line # Cinthe dye Tae JS Sho bassin! pomcé, And (bythe 11. of the Spt) ffonvthe pointes \Acii od: 5: 00 Neasastos Dd. ja Dand The fecend Tbr Cafe, The third tafe. ote" An othep wore ready ‘Way. 7 + . ~ Y 4 . ¥ 2 . Phe third Booke PandLrayfevpyncothelines..7 B and BC perpendicular lynes DF and LF..Now Demonftra- forafmuchas either of thefe angles 8 DF,andBE Fis aright angle,a right line produ- Gion. ced from the point D tothe point Z, thal deuide cither of the fatd angles: and foraf- Hiuch asit falleth vppon therightlines ‘D F and & F,rtthalt make the inward angles Om ondand-the felfe fame fide, namely, the angles DEF . and E_D Fleflethen two right angles. Wherefore (by ea SB the fitt peticjon ths liies DF and E F being produced wo: . fkallconcutre, betthem tondurrein thepomtF. And = / f fora(much asacertaine \nightlineD F deuideth acer- Dhol suman BA taine night lyne eft & into, tivo equall partes and per- 4 \/ NGA ss pendictilarly, therfore (by the corollary ofthe firlt of pe: 4 this bodke}in theline Dv is the.centre of the circle, & | Sor by the fame reafon the centte-of the felfe fame circle A fhalbe in the rightline E F. Wherfore the centre of the | 3 circle wherof 4B C is a {eGtion,is in the point F,which is comm to either of the lines 1 Fand E F.Wherforea {ection ofa circle being gené,namely,the fedion AB C,there is de{cribed the circleof the fame feGion: whithwas required to be done: “arg. a And by this laft generall way, if there be geucn three pointes, det howfoeuer, fo that fn additions they be not all three in one right line,aman niay deforibe acircle which {hall pafle by all the faid three pointes. For asin the exampic before. put, if pow fuppofe onely the 3. pointes 4,B,C,to be geuen and notthe circumference 4 B Cro be dtawen,yetfollowe ing the felfe fame order you did before, thatis, draw-a right line frome4 toBandan other from Bto C and denide the faid right Imes mto two equall parts,in the points D and E,and ere& the perpendicular lines D Fand £#cutting the onethe other im the point F,and draw aright line from F to B:and making the centre.the point F, and the foace F B defcribea circle,and it thall paffe by the pointes 4 & C: which may be pro ned by drawing right lines Roni to Fand from FtoC. For forafmuch as the two fides A°D and DFof the triangh: 4D Faré equali to the two fides®Diand D'F of the trizhgle B/D P( for by fuppofition the line e4:D.isequall to the line DZ, and thelyne D Fis cémmon to them both)and the angle4‘D Fis equallto the angle BDF. (for they ave both right angles )thertore(by the asofthe'firft rhe bafe 7 Fis equall-to the bale f F.Aad by the fame reafon theline FCisequall to the line FB. Wherefore thefe three lines F.4,F.B and/F Careequall the one to the other. Wherefore:makyng the cenrrethe point F and the {pace F 2, it fhalhalfo pafle by the pointes 4 andC .. Which was. required to be done, Thisipropofitionis very neceflary formany things as you fal. ekter warddéee, * iKDh acl | sighir ade VELpAgg BARI Campane putteth an otherway, how to defcribe the An othereon- whole circle of a {eid geuen. Suppofe that the fection fiructionand be AB Jtisrequiredto defcribe the whole circle of the demonfiration fame fection, Draw in the fection twolines at.alfaduen- of this Prope~ tures. 4C and B D:which deuide into two equall parts fition after, .ACinthe point E,and BD in the pointe F.Then from Campane the Wwe, pdintes of the denifions draw. within thefeci- * on,two perpendicularlines EG and F H which let cutre tha obethe ocherin the pottit And the centre of the circle {hall be in eithor of: the faid_ perpendicular lines One os : bythe toroljaryofthe AMt ofthis booke: Whertore the point RKis'the centre of the cir i Ht ie ial: tt ti Hb - Hd. if ib Mie ee + , > ' —— — ee cle:which was required to be done. elise’. _ BUtifthelines EG & F 4 do ndt cut theone theo-+ ther, butmake one right line as dothG Ain thefecdd figure: which happenetitwhen thetwo lines. 24 Cand B D are equidiftant. Then the lineG#, being applyed to cither part of the circumference geen; thalt patie: . by the centre of thé circle, by thefelfefame Corollary. a. For thelines EG-and F A-cannotbe equidiftant, For therrone and the felffanie circumference fhould haue = Nog two Centres. Wherfore theline ZG beingdeamiedin« A: & as Se ta. of Euctides Elementes. Fol.:eo, to two equall partes in the point X,the faid point K thall be'the centre of the fection. Pelitarius here addeth a briefe way how to finde out the centre of a circle,which is commonly vied of Artificers. Suppofe that the circumference be ABC D, whofe éentreit is required to finde out, Take a point in the circumference geucn which let beA, vppon which defcribea circle with what openyng of the compatie you will,which letbe EFG. Then-take an other pointin the circumfe rence geuen which let be B, vponwhich defcribe an other circle with the fame opening of the compaffe thatthe Cire | : cle BF G was deferibed, andJetthe fame be E H G, which let cat the circle E F Gain the two pointes E andG. (Ihave not here drawen the whole circles, but onely thofe partes of them which cutthe one the other for auoyding of confufion ) And drawe from thofe cehtres thefe right linesAE, BE, AG, andBG,which foure lines thall be equall, by reafon they arefemidiameters of equall circles, And draw ‘aright line from Ato B; and fo fhall there be made two Ifofceles trianelés AE BsandAGB vato whom theline A Bis a common bafe. Now then deuide the line A B into two equal partes in the point K which muit nedes fall betwene the two circumference E F Gand EHG, otherwife the part fhould be greater then his whole, Drawe aline from EtoK and pro. duce itto the pointG. Now you fee thatthere are two Hofceles triangles deuided into toute equall trianglesE AK, EBK, GA KandGBK’, For the ) two fides AE and AK of the triangle ALK are equalito thetwo fides BE andBK of the trianele B E K,and the bafe E.K'ls common to them both, Wherefore thetwo ane glesat the point K of the two triangles AE K and B EKare by the 8.of the firft equalls and therfore are right angles.A nd by the fame reafon theiother anglesatthe poynte K “are right angles, Wherfore EG is oneright lyne by the 14) of the firft; Which foraf= Imuch asitdeuideth theline A B perpendicularly, therefore it pafleth by the center by by the corollary of the firft of this booke. And fo if you take two other poyntes ,name« Cand_D inthe circumference geuen,and vpon'thé | | ‘defcribe rwo circles cuttyng the one the other in the pointes Land M,and by the faid poyntes pro- duceatight line, it thall cutte the lyne EG beyng ptoducedin the pointe N,. whith fhall be the cene tre of the circle by the fame Corollary of-the fir of this booke,if you imagine the right line C D to be drawen and to be denided perpendicularly by thelyne LM, whichit muftneedesbe as we haue before proved. And here note thatto do this me. chanically not regardyne demonftration , you neede onely to marke the poyntes where the cir- cles cut the one the other,namely, thepoyntes E, G,and LM, and by thefe poyntesto produce the linesE G and. M till they cut the one the other, and where they cut the onethe other, thereis the © “centre of the circle,as you fee herein the feconde figure... = gr oe A ready way to finde out the center of 4 circle commode by ved a~ monzs arti- ficers. alae accents Sa acelin Oa SSS es ———————— ae —— mm — nee” - Ss a a a Se Sass. SS SS = oo) So Seniaticet pSB oe ae = 7 = ~ ‘ . - * — . —_— 4 ‘ . " - * - ‘ re Y ¥ » - iu ‘ r Wear Ean 7 - § a 3 as ;, : ek ee eee en a sm , Se eee eee z 7 “ Sts = - . ce . ‘ } = a hes i a ta ence - =2 a < S male tak ee LD a — sharing A tiehad ; - ei Pe i tA F ee — = ne ee Bo = : ~, — ~ —_— : : — == — nw yi —— aes ‘= > - - = —- ~ ~ ~ — ~*})).4. a Per v 5 ~ — ~- ~ 7 —_- a « + —— - —— ~ — ™ ¢ ; Ti ; - my pal ms > Be —~- 3 -- ; ~ - = Tes Oo = = =. : —_— = : = ; —SS ; , 7 es ake ae ¢ ——. 7 see - _ Ste > oe eS R= oe + . ne a Be ee ee yn ree 3 sine apes arn’ = a earn = — _— e+ —- ————— - . = a inte > a a a a Coy SSS Re Ak sa - = A —- ant ee eS EP ~ a =% am bh Lai + < e a . - ee ~ — - -- ———. ~— ——— — bs t - ne on Ss _~ — ta ES — — ee ee = _ - = = ae = — — <= = — — rah 6 = os 2 2 33 a i “ se - Ee = = ~ - ~ - — amas © > pe ee a eee ae Se we a — -- = = — ==-5 eas ee = = ae eee >= —— > — ~ : - = ‘ ——s — 7 v= TES a = _—~ ——— = = —— — = : = === —- == —— _ ot ov ees ———e a - ———* — rs P ————— ee saan ww ee ———— —— ~ —— = — a - —___—_—— = ee a - = = es —3 - :- = ————— = = = ——_— = _ = = anet Suenseeeenie 7 z= — - = : = — - -- —————— — ms + ’ ~~“ -— 6S-- i 9 a ag Be = ~ “2% - > 22 ~— te Sere ee _ — = . = — == . — = . ee ° - a = = — = = = ~ = re . * -_ > Ss > : “+. _ T a Ar eR tS - - = ae aa - — oe _— + > m= oe er — 2 es jan: ; = S : — a = — —— = — = = = — = —= -4 5 Poa Ss to Pes 7 2 _ = = — ee ee Se ee ee See eS — > . ~ a - : == — ——— - —_—— — . —— — —————— — —— = - - ne —=s> po a be a Se : = 2 ~~ = . an 2 ae —————— . =—— == —== = =— ~ —— : : = = - 2 2 = : Sa = : = = es nema mealies = ~ a en ee ee ee arr oo oa oS we ae = et et — es : ————— = —— - ; = =o so — —— =< =~ Mase = a —_ eS ——s SS ee eee ou — pars a ——— ——" ee ———— == ——- ——— = = —— — SS = 4 ni as + eee aa +--+ ~ ea ae 6 ems en am a ee ea a aa a a ce : y . rs Con firnttion. Demenffra- wos > ‘7 eT hethird Booke*: — $997 be 23: Theoreme.. Lhe 26. Propofition.. Equal angles in equall circles confiftin equal circteferences, whether the angles bedramen,from the centres,or from the circumferences. {U4 ppofe that thefe circles ABC and DE F,be equall. And from their ie centres namely the pointes G and H, let there be arawen. thefe equal and ANS WANS OR angles BG Cand E HF: and likewi/e from their circumferences ved equall angles BAC and ED F. Then 1 fay, that the circumference B RC is equall to the circumference ELF . Draw( by the firft petition) right dines from B to C, and from E to F. And for a/much as the circles A BC and DEF are equall,the right lines alfo drawen from their centres to their circum ferences J are (by the firft definition of the third.) equall the one to the other. Wherefore thefe two lines BG and | | G C,areequall to thefe two lines E Hand HF. And the angle at the poynt G 1s equall to the angle at the point FA: 1Vherfore ( by = 4. of the firft the bafe is +h iW bale EF. And a -\ “for-afmuch as the angle at K TE the poynt A ts equallto the angle at the point D, therefore the segment BAC as like tothe fegment FE D F, And they are defcribed vpon equall right nes BC and EF. But like feomentes of circles deferibed vpon equall right lines ave ( by the 24. of the third ) equall the one to the other. Wherefore the fegment BAC is equall to the fegment EDF. And the whole circle. A BC is equall toy whole circle DEF. wherefore (by the third common fentence ) the circumference ree ~ mayning BK C is equall to the circumference remayning E L F . Wherefore equal angles in equall ciréles confift im equall cireumferences -whether the angles be drawen from the centres or from the circumferences: which was required to be demonstrated. ae ik ea | SS 4pT he 24.7 heoreme.. The. 27, Propofition. Ineguall circles the angles which confifi in equall circumfe- rences,are equall the oneto the other , whether the angles be drawen from the centres or from the circumferences 4, p68? Sah Suppofe . (At of Euclides Elementes. Fol.tot. =75 Vppofe y.thefe circles ABC,and D E F, be equall. And dpon thefe Xequall circumferences of the fame circles namely, vpon BC and E F, éj let there-confift thefe angles BG Cand E HE drawenfrom the cen | tres and alfo thefe angles BAC and EDF drawen from the car: cumferences . Then Lfay, that the angle BG C is equall to the angle EB F, and the angle B AC-to the angle ED F.1f the angle BG C be equall tothe ans le EHF then itis manifest that the angle BAC is equall to », angle ED F (by the 20. 0f the third). Bue if the angle BG C benot equall toy angle b IB, thenis the one of them greater then the other . Let theangle BO C.beigreater And ¢ by the | 24 of i the firft) Dpon the right line BG, and ynto the point genen init G;' make bnto the ancl BAF * an equall ans | gle BOK.But © oS! S (by the 26. of prnerk »y third )equall . angles inequall circles confift ype equall circumferences whether they be drawen fromthe-centres or from the circumferences . Wherefore the circumference B K is equall tothe circumference EF, But the circumference E F is equall to the ‘circumference BC: Wherefore the circumference B.K_alfo is equall. to the cir cumference BC , the leffe to the greater : whichis impofsible . Wherfore the ans gle BGC is not ymequall to the angle E HF: Wherefore it is equall. And( by the 20. of the third the angle at the point A is the halfe of the angle BG C:and (by the Jame) the angle at the point D is the halfe of the angle k HF. wheres forethe angle at the point A 1s equall to the angle at the point D. Wherefore in equall circles the angles which confift in equall circumferences are equall the one to the other, whether the angles be drawen from the centres or from the circumfee __ rences : Which was required to be proued. | . SaThe 25. Theoreme,. The 28. Propofition. Jnvequall circles, equall right lines do cut away equall cir- cumferences,the greater equall to the greater and the lefSe e- quall to the lefSe. Da.iy. Suppofe Demenfirae tion leading — to an impof= fbilitie. Conftrattion. Demonstra- sion. tres are equall . here» fF: Yagghk fore théfetwolinesBK © | \ The conner/e of the former 4S Propofition. Con/tru ction Demonstra- $107, at Fhe third Booke ° Pane AP ppofe that thefe circles A BC, andD EF beeguall . Andin them let Bay . there be drawen thele equall right lines BCand EF, -whichdet cut away =e thefe circumferences BA Cand D EF being thegreater , 7 alfo thefe crrcamfperences BGC and E HF being the leffe » Uhen £ fay;thatthe greater cipcumference BAC isequall to the greatercircumference EDF: and the leffe circumference BG -C is-equall to the leffecirumferenceE HF . Take (ay.the Jif of the third) the centres of the circles,and let the fame bethe pointes.K and L.. And draw thefe rioht lines, KB, KC, L E,,and- LE. And for afmuchas the circlésave equal ther fore(by the firfrdefinition of the third ) the lines swbsch are drawen fro the cena and JK _C, are ,equall to © 4 | thefetwolmes E Land 2 DOE OSG ; LF. And (py fuppofitis TT on )the baf $ BC seal is ee oS to the bafe EF. Where- fore( by the 8 «of y firft) | the angle-B K C 1s equall to the angle ELF. But (by the 26.0f the third) eguall angles drawen from the centres ,confifi vpon equall circumferences.W hers fore the circumference BG Cis equall to the urcumference E HF >and} whole tircle: ABC isequalltothewhole arcle DEF. herefore the ctreumference remapnine BA C,is( Ly the third common fmtence) equall tothe circumference Yenianiigeh DP. wherefore m circles equall rightiimes do. cut.away.equall trrcimferences, the greater equall to the greater andthe leffe equall to the f €: “which Was required tobe proned. } ) én third) the centres of the circles., and let them. be thepointes Kiand Ls, and draw thefe right ines KB, K C, LE, F..And for ofmuch as the circumference BGC is equill to the circumference E-Et F theynsle BKC is equall to$- angle E LF (by the 27.0f the third). And for afmuch as the circles ABC and DE Fare equall the one to the other therefore (by the fir] definition of the third )the lines BEE ¢ Se BS which Fol.to% of Enclides Elemsentes. ibibo Soko drame from ogre = the. centtes > pk are “equall. | W hevefore ‘thefed.lines Pree | BK and oN | KC > are eC f qual 10. ak: Pi aks | the[ey.fings oS y LE and gee Ba LF er they i unl oid ‘Sy The 4. Problene, The 30. Propofition, Lodenidea circunifercice genen Into two equall partes. =A V ppofe that the circumference geuen be ADB. I tis required todes S| ude the circumference AD B into two equall partes. Draw aright | line from Ato B~ Ant (by thex0. of the first) deuide the line AB ‘into two equall partes ia the point C. And ( by the 11. of the firit from the noint C rayfevp rnto A Ba a tag line CD. And draw thefe right lines AD and D B.And forafmach as the line A Cis equall to the line C Bee the line C Discommon to them bith stheres : fore thefe two lines AC and CD aye equall tothe/e twa lines B.C and C D. snd by the 4.peticton,) the angle ACD is equal to the acl BCD, for either of them is 4 right right angle. WV berfore (by the 4. of the first) the bafe AD js equall to the bafeD B. ‘But equall right lines do cut away equill circums is “Ferences the sveater oo tothe sreater, <7 thetefse cquall to the lefse( by thez8\ of the third) And either of thefe circume Ferences AD atd'D Bis lefse thar a W Aordiathen sh the circumference A Dis equal to the circumference D'B.Wherfore the circumference genen is denis ded nto two equall partes: Which was required to be done. oa | | tT he 29.7 beoreme. : T he 31. Propofition. ey na tirele an angle madein the femicircle 8 a right angle: = —— Dad iti. but Confiructione Demonfire- ROR. The jirS part of thes Theoe PEM e Sevond part. T bird part. “Lhe third Boole’ but an angle made in the fegment greater then the femiciréle wsleffe then aright angle,andan anslemade in the fegment lefJe then the femicircle, is greater iben aright angle. And moreouer the angle of the greater fezment 1s greater.then a right angle: and the angle of the lefe Jegmentss lefSe'then a right angle. : NO eee 5A p pofe that the circle be ABC D,and bt the dimetient of the circle be Ray the right line BC, and the cétre therof tve point E. And take in thee =" micircle a point at all auentures and let tre fame be D. And draw thefe right. lines BA,A CA D5andD C. Then I “ere ee faythat the angle m the femicircle BAC, 2 namely, theangle BAC 15 a right angle. And the angle AB C-which is in the fegment AB C being greater. then-the femicircle,.is lefse then aright angle. And the angle AD C whith isin the fegnent AD C being lefse then, the femicireleas greater théa right ane cle. Draw aline from the point A to the point # and extend the line B.A nto the point F. And joralmuch as the line BE 1s equall to the line E-A, ( for they.are drawen jt the centre to the RAMEE) therfore the ans gle EABis equalltothe angle EB A (by the 5.of the first). Againe forafmuch as the a line A Eisequall tothe line EC, the angle ee AC Ets (by the fame )equall to the angleC A E.wherfore the whole angle B Cis equall tathefe two angles ABCand AC B.But the angle FAC. which is an outward angle of the triangle ABC is (by th: 22. of the first) equall to-the two angles A DCex AC Bwherfore the angleB A Cis equall to the angle Fe _ ACWherfore either of them is aright ingle. Woerfore the angle BAC which isin the femicircleB ACis aright angle. and forafmuch as (by the v7. of the firft) the tivo an gles of the triangle A BC namely,ABC and'B AC are le[Se then tworight angles, and the angle B A Cis aright angle. U’herfore the angle ABC isle[Sc then a right angle and it is in the fegment ABC which is greater then the femicircle. _ And forafmuch as in the circle there is a figare of foure fides, namely, AB CD. But if-within a circle be deferibed a figure of foure fides, the angles therof ‘which are oppofite the one to the other are equall'o tworight angles (bythe 22 of the third) Wherfore (by the fame) theangles ABC and ADC areequal of Euclides Elementes. Fol.103. to two right angles.But the angle ABCs lefSethena ri¢htangle IV berfore the anole reniayning A DC is oveater.then a richt angie, and it ts-im.a fegment phichss lefSe then the femicirc.e, Now alfol fay that theangle of the greater feyment, namely , the angle The foureh which is comprehended pnder the curcumference ABCand the right line A C pare. is greater then a right ancle and the angle of the lefse fegment comprehended pider the circumference: A D.C, .and.the right line A Cis lefse thé aright angle sy which ¥ : may thas be proved. Forafmuch as the angle comprehended pader the right lines. BA and v ACis aright angle, therfore theangle come 2S gaan rehended ‘vnder the circumference A BC #7 Sg Sr ts nates | and the right line AC ts greater thenaright ff. YEN XN anole: for the whole is ener greater then his / i _ TA part (by the 9. common fentence. / N bees i/ 4 es My \ ee : ; Entinanehieatel ' OE eee eee eee, at ~ Againe forafmuch as the angle compre* Bierce: pray: © Thefift and hended vnder the right lines AC and A es j “ait part. is a right angle, therfore the angle come : prebendea vader the right line CA and the | incumference ADC 1s lefsethen @ right an ~ lew berfore ina circle an angle made in the femicircless aright-angle bat an angle made ; in the fegment greater then the fenucircle is lefse then a right angle,and an ane ole made in the fegment le[Se then the femucircle, ts greater then a right angle. And moreouer the angle of the greater Jegment is greater then aright angle: ex the angle of. the lefs e Jeginent is lefse thena right angle : which was required to be demonttrated: <1 g9An other demonStration to prone that the angle BA C 1s aright angle.Fors An other De- “afmiuch as the angle AE Cis double tothe angle B A E (by the 32. of the firf}) monitration “for it 1s equal to the two inward ancles which are oppofite. But the mwarde ans — fick — gles are (by the 5.¢f the first) equall the one to the other and the angle AE Bis femicircless double to.the angle E AC. Wherfore the angles AE Band A EC are double to "8 ang! theancle BAC But the angles AE Band AEC are equall to two right ans wwies: aides the angle BA Cis a right angle, Which was required to be dee monstrated. \. . SeCorrelary. ~ Hereby ivis manife/t,that if in a triangle one angle be equall «4 cwrotery tothe iio otber angles femayning the fame angle ia right —_ 2GEA 4 angle: | it a ae a Bit ~ or =- or — — _ —-— - — Te = poe _ 4 - pes > - a Pee = oF - ~~ - = Z eS oe = jab ——. = - ; ~ + rr on PES = ee i ¥ = 7 cemenieec was a EE SL ES BS SS ee ae Rewetie oa>ate ; 2 - : rH : : e x — ~ ~ > —- _ ae + ~—s, == ——— ee = Sa Ee hs yr : sae wet ros eee ie rancindaetian aa Ta RE BEES OE eager ai TES = seals — - ees = pa >= F “s ; - SSeS: sie —— ee — ~ ~ yea a — —— oe PSR Oe a ee i ie oe. ~~ = a . c ; to an ablurdi~ ite » Lhe third Booke angle: for that the fide angle to that one angle (namely, the “angle which is made of the jide produced without the trian: _ Sle) ts equal to the fame angles but when the fide angles are “equall the one te the other,they are alforight angles. gy An addition of Pelitarins. Ifin a circle be infcribed a rectangle triangle, the fide oppofite vnto the right -angle fhall be the diameter of the circle. ean fin addition Suppofe that in the circle ABC be infctibed a “of Pehtarins. re&angle triangle ABC, whofe angle at the point B let bea right angle. Then fay,that the fide A C is the diameter of the circle. For if not, then thall _ thecentre be without the line A C,asin the point Demonflva-% E, And draw aliue from the poyntA tothe point siontea dng §=—_-E & produce it to the circumference to the point DD: and let A E D be the diameter: and draw aline fe from the point B to the point D.Now(by this'31, + | Propotitio) the angle AB D fhall be aright angle, and therefore fhall be equall to the right angle ABC,namely, the part to the whole : whichisabe furde. Euen fo may we proue, that thecentre isin no other where but in the liae AC. Wherfore A Cc is the diameter ofthe circle: which was required to be proned. $320 q An addition of Campane. - By thys 3x. Propofition,and by the 16. Propofition of thys booke, itis mani-- felt,that although in mixt angles which are contayned vnder a tight line and the circumference of'a circle,there may be geuen an angle lefle & greater then a right ” angle,yerican there neuer be geué an angle equall to aright angle.For euery fecti- on of acircle is eyther a femicircle, or greater then a femicircle, or lefle,but the an- gle ofa femicircle is by the 16.ot thys booke, lefle thena right angle, and fo alfo is teangle ofa leile fe@ion by thys 31.Propolition : Likewife the angle ofa greater icdtion, is greater then aright angle, as it hachin thys Propofition bene proued. . Seo 7 be 28.Theoreme. The 32. Propofition. _ © farightline toucha arcle,and from the touch be dramena ., vight line cutting the circle: the angles which that hne and the touch line make,are equall to the angles which confiftin the alternate feomentes of the circle. | END V ppofe that the right line EF do touch the circle ABC D in the KS © A| point. B: and from the point B let there be drawen into the circle XQ ABCD a right line cutting the circle and let the fame be BD. . memes Then I fay, that the angles which the line BD together with the ue touch An addition of Camp ants of Eutlides Elementes. Fol.to4. touch line EF dovaake, areequallto the anslés*which arein the alternate fer amentes of the circle,that is, the angle FB D is equall to the angle which confi Jtechin thefegnent B.A D and theangle EB Dis equalltotkeangle which cou/isteth i.chejeyment. BCD... Raife vp (by the ars of the fuift framy point Conftinétion. 'B vito the right line EF a perpendicular line BA. And in the circumference BD take aypotut at.alladuentuees, and let.the fame be C. And draw thefe right lines ADD Cand CB. And for afmuchas.a certaine right line EE tone Demonfira- cheth the arcle, ALC, in.the pot B, and from thepoint B where the touch ts, = as. rayfed Dp PuLO thé touch line.a perpen dicular ‘BA: I herfore( by the 19. of the third) in the line BA is the centre of the circle ABC D.Wherforey ancle A DB being in the femicircle, is( by the 31. 0f the third a right angle. Whereforethe ane | glesremayning BAD and ABD,are .\{. equall to one right angle. But the angle. \... . ABF is alright angle Wherefore the an- ..- gle A B Fits equall to the angles BAD and ABD. Lake away y angle ABD ‘which is contmon to them both.W herefore the angle remayning DBF 1s equall to the angle rernayning BAD, which is in 3 theatternate feomentofthe circle. dnd forafmuch as.inthe circle is. afigureof Lower fides putmey AB. CD, therfare(hy the 22..0f the third) the. angles which are oppofite the ane tothe other, areeguall to. tworight angles, Wherfore the arte gles BAD. and BCD, are equall.to two right angles; But the angles DBE and; '‘D B Ey are alfoeguall totwo right angles... Wherefore the angles DBE and DBE, ave eguall to the angles BA D and BCD. Of which we bane proned that the angle B.A Dias equallto the angle. OD BE. Wherefore the ane glevemayning DBE, ts equall to theangleremayumg D CB, which isin the alternate fegment of the circle namely,in thefegment D.C BIf therfore aright line touch a carcle,and from the touch be drawen.a right line. cutting the circle: the angles whighthatluse and thetoach line make, are equall to Zangles which gon fi/t.niy altergate fegmentes.of the.circle ; which was required to be proued. } i rl SW i aig iY j i ) (i a 0 i ‘fi } 7) re Hs PH Ht _jaln.thys Propofition may beawo cafes «For the line drawen fromthe touch and Two eafes ix Eutting the circle, may. eyther pafle by the centre ornot. Ifit pafle by the centre, *his Propefi- rhetris irmanifett (by the 18. of thys booke) that itfalleth perpendicularly vpon *#s therouch tinesanid dcniderh thé-circle into two equall partes, fo that all the angles inechefemicirclejare by the former Propofition,rightangles, and therfore ¢quall to the alterndteangles made by the fayd perpendicular line and the touch line. Tfit pes not by the.centre, then, followe the conftruction and deinonfration be-. ore put. =: | | —_—- ae _ Se ee pe = ae : —— SgyT he es ~ ey — ay essa = = yale rtf. . -s _——_ - — = — -_— ~ + . LE eae ay Se : - < « = et a - a ’ = weal ee . = ea ee a a ~ p ~ Tne -—— oo : ahi Me — a - — - = seme = . oad —— e

this Propof- sae. in the farfede/cription. And( by the 23 °-~ The fireafe. of the firft) pon the right line 4 B* Conftitizon. “and tothe pot in it A defcribéean * angle eqnal to the angle C, and let the’ fame be DAB. Wherfore the angle > 4 D ABis an'acute angle. From) the 62S 2° point hraife bp( by the 11. of i firft) © Dnto te (me AD a perpendiculer line Af: —Andby thexxof the firft ) 7 denide the line A°B into two equal partes in the point F. And (bythe 41, 30.8 EME eS. SANTOS Shggteas of the Jayne’) from the point Fraife'vp ynto- the line AB a perpendicular ne EC and dea line from GtoB. And Torafmrch asthe line_A F is equall to the ‘ hn ths yn VaR ea es 6 Wha | PS Osi rs ‘LS & mr - “ SS sion, dnnte’F Bind the line FG ts common to thon both: ‘therfore thefe two lines APF and EGO ave cynall to thefe tivo tines FB and FG*and the angle“ AFG ts (by the 4+ petition) eqnallto the anoke GF Be Wherfore( by the 4. of the fame) the Demonjira- oS i 5 Pppofe that the right line genenbe AB and let the rettiline anole oe | sen beC.1t isveqmred dpon the right line gent A Bto defcribe-a fegt A ment of a-circle which fhall contayne an: angle equall to the angle C. = Now the angle Cis either an deateancle, or a right angle, or an’ obb am ey . a ¥ uss DAVE Pale AG is equall tothe bafe GB. Wherfore making the centre G andthe [pace = 47 & ie " F pL) em ie ; o ~ f Fh hy Ae bee f +5 . TF Py ss yee ; eth og cy GO PLdeferiber by the z.peticion ) a circle and it hall paffe’ by ‘the point B* des a &’ a geet PC ANTIN © 7 Sok P : m7 r *-. : t dyno arth lneA D wakind toge e +9 ; <> « i a) Wiles Hherfore(by the covrellary of the Jeribe uch acircle (let the fame be A BE: And draw aline fromE to B. Now jobufmethas from the ende'of the dianteter AB namely, fromthe point A$ they ‘with the right line AE aright une 6. of the third) the line AD toucheth the cre AD ES And ford{much asa certaineright'line AD toncheth the eh _. ABE, «7 from the pomt.A where the touch is is drawen into§ circle a certaine waa, Legnt ine AB therfore( by the 32: of the third) theangle'O A Bis equal to the wots WHA E Bwhich is ut the alternate Segment: of the circle. But the an gle DAB 4s equal tothe.anele Cwherfore the angle. C is equallto the angle A E B.v-here fore vpobtheright ine geuctd Bis doferibed afegment of a circle which cons tayneth theanele AE B,whichisequallte the angle geuen namely,toC, 03 Thefeend ~° “Buknewfuppofechat the ang Chearight angle. Itis a iodine reduired "Ya ; i £4fe. of Euchides Elementes. Fol, 105. pon the right line A B to defcribe'a Seg Sig ment ofra-corcle,Wwhich fhallcontayne an Od: angle equal tothe right angleC ‘Defcribe.. e. ; againe ppon the right line A Band to the point init A anangleBAD equal to the rectiline angle geuen C (by the23. of the first) asitts fet forth in the fecond de- , | Jeription And ( &) the 10.0f the firSk) des” uide the line AB into te equal partes in the point F. And making the centre the : point Fand the {pace F Aor FB defcribe | (by. the 3 «peticion y circle AE B. ther. P Demon ree forethe right line-A D toucheth the cirs — cle AEB: for that the angle BAD isa right angleW bherforej angle BAD is equall to the angle-which is in the segment AEB, for the angle which is in a Semictrclets aright angle( by the 31. of the third ) Lut the angle BA Dis equal to the angle C. Wherfore there is againe defcribed vpon the line AB a /egment of a circle namely,A EB which containeth an angle equall to the angle geuen namely to C. But now [uppofe that the angle C bean obtufe ancleV pon the right line AB ; and to the point init A defcribe ( by the 23. of the firft) an angle BAD equall es to the angle C; as it is in the third defcription, And from the point A rayle Yp dnto the line AD a perpendiculer line AE Confiruttion. (by the 11. of the firft) And agayne by the 10. of the first) denide the line A B into tivo equall partes in the point F, And from the point F rarfe vp vuto theline A Ba per pes dicular line FG (by the 11. of the fame) o> drawe a line from G to'B. And now fora/: much as the line A F is equal to the line FB, and the line FG is common to them both ; therfore thefe two lines A F and FG are ¢: guall to thefe two lines BE andEG: and the aigle A FGis (by the. peticion ) equall to the an cleBEG: ‘wherfore (by the 4. of the fame) the bafe AG is.equail to the bafe G B.Wherfore making the centre G and the [pace G A defcribe( by the 3. peticion )a circle and it fall pafse by the point T: let it be defcribed as the circle AE Bis. And forafmuch as from theende of the diameter A E is drawen a perpenatculer line AD, therefore (by the corvellary of the 16. of the third) the line AD toucheth the circle A EB or from the point of the touche namely, A is extended the line AB. Wherfore (by the 32. of the third ) the angle BAD is equall tothe angle A HB which is in the alternate Jegment of the circle. But the an leBAD is equall to the angle C. | Ee j. _ Where Conftixttios. Demenfira- ston. wat. : bax % Conftrnébion. Demonfira~ £20%e of the ton che namely, 'B, is drawn into the eS Thethird Booke’ 7 @ W berefore the-angle which is in the Segment: AH Bes equall to the an rote iC: Wherfore vpon the right line geuen A Bis defcribed a Jezment. oft circle AL iss Bowhich contayneth an angle e equall tothe angle even, “ete »C: which was required to be done. The 6.5 Probleme. The 34. Propofition. From acircle geuen to cut away a fettion which fal containe. Av anole equall to a rechiline angle geuen. } ct hex fh V ppofe that the circle geuen be AC and let the reétiline angle gener sg JANG E be D. Itis required fro the circle A BC to cut away a fegment “which feo fae ve frall contayne an angle eq juall to.the angle D. Draw( by the 17-0f the- Ree A hiy d)a line touching t! ve circle, and let the Jame be EF: and let it tonche in the point B. And ( by the 2 3. of th be first pon thevright line EF and to the point init 'B mejevive the angleF BC eo all £0 tbe an ole DD. Now forajmuch as a cert ayne right line Et F toucheth the cire . . cle ABCin the p Lied B: and fr ora y point circle acertaine right line B C; therefore (by the 32. of the third )the angle FBCis equallto the angle BAC phich i isin the alternate fegment. But the angle F BC 1s equall to the angle DW herfore e the angle B AC which confssteth in the be Jegment BA Cis San to the angle p, wheres _ fore from the circle geuen A BC is cut away a fegment B AC, which containeth anangle — to the rectiline an ge geuen: which Was required to be done. Tbhe29.T heareme. The35.Propofition. Afi inacircle two right lines docut the one the other,the rect- angle parallelograme comprehended under the fegmentes or parts ofthe one line is equall to the rettangle parallelograme comprehended under the Segment. or partes of the other line. i e) Et she circle be ABCD, ere: init let shefe two right lines AC pe aes B Deutthe one the other in the point E. Then Lfay that the rectangle. ay ” parallelagramme contayned ‘vnder the partes A EandE Cis equal to ‘thee of Euchdes Elementes. Fol.106. the re&tangle parallelogramme contained ynder the : ioe | - = partes E and EBLor if the lime AC and BD es be drawen by the ce ntre then ts it manife/t that for peck as much as the lines A E and ECare equall to the lines DE and E B by the definition of a circle, the reciangle parallelograme alfo contayned ynder the . ~\ lines AE and EC 15 equall top rectangle paralles oe loovame contained vnder, the lines DE.and EB. *® two But now fuppofe that thelines AC and D B be not extended by the centre and take( by the 1. of the third) the centre of the cire cle ABC D,and let the fame be the point F, and from the point Fdyaw to the right lines AC and DB perpendicular lines F Gand F H (by the 12, of the first) and draw thefe right lines F BF Cand FE. And fora/much as a certaine right line F G drawen by the centre ,cutteth a certaine right line AC not drawen by the centre in {uch forte.that it maketh right angles, it therfore deuideth the'line AC into two equall partes ( by the 3. of the third). Wherfore the line AG is equall to the line GC. And foraf: much as the right line AC is deuded into two e- quall partes in the pomtG, and into two “ynequall partes inthe pomt E: therfore (bythe 5. of the fecond) the reclancole parallelos gramme contained yuder the lines A E-and E C together-with the |quare of the Aine Gis equall to the fqguare of the lineG C, Put the [quare of the line GF common to them both ,wherfore that which is contained pnder the lines A Es E Ctogether swith the [auares of the lines EG and G F is equall to the {quares of the lines GF er GC. But vntoy /quares of y lines EG e GF is equall 3 '[quare of) line F E (by the 4:7.0f the firit):and to the fquares of the lines GC andG F is equall the /quare of the line F C( by the fame) Wherfore that which is contaia ned vader the lmes A E and EC,together with the fquare of the line F E is es quall to the {quare of the line FC, Bat the line F Cis equall to the line FB. For they are drawen from the centre to the circumference. Wherfore that which is contained Ynder the lines AE and EC together with the [quare of the line FE is equal to the {quare of the line F B.And by the fame demonstration that which 4s contained Ynder thelines D E and EB together with the fquare of the line F Eis equalt to the /quare of the line F B. Wherfore that which is contained buns der the lines A E and E Ctogetherswithithe fquareof the line EF is equall.to that which is contayned ynder the lines DE and EB to gether with the [quare of the line E FT ake away the [qnare of the line FE whichis common to them both. Wherfore the rectangle parallelogramme. remayning which is contayned dander thedines A E and Cts equall to the rettangle parallelo gramme remay* Mug, Which ts contayned ader. the lines.D E and EB. If therefore ina carele tworight lines do cut the one the other: the recangle parallelogramme compre- Fe. y. ended Two tafesin this Propofi- $1072. Firfi cafes Demonstras $10 ite The fecond c o— « = fan in oy = . , — . ~ = ea - #2 eae +" ~~ x Po - % z mad 7 . “ — ee - ™ . == ~ = < — : ie . -- 2 = ee a = = nile ne ‘ SS = —— St ~ Ge SS ee : —— = a oe —--- oe - _ ast = oe = er — cote : a /. Se ee - _ — —_—. - _ ~=8 - he a 8 ee ne See ee x a = - ~ . Stress =-3- — - — o = ——---- 5 em ee Sneemalche ews ~ - > = : = ee st - : = —— at ee — = SS en -— he a a = —— = - _ - =. hen ~ Oe aS LaF Ser = “ ; : - ane aee —- — = = — — ——— — ———- — = OT Slo. ERs ye rare x = : a —=- —— - rn —— z Se eEes = ———- - - ———— — - a >. aoe = : "ya > —) ‘ 7 ——— ~ _ 3 : i ==> 3 — — ~ = = = > x ™ — —_ ed ra of ~ ~ me - . a —_ “— . : a = — — - - _ ee e~ 2; Se ae. = --res : = = pit mn Sr oi ‘ SS ~ z = r ~ ae : —— ae 2 4 = oe — - a ° — — ‘ = : a ~ aoe 2 — : —s = — _- - ah | Eo ——— -= - —— —— = —--- = —--——- -— = === =a = anes = = - ra ieee ee ae —<— ——— = = = — = < ———— == = — = —————— = = y= ; rs SS = . ——— == ———— SS —<$<—<—<—<—<—<—<—<—— — — —_————- ae — vs : = Sess : - : 4 = + " on * : - : — —— 4 eo = eee a Elen — ——_ Ls Se ee ae nied. Slee tetal - = atin ea ee - = — - == — mn ee eS a SSS eS: = ——— ese —— = ? <=: aseieseers aaa c - = = = aia : — a —— = ‘was required to be demonstrated. Three cafesin ehis Propoft- bi65%0 The thard cafe. -and £ C together with the {quare of the line EF, ; BE and £F.-Take away thefquare of the line : ‘The third Booke hended vnder the fegmentesor parts of the one line is equall to the rettangle piae rallelograme comprehended ‘vnder the fegmentes or parts of the other line-which In thys Propofition are three cafes: For eyther both the lines pafle by the cen- tre, or neyther of chem paffeth by the centre : or the one pafleth by the centre and the othernot. The two firlt cafes are before demontirated. But now let one of the lines onely, namely, the line e4C paffe by the centre, which let be the poynt F,and let it cut the other line, namely,,2. Din the poynt-Z.. Now then theline 4C deuideththeline BD eyther intotwo equall partes,orinto two yn- equall partes . Fyrftlecit deuide it into two equall partes: Whereforelalfo it denideth it rightangledwyfe by the 3,ofthys booke.\Drawe aright line from B. to F. Where- fore BEF isaright angled triangle. And for almuchastherightline 4C,is deuided into two equall partes in the poyn: F,& into two ynequall partesin the poynt £ « Ther- fore the rectangle figure contayned vnder the ae) lines e-f F and EC together with the fquare of ea the line Z £,is equall tothe fquare of the line FC a Pin: (by the 5. of the fecond). but vnto the {quare of Pak the line FC isequall the fquare of the line BF {for that the lines F B and F Care equall): Ther- fore that whichis cotayned vnderthelines 4E PAY t ; } | \ | he isequalito the fquare of the line BF. But vnto the fquare.of the line B F, are equall the fquares L ofthelines BE and E F (bythe 47. of the firft). = 7 ee re Wherefore that which is contayned ynder the lines E and EC togetherwith the fquare of the line EF, isequalltothe {quares of the tines it bg — Fons we Zs : E F whichis cofamon to them both: Wherefore that which remayneth, namely, that whichis tontayned viderthe lines e# Land EC, is equalito. the refidue, namely, to the fquareoftheline BE. But the {quare of the line Eis that whichis contained vn- der thelines BE and E D.:.for (by fuppofition) the line 2 E 1s equall tothe ine ED, Wherefore that which js contayned ynder the lines 4 E & E C,is equalltothat which is contayned vnder thé lines B E and £ D: which was required to be proued. But nowdet the line. £C paffing by the centre, _ AA denide,the line BD not paffing by the centre,vn- equally in the poynt E. And fro the poynt E raife yp vito the line eC a perpendicular linesE H, whichiproduce on the other fide tothe poynt.G. Wherefore (by the 3. of this booke) the line EH is equall to the line EG’. Wherfore as we haue'be- fore proued,, that which is contayned vnder the lines\.4 £ and EC,is equail to that which ts con- tayned ynderthelines GE & EA: butthat which ‘Is contayned vnder the lines BE and ED, ts alfo equallto that which is contayned vnder the lines G EandsE H, by the fecond cafe of thys Propofiti- on: Wherfore that whichis contayned vnder the lines'e4 E.and EC, is equallto that whichiscon- tayned vader the lines B E and E D : which was agaynerequired to be proned. oe ‘ .» Amongeft all the Propofitions in this third booke,doubtles thysis one of che chiefeft. Forit fetteth forth vnto vs the wonderfull nature ofa circle’. So chat by of Euchdes Elementes. Fol.to7e itmay be done many goodly conclufionsin Geometry , as fhallafterward be dé Jared when occafion {hall {crue. - S@The 30.Theoreme. The 36. Propofition. Tf without a circle be taken a certaine point, and from that point be drawen tothe circle two right lines, fothat the one of them docut the circle, and the other do touch the circle: the rectangle parallelogramme whichis comprehended vnder the whole right line which cutteth the circle, and that portion of the fame line that eth betwene the point and the vtter cir- ciiference of the circle, 1s equall tothe [quare made of the line that toucheth the circle. = gl V ppofe that the circle be ABC: and without the fame circle take ae Wiel Wee ny point at all aduentures and let the fame be D. And from the point Atee,| DP let there be drawen to the circle two right lines DC A and DB, == and let the right line DC A cut the circle A C Bin the point C,and let the right line BD touch the fame. T hen I fay, that thereétangle parallelos gramme contayned bnder the lines AD and DC, is equall to the /quare of the 4ne BD. Now the line DC A is either drawen by the centre or not. First let it be drawen by the centre. And (by the fir[t of the third let the poynt F be > centre of D the circle A BC, and drawe aline from F to B. Wherefore the angle F BD is aright angle.And for afmuch as y right line AC is denided into two “equall partes in the poynt F,and ynto it is added direEhly aright line CD, therfore( by the 6 « of the fecond )that which is contayned ‘vnder the lines AD and DCtogether with the fquare of 5 line CF, is equall tothe fquare of the line FD. But the line F Cis equalltothe line FB, for they are drawen from the centre toy circum ference: Whers fare that which is contayned vnder the lines AD and D C together with the [quare of the line FB, is equall to the {quare of the lineF D. Buty |quare of the line FD, is ( by the 4.7. of the firft )equall to the {quares of the lines FB and BD ( for the angle F BD is aright angle ). Wherefore that ~which is contayned ‘onder the fines ADand DC together with the Square of the line F B, is equall to the w Ee.iy. [quares Confirnflions Twoctafes in this Propofi- 10%. Lhe fir cafe. Demonfrea- » 630% SS — a ~_ = = ate ~ See arte o = = Se 2-0 mE = s < ag es q - ap Ie an 7 = Ss = - aS a — = a = en Cee * Se ee ae eh a SI a le ee tet peeN —— ———— eae ~ — — - - - ve “ = ~ nee e eS ee =< : = - —— - = ws — J is? : ‘ 3 rt > — Thc - a _ = ==> — = =m — — a Sey “ a ——— . ————— “ i Pete ~ = >. a : Z —, = ~ ast —= Se 7 ——s aes : tae ee = Ses e a wise eS ie Mc - = = _—— = — ——— - — - roy ww z x . np aeons 5k ————- : a ee eens ———S . ~— = ~ Se a é ts = = Se -_ ———— _— ~ —- SS See —— > Se i A ~ Fe = > ——- z = — - — = — Sere Sys SS a SSS Se apes. ms ——F. ee SE ESS == == Sa ——: ==: a - a = ne so Fx ab ~ ites Sly eee. - _— = bi = = Bs . : * 3 3 OT 2 eee 7 aA a, Ore : ane —, -$ oe ae a = 2S OO = =~ = = —— -————— - — — _— -- — ——— age —aeae : a ae = : -- . -- +> ~~ “ae 4 : — ———— ee - = 2h. ee EEE = = = = = some = +. tea ¥ Sob .5 ene. geeeroe call = ~ = Ss sa —— SE SS SSS SSS =~ = == — = —= — == = ~ = ao — ————————— —— Ts . + yn ey ES ee Bee oe -e . . —EE = i = — . ———— - — — —— — = 7 ——— 5 — 3 E — =~ = . — ilies =tas sansa: = a Soe = —- = — : eens a — — awe - . = 4 Ste I: anes ae ier ==> 27 Et TE 2 . — —— : : \ >, The fecond ezfe, Conflructtoss Demonitra= tion. The third Booke squares of theines F Band BD. T ake away the [quare of the line FB which is common to them both . Wherefore that whichremayneth, namely; that which iSeontayned yider the lines AD and D C,is equall to the [quare made of the line DB volhchtoucheth the circle. = But now fuppofe that the right line DC Abe not diawen bythe centre of the circle ABC. And (by the first 0, the third) let the point E.be y cene | tre.of the circh ABC. And fromy poynt E,draw : (by the 12.0f the firft) buto the line AC a pere | | pendicular lim E Pyand draw thefe right ines psy t | | : | BEBE Cand ED. Now theangle EF D isa i Sie right.angle. ..dnd for.afmuch asa.certaine right, fv i a8 line EF draven by the centre ,cutteth a certayne Bh. \| disc: other rightlie AC not drawen by the centre,in | Pak ' \ uch fort that't maketh right angles , it deuideth >| re | at by y third of the third) into two equall partes. \ / Wherefore theliane A Eas equallto theline FC... ~ : are éind fora muir as the | right line ACs deuided ead eae puto tbo equal partes inthe poynt F,2x bnto it ts be added diveéil; an other. right hne making both one right line therefore ( by the «of the fecond ) i that which isiontayned vnder the lines D Aand D C together with the fquare - of the line F Gis equall to the {quare of the line FD: put the fquare of the-line F E common io them both. Wherefore that which is contayned ‘vnder the lines D Aand DC together with the fquares of the lines CF andF E,is equall to the [quares ofthe lines F Dand & E.. But tothe fquares of the lines F D and FE, ts equallthe [quare of the line DE ( by the 47.0f the firft ) for the angle EF Dis artebtangle. Andto the fquares of the lines C F and FE, is equall , the fquare of ibe line C E (by the fame). Wherfore that which is contayned vue der the lines 4 Dand D C together with the fauare of the line EC, is equall to the fquare of beline ‘ED . But thé line EC is equall to the line E B: for they ave drawen from the centre to. the circumference, Wherefore that which is cone tayned ynderthe lines. A D and DC together with the [quare of the line EB, 4s equal to th. fquare of the line ED . But to the fquare of the line ED, are ea ‘qual the fquares of the lines B Band BD ( by the 4.7. of the firkt)for the ane gle EBD isa right angle: Wherefore that whichis contayned vader the lines ADand D6 together with the fquare of the line EB, is equall to the [quares of the lines EBand BD. T ake away the [quare of the line EB: which is come mon to them loth : Wherefore the refidue, namely, that which is contayned Yne der the lines ADand DC, is equall to the [quare of the line DB. If therfore withouta ciwde be taken a certaine point,and from that poynt be drawen tothe circle tworight lines, fo that the one of them do cut the circle, and the otherdo : hee ea “touch AY of Euchdes Elementes. — Fol.108. vouch the tarcle': the reGansle parallelooramme which is tomprehended bnder the whole right line Awhich cutteth the circle-and that portion of the fame ling that letl>berwenest be poynt and the btter circumference of thecircle,is equali to the /quare made of the line that toncheth the. circle ; which was required to be demon/trated, ) q I wo Corollaries ont of Campane. If frombneand Weyefefanie poyat taken without arcirtle be. draven sntothe circle lines how many foener : the retladgle Parallclogransmes contayned under enery one of them and hys ontward part, are equal the one to the ether. And thysis hereby manifeft, for that euery one of thofe rectangle Parallel eramimnes are equall to the fquare ofthe line which'ts drawen from that poynt at If two lines drawenfrom one and the felfe fame point do toxch a circle, they areequall the one to the other. Which although itneede no demonftration, for that the fquare of eyther of themis equallto that which is contayned vnder the line drawen from the fame poynt andhys Outward part: yethe thus prouethit. Suppofe thatthere be acircle BCD, whofe centre let be £, and without it take the point 4. And from the poytit.4 drawe two lines 4 Band HAN ef ‘Dy whichlettouch the circle in the poyntes Pes eS Band D. Then Lay,that they are equall. Draw i 3 . thefe rizht lines EB, E D,and AE- And by the 18. of thys booke,eyther of thesamgles at the poyntes Sand ‘D isa right angle. Wherefore(by the 47. of the firft).the {quare of the line e7 E£, is cquall to the two {quares of thelines A Band EB: and by the famereafon,to thetwo {quares ofthe lines 4D and ED. Wherefore the two fguares of thelines 4B and EB,are equall to the two fquares of the lines ef D-and £ D-And forafmuch as the fquares of the lines EB and F Dare equail, therefore the two other fquares ofthe lines 4B and 4D are alfoequall.Wher- ee fore the line 4B is equall to the line 4 D:which Senn a was required to beproned. : G The fame may be proued an other way : Draw a line from BtoD. And (bythe 5.of the firft) the angle EB D isequallito the angle ED B . And forafmuch as the two an- gles ABE and e4 DE areequallnamely,for that they are rightangles : if you take from them the equallangles EB D&E DB, the two other angles remayning, namely, the angles 4 BD and 4 DB thallbe equall, Whetefore( bythe 6.ofthe firft) the line AB is equalltothe line 4D. ¢ Flereunto alo Pelitarins addeth this Corollary. From & poyne geuen without acircle,can be dratyen unto acircle onely two touch lines. eS ST Vas bwind at : LA The former defeription remayning,] fay that from the poynt A canbe drawen vato — | Ec. iiij. the First CoreZaq rye Second Co rollarys Third Corof- lary. ees Pl be ee ae! ) bat Se | This propoft- tion ts the co- nerje ofthe former. Couftruttion. Demonfira- ston. ' circle. And that in fuch fort, that that which Aet the fame be DE. And ( by the firft of the ine.4B Ddoonot p Which was required to be And therfore the uareo $y The ende of the th is equallto the fq rig linese4 Dandc1B b foe “SS dees ce = at = o 2 G "SS o>} 3 © S — 2 o ao, z Butif the! Pe wy 4 $2 = @ WY vu fal a4 a+ a x a] os ie & vv core a J = S te oa : : ae ee i > Se ae are ee an ——— on = 7 ee Sm ee | | a Fol.t10. qi Tbe fourth booke of hue clides Elementes. © WARD HIS rovrru voox & intreateth ofthe infcrip- The i a ayy \ ap Ot? wa 4. hae dh rigat lined figure,and how a right lined figure may fo be infcribed or circumferibed within or about the \aotber.Alfo it teacheth how. triangle, a {quare, and Alay certayne other redctiline figures being regular may be 2s inicribed within a circle. Alfo how they may be cir- cum{cribed about a circle. Likewife how a circle may be infcribed within them. And how it may be circumf{cribed about them:And be- caufe the maner of entreatie in this booke is diuers from the entreaty of the for- mer bookes.he vierh in this ocher wordes and termes then hevfedinthem. The definitions of which in order here after follow. Sp Definitions. Ar. tiline figure is Jayd tobe i nfert hed ina rechi ine figure, irl defn when euery one of the angles of the infcribed figure toucheth vom euery one of the fides of the igure whertin it ts infcribed. . As the triangle 4B Cis infcribedin the triangle D EF, becaufe that euery angle of the triangle infcribed namely,the triangle e-7 BC toucheth euery fide of the triangle within which it is de{cribed namely,of the triangle DEF, As the angle C_4B toucheth the fide E D -the angle 4B Croncheth the fide D F, and the angle 4C B toucheth the fide E F.So likewife the {quare 4 B CD is {aid to be infcribed within the {quare E FG fH. for eucry angle of it toucheth {ome one fide of the other. So alfo the Pentagon or fine angled figure 4B CDE isin{cribed within the Pentagon or fiue angled figure FG HJ K,Asyoufeeinthe figures. = , Likemife —— on . « “ . a — : R it . : St coe er pita MOET SS E ——— HS — - aa e aes . Pee FO ee ; —— EE ———— SS SS = cz X Se a ely ee. re — - —- ee ser = oe —— i. = ~ = - _ a : - — = - — es om ——— — a Se a ' cai — a xs = > =< - = ——— > rae = : . ~ . = ——————<$— ————— ———— - —_— SS ne ee ee ce ee a te = ut tee ee ~ roe = => — - . Second de- fisition. The infertpti- tion and ¢ir- cursfcripticn of rethilsne jignres pertai- 9th only to veguiar fle AHTCSe The third dea finitions The fourth Booke Likewife a retliline figure 1s aid to Le circumfcribed about a retliline figure, when euery one of the fides of tbe figure cir- cum/cribed, toucheth euery one of the angles of the figure a- bout which itis circum|cribed. As in the former defcriptions the triangle D E F is faidto be circumfcribed abour the triangle 4B C,for that euery fide of the figure circumfcribed,namel y,of the trian- gle‘D E F toucheth every angle of the figure wherabout itis circumf{cribed.As the fide D F of the triangle D E F circumfcribed,toucheth the angle 4B C of the trian gle ABC about which it is circunifcribed:and the fide EF toncheth the angle B C A,and the fide C Dtoucheth the angle C 4 B.Likewile vaderitand you ofthe {qguare EF GH which is circum{cribed about the fquare 4 2 C D: for every fide of the one toucheth fome one fide of the other. Eué fo by the faine reafon the Pentagon F G AJ Kis circumf{cribed 2- boutthe Pentagon BCD E, as you feein the figure on the other fide. And thus may you of other reGtiline figures confider. By thefetwodefinitions it ismanifeft,that the infcription and circumfcription of rectiline figures here fpoken of, pertayne to fuch rectiline figures onely, which haue equall fides and equall angles,which are commonly called regular. Itis alo to be noted that rectiline figures only ofone kinde or forme can beinfcribed or circumicribed the one within or about the other. As a triangle within or about a triangle: A {quare within or aboutd fquare: and fora Pentagon within or about a Pentags, & likewife of others of one forme.Buta triangle can not be in{cribed or circumf{cribed within or aboute a fquare : nor.afquare within or about a Penta- gon.And fo of others of diuers kyndes. For euery playne redtiline figure hath fo many anglesas it hath fides. Wherfore the figure infcrided muft haue fo many an- gies as the igure in which itis infcribed hath fides : and the angles of the one (as 1s fayd.) muft touche the fides of the other.And contrariwife in circumfcription of figures,the fides of the figure circumfcribed muft touch the angles of the figure a- bout which it is circumfcribed. Aretliline figure is fayd to be infcribed in acircle,when ene- ry one of the angles of the inferibed figure toucheth the cire cumference of the circle. A circle by reafon of his vniforme and regular diftance which it hath from the centre to the citcumference may eafily touche all the angles of any regular reéti- line figure within it: and alfo all the fides of any figure without it. And therfore a- ny regular rectiline figure may be infcribed within it,and alfo be circumfcribed a- boutit. And agaynea circle may be both infcribed within any regular redtiline fie gure,and alfo be circumfcribed abourit. As the triangle A 2 Cis infcribed inthe circle. 4 BC : for that euery anele toncheth fome one pointe of the circumference of the circle. Asthe angle C4 B of the triangle «4B Ctoucheth the point 4 of the circumference of the circle, And the anglee# 4 c ° 4 i a i, Te al ia a ef tence of the circle And alfo theangle 4 C B of thie tri- ne aN “angle toucliech the, pointe & of the circumference of Ny \ the circle, Inike manner the {qaare 4 DE F is inferi- we ‘ bed inthe fame circle 4 BC: for that euery angle of [| , | es Be es Se of Euclides Elemente. Folin. of the triafgletoucheth thé pointe Piof the circutfe- A eee {7 3 the {quarein{cribed,toucheth fome one poynteof the FK_ , Pp circle in whichitisinfcribed. And fo imagine you of . \ ¥ y j redilined figures. ver | ce LYE eA circle is fayd to be circumfcribed a about.aretliline igure, whe the.cir- cumference of the circle toucheth cuery one of the angles of the figure about which it 1s circum/cribed. Asin the forinef example of tlic third definition. The circle:A D E F is circum(cri- bed about the triangle 4 B C,becaufethe circumference of the circle which is circum- {cribed toucheth eueryan gle of the-triangie about whidhitis circum {cribed : namely, the angles CAB, AGCiand BCH. Likewife'the fame circle _4 D E F is citcumf{cribed about the fquare 4D E F by the fame defimtion,as you may fee, The fourth dé jinssiotl s Aucirele is fay £0 be iaferibed tn areililine figure, when the reff de- circumference of the circle toucheth every one of the fides of * wwo¥Pe floure within which 1615 inferited. .) OAs thé tircle 4BC Disinfcribed within the triangle EEG pbecaufe the circumference ofthe circle toucheth enery fide of the trianglein which it is infcribed: namely the fide EF th thé point B, atid thefide GF in the pointe C,andthe fide G Einthe point D. Likewife by the fame reafon the fame circle is inferibed within the fquare HJ KL. And fo may you indge of other reGtiline figures. A rettilined figure ws faidto becircum- © feribed about acircle, when enery one of tthe fides of the fr- gurecircum/:ribed toucheth the circum sference of the circle. Asin the former figure of the fift definition,the triangle E FG is circumfcribed a. bout the circle_4 8 C D,for that euery fide of the fame triangle beyng circumicribed toucheth the circumference of the circle,about whichit is circum{cribed, As the fide EGofthe triangle E F G toucheth the circumference of the circle in the point‘D: and the fide E F toucheth itin the point & : and the fide G F inthe point C.Likewife alfo the {quare HI K Lis circuni(cribed about thecircle 4B CD, for euery one of his fides. toucheth the circumference of theeircle, namely, inthe pointes 4,8,C,D, And thus confider of all other regular right lined figures (for of them onely are vnderftanded thefe definitions ) to be circumfcribed aboutacircle, orto beinferibed within ac¢ir- cle: or ofa cirele to-be circumferibed or infcribed abcut or within any ofthem, Rk e wk om bu ; ts oe . \ “it He : > ee nwt? Ve may ns a ae , F A e #® a The fixtdes Bi1ONe Sexenth de- finition. Con ftrettion. Two cafesin this Propof- ¢10n. Firft cafes Second cafer Denon ftra- bite « fir St yout Dnto the tine D an var he fourth Booke : | A rightlyne 1 fa yd to be coapted or applied ina tiréle, when the extremes or endes therof,fal uppon the circumferenc eof be Cli Cie rye R ¢ 1. s : Sear vec (iurd ¢ } ee s : : >{ ; 7 ” & me oe As theline 5 \ 20 1a} @2tO DE coapted or to be appli- a — edtothe circle AB Cytor thatboth his extremes fall oe | a vyponthe circumference of the circle in the pointes B, peices * andC, Likewifetheline DE. Thisdefinitionisverys “D P= 1° » \e neceflary,and is properlytobe taken ofany lyne ge- is uen to be coapted and applied into'a circle; fo thatit * 7 | exceede not the diameter of the circle geen, . A es 4 Ae ee The 1. Probleme.:..:T he ts P ropofition. In acircle geuen,to apply aright line equall vnto aright line genen, which excedeth not the dtameter of a circle. o- & | SZ V ppofe that the arclevenen:be A BC) and let the'right Ny 44) lineg Why, SN be D.Nowitis required onthe circle genen AB C;to ape is Gre ply a right line equall ynto the right line D* Draw the | ; i 7 < \ \ * 2 P\ y past Moet = te ¥ . ff if a" mines ee || that done ‘which was req PP wn ~ wre ae ET Ret ema non: fees Pier, Cs Uo. Os ~- A SAS ‘ -f\ NC Gee ~ 7 Hy Pg pee, ~ 4 7 Pit omall Pepe ee 1 Eb . wo ~-F (WO thos ; > aa* $e hn Shc aeeacaes Oa ns AB Cis applyed a right line D. But if not, then ts the line BC greater then y line D. And{ by the third of the equallline CE. Andmakmg the centre C,and the space CH, deferibe ( by the third petition a circle EG F ,cut- ting the circle A BC in the poiat Fey dra a line-from CtoF. And for dfmuch as the point C 15 y-centre of the circle EGE, therefore ( by: | ais the 13. definition of the firft) the line CF ts equall bntothe line CE . But the jine CE 1s equall vnto the line D. Wherefore( by the firft common fentence) the ne _<" AK B,and AMB,are ** 8 equall to two right angles. — Ape ns | And the angles DEG 7 D E Fy are( by the 13, of the fir/t equall to two right angles . Wherefore the angles A KB, and AM Bare equall pnto the angles DEG,and DEF: of which two angles the angle A KB is equall ‘vnto the angle DEG: Wherefore the angle remayning, A M B, is equall ‘nto the ane le remayning {DE F.Indike fort may it be proued, that the angleL N M, is i ‘ GYTANY 5-4 7 ae ‘oe oe ty : ' 2 Lg 3 > equall.to the agh D FEW berfore the angle remayning M LN jis equall pne tothe angleremayning EDF. Wherefore the triangle L MN, 1s equiangle nito.the triangle DE F: andit is defcribed about the circle ABC. Wherefore ‘about acircle-ceuen ts defcribed.a triangle equiangle “bnto a triangle genen: ‘Which was required to be done... : ~~ <> ‘ qT An other way a ifter Pelitarius.. ~~ 5% i ; In the circle-A B.C infcribeatriangle G H K equiangle to the triangle & D F (by the former Propofition) : fo that let the angle at the poynt G be equall to the angle D, and let the angle at the point H be equall tothe angle E: andletalfothe angle at the poynt K be e¢ uall to.the angle F * Then drawetheline LM parallel to the line G H, which let touch the circle in the poynt A (which may be done by the Propofition ad- ded of the fayd Pelitarius after the 17. Propofition) . Draw likewyfe theline MN pa- rallel ynto the line HK and touching the circle in the poynt B; Andalfo draw the 3 ’ S of Euclides Elementes. Fol.ttte LN parallel ynto the line G K and touching the cir- cle inthe poynt C, And thefe three lines fhall vn- doubtedly concurre,as in the poyntes L,M,andN, which may eafily be pro- ved, if you produce onet- ther fide’ the lines GH, G K, and HK, vntill they cut thelines L M,LN,and MN,inthepoyntes O,P, Q.R,S,T. Now! fay, that thetriangle LM N-circt {cribed about the circle ABG, is equiangle to the triangle DET. For it is mianifeft,that itis €quian- gle vnto the triagle GHK, | ee Se ce by the proprictie of parallel lines . For the angle MT. Q isequall to the angle.at the poyht G of the triangle GH K (by the 29. of the firft)-and therefore alfo the angle at thé poynteL, is equall to the felfe fame angle at the poynt G (for the angleat the point L,is bythe fame 29.Propofition, equalltotheangle MT Q_). And by the fame reas fon theangle atthe poynt M,jis equall to the angle at the poynt H of the felfe {ame tri- angle sand the angle at the poynt N, to the angle at the poynt K. Wherfore the whole triangle L MN;is equiangle tothe whole triangle G HK: Wherfore alfoit is equiangle tothe triangle DEF: which was required to be done. Demon ira- ti0n. SaThe 4. Probleme. The 4. Propofition. In.a triangle geuensto defcribe a circle. ake V ppofe that the triangle geuen be ABC Itis required to defcribe acre Hata Sicle inthe triangle AB U. Denide( by the 9. of the first )the angles AB Confiructione eC and AC B into two equall partes by tivo right. lines BD and CD. And let the/e right lines meete together in the point D. And (by the 12+ of the firSt) from the point D draw ‘pnto theright lines AB, BC and CA perpendie cular lines namely {D E,D F,and DG.And ‘forafmuch as the angle AB D 18 aa equall totho angle (BD, and the right angle. BE Dis equall vnto the right angle BED. ~~ Now then there are two triangles E BD.and | FBD haning two angles equall to two angles. and one fide equall to one fide, namely, B D which is common to them both, and fubtendeth one of the equal angles. herfore( by the 26.0f the firSt) the reSt of the fides are equall vnto the reft of the fides. Wherfore the line DE ss equall ynto the line'D F : and by the fame reas fon alfo the line DG is equall vatoy line DF. oe Demonfira- £70H leading fo an timp 6f- froilttie. ‘Three cafes im thes Propoft- P20%s TheprsPoafes DG Aefcribe a circle and it will pafse through . iV herfore thefe three right lines D E,D Fie DG are eqtall the one to the other( by the first common fentence ). Wherefore making the. cens tre the point D ,and the | pace D E, orD For thepomntes-E,F ,G, and-will touch y right lines AB,BC,and C A, Forthe angles made at the pointes E,F,G,are right angles’. For ify circle cut thofe right lines then frothe end of the dix ameter of the circle hall be drawen a right line making two right angles ez falling within the ® F Cc circle: which 1s impofsible, as it was manifest ( by the 16. of the third)..W heres fore the circle defcribed,D being the centre therof,and the {pace-therof being ete Cc ther D E,or D For DG, cutteth not thefe right lines AB,BC,e¢ C-A. Where fore (by the Corallary of the fames).it toucheth them and the circle ts.defcribed in thetriangle A BL. Wherfore sm, the triangle genen ABC,1s deferibed acrcle EF Gi yhich was required to be-done. 12 tek be S Probleme, The 5.Propofition. 2A vout atriangle cenensto defcribea circle. po ied V ppofe that the triangle geuenbe ABC. It is required about the triane ; Boy Tiigle A BC todefcribe a circle. Denide ( by the ©. of the first) the right t= limes AB and A Cinto two equall partes inthe pointes D and E. And fromrbe pointes Diand E (by the w0f the firft rayfe vp vnto the lines A Ber A Crwoperpendienlar lines D Fand E F.Now thefe perpendicular lines meete together either within the triangle A BC or in the right line BC or els without the riolit hie BE, | First let thent meete. together within the trimiele in'the point F’ And ( by9 firft peticion ) drawe right lines from°F to B; from F to C,and from F to A. And foraf: much as the ne AD is equall Ynto the line D B,and the line D Fis common, bne to them both,and maketh the angles on ech fide of bimright angles ,therfore( by the 4. bg of the firft the bafe A F is equall pnto the. > \ bafeF B. In like forte may we prone that the hneC Fis equall ynto the line AF. pe Wherfore the line F'B is equall ynto the line C F.Wherfore thefe three right lines F A,F Band F C are equall the ane oi | the | ae ee - > of Enclides Elementes. Fol.1%. the other.W ber fore makinie the centre the port F yand the {pice FA or EB, or ECdefcribea ctrole and it {hall pafse by the poyntes AB C.And/o there is a care cle decribed ahaut the triau: ele A BC, asye fee mt the just de{cription. “But now fuppofe that the right lines, @ Fand E F do meete together Ypon the SY Fig * — f contl right line B C in the point Fas itis in the Fe eS =. fecond defcripizon sand draw.a right Lyne en ee from A to F and in like forte may “we ao ~ a prone that the poynte F is the centre of. 1K | a =e the circle defcribed aboute the trianole © esdiicer SACRE | SOR har thea \ = “Byt mow fuppofe that the right tmes \ DF and EF domeete together without moe? ae PES thoxyitngle AB Cin the poynt FA Spe eS RRP Sy Aad as it is in the third. defcription draw right. lines from F to. A,from Fto B,and from F to Pe rey C. And forafmuch as the line AD is equall aot yer { oe ynto theline DB, and the line D Fis come OF re honda! ie mon bnto them both and maketha right ans eames eas gle oneche fide of him wherfore( by the 4+ of | ee f the first the bafe A Fis equall ynto the bafée. -\ | BE And inlike fort may we proue that the \ ‘% line C Fis equal bnto the line A F.w berfore as Plan si agayne making F the centre, and the [pace F Sls igi A,or FB, or FC, defcribe a circle and it fbal- ets pafse by the pointes A,B,C, and fois there a civcle defcribed about the triangle ABC,as ‘ye fee it sin the third defcription. W. herfore about a triangle geuen ts deferibed a circle : which was required to be done. ; Ses San Correlary Frereby itis manifeft,that when the centre of the circle fal- leth within the triangle,the angle B. AC being in a greater Jegment of acirelets lefse thé aright angle. But when it fal- leth vpon the right line B Ctheangle'b A Cbeing ina femi- circle is aright angle. But when the centre falleth without the right line B C,the angle BAC being ina lefe fegment of acirclessgreater then 1 right angle. Wher fore alfowben the . angle senenis leffe then a rightangles the right lines D E Pf. and | Fhe fourth Booke and E-F will meete together within the Jayd triangle. Bue when itis a right angle they will meete together ‘ypon the line BC. But whe itis greater then aright angle,they will meete together without the right line BC. ' SmThe 6.Probleme. The 6. Propofition. Fn acircle geuen,to defcribe afquare. BCD to defcribe a fquare. Draw in the circle ABCD two diameters making right angles and let the fame be AC and BD, and drawe right lines from A to B.from B toC,fromC toD, A and from D to A. And fora/much as the line Demoniira- {B Bis equall ynto the line ED (by the 35. hig definition of the firft) for the point E is the centre. And the tine E Ais common.to them both, making on eche fide a right angle; thers F fore( by the 4. of the firft) the bale A Bis e- quall ynto the bafe AD. And by the fame reafon alfo either of thefe lines BC and CD is equall to either of thefe lines A Band AD: wherefore ABC Dis a figure of foure equal | fides.I fay alfo that itis areftangle figure. For forafmuch as the right lineBD asthe diameter of the circle ABC Dtherfore the angle B.A D beyng inthe fee micircleis a right angle ( by the 31. of the third) And by the fame reafon euery one of thefe angles ABC,BCD andC D Aisa right angle.Wherfore the foure jided figure A BC Dis a rectangle figure and it is proued thatit confifteth of eguall fides.W. herfore (by the 30. definition of. the firft) it i$ a [quare , and it is defcribed in the circle. A BC D: which was required to be done. iG V ppofe that the circle ceuen be ABCD. I. tis requiredin the.circle A SY Contruétion. © 'Lbe'7.Probleme. The7:Propofition, as eAbout.acirele genen,to defcribe 4 [quare. | | GEN GAY bpofe that the circle geuen be ABC D.It is required about the cirs BAN a fle ABC D to deftribe a fquare. Draw in the circle ABCD two Weg Uameters making right angles where they cut the one the other, and we det the fame be AC and BD. And by the pointes ABD, drawn Conftruttion. of Euclides Elementes. Fol.1th, (bythe v}-of the third ) right lines touching the dircle A BC Dyand let. the fame be FG, > aes GH,H K,and KF. Now forafmuch as SRURS SIRES the right line FG toucheth the circle ABCD.» ° in the point A, and from the centre Eto the point A where thé touch is is drawen aright &: | line EA therfore( by the 18 of thethird )the Ry otras ra ee angles at the point A are right angles and by : t20ns : . Demon|tra= ay if the fame reafon the angles whichare at the | porntes B,C,D, are aljo rig ht angles . And ee | forafmuch astheancle AE Bisaricht ane Bot 2) 3 * H ¢ i ole,er the angle E BG is alfoa right angle, therfore ( by the 28. 0f the farst the tine GH | | : is aparallel‘pnto the line AC: and by the fame reafon the lyne'A Cts d parallel nto the lyne F K. In like forte alfo may ‘we prone that either of thefe Imes G EF and HK is a parallel vnto the lyne BE D:Wherfore thefe figures G K_, GC; AK 5F B;and BK are parallelogrames/ Wherfore (by the 34- 0 the firft the line G Fis equall ynto the Ime HK and the line G His equall’pnto ghe line F Ki And forafmuch as the line A Cis equall pnto the hneB D) but the line’ AC is equall buto either of thefe lines G Eland FK sand the line B Dis equall to either of thefelmes G Fand HK. Wher fore either of thefe lines G Hl and F K is equall toeither of thefelinesG F and H K» Wherfore the figure FG AK. conjisteth of foure equall fides: I fay alfo thatits a retlangle figure. For foraf? muchas G BE Aisa parallelogramme, and the angle AEB is a right angle: therfore ( by the 34. of the first) theangle AG Bs right angle. In like forte may ‘we prone that the angles at the poyntes H, K, and F are right angles. Wher fore RGF K is a rectangle foure fided figure anditis proned that it cone fisteth of equall fides: wherfore it is a {quare an d itis defcribed about the crcle ABC D.Wherfore about acivele genen is deftribed a [quare: which was requte red to be done. SeThe 8.Probleme. The 8.‘Propofition. Fn a fquare geuen;to defcribe a circle. Are Vppofe that the quare geuenbe ABC D. Leis required in the [quare ) we TA BCD to defcribe acircle. Deuide (by the 0. of the Firft,). either of Conftrnctions Let thefe lines ABand AD into twoequall partes inthe pointes E and &, And by the point E (by the 31. of the firit draw a line B Ei parallel ynto either of thefe lines A Band D C: and (by the fame ) by the point F draw aline FR. parallel puto either of thefe lines AD and BC. Wherfore euery one of thele fi- enue gures Confirnftion. Demonftra- 80% ‘Z be fourth Booke gues AK, KB, AH, HD, AG, GC, BG 4nd.G D is a parallelograme, and the fides which are oppofate the-one to the other, are (by the 34. of the first) equall the one to the other. And forafmuch as the line A Dis equall pnto the line AB, and the halfe of the line A Dis the line, AE, and the halfe of the line A Byis the line: a i, AF, therefore the line A E is equall vnto the, bia seer opt tine 4 F': wherefore (by the fame.) the fides. | which are oppofite are equall. Wherefore the line F Gis equal nto the line E Gln like fort may we proue that either of thefe lines GH, 2: ¢ and G K 1s equall toeither of thefe lines FG «| andG E. Wherfore (by the firSt common [ene tence ) the/e foure linesG E,GF, GH 3 and GK are eqnall the oneto the other. Wherfore.| | making the centreG,and the {pace either. GE 5B Hi oO F350 Hor GK, defcribea circle and it | | will pafse by the pomtes BF, F1,K., and-will touche the right lines AB, BE, € Dyand D A.Forthe angles at the pointes E FAA, K are right angles.Forif the circle do cut the right lines AB, BC ©: D3and.D A, then the line which is drawen by the ende.of the diameter of the circle making right angles fhould fall ‘within the circle, which is impofsible (by the 16. of the third Ww herfore the cene tre being the paynt G and the {page beyng GE or G For G Hy or G-K ifa-cire cle be defcribed it fhall not cut the right lines AB, BC sC D,and DA. Wheres fore i fhall touch them. And it is defcribedin the ‘{quave ABCD: ‘wherefore in afquare geuen is defcribed a arcle:-which twas required to be done. 7 wont» St The g. Probleme The 9. Propofition.. cw bout a fquare genen,todefcribe acircle. | agi,’ ppo/e that the fquare geuen be AB Iayic D. It is required about the fquare eA BC Dito defcribea circle, Drawe_.\° right lines from Atol and from D to Bye let them cut the one the other in the poynt- Es [7 And forafmuch as the lyne DA is équall’yns? | to the lyne AB, and the line AC is common Dito them both therfore thefetwo ines D'A™ and:AC are equal ynto thefe'twoljnes BA - ~~ and AC, the one to the other. And the bape ses DC isequall vnto'the bale BC. wherefore Z 4 (by the 8. of the fir/t) the angle D AC is equall ynto the angle B.A C. wheres fore the angle D AB is deuided into two equall partes by the line AC. in . tke = ed oR 2 - donchessbe £,FG54. Anddrawing thefe two of Cnglides Elementes. Fol. like ortinay we proste'tha Fenery one of thee anglesA BC, BED, anal DA is denided into two emuall- partes by the ye opt lines AC and DB..And foraf- much asthe anvle PBs byrall prto the angle ABC: andof the ang lkeDA Brhe dole POF Bis ae yurye Lind of the angle ABE the angle EB Ais the halfe: I herfore the angteti-A. Bis equall nto the ark EBA: wherfore ( by the 6. of the farft ) the fide £ Ars equall nto the (ide E B. In like forte may Wwe proue that either o f thefe right lines E Aand EBs equall ‘pnto ether of thef ¢ lines EC and E D.Wherfore thefe foure lines EA. “BBE Cand E Dare ¢ J sualvthe onetothe other.W herfore making the centre E, and the [pace any of thefelines E A, EB, EC, or ED. Defcribe a cirtle andit will palfe by the pointes A. B,C,D, and {hall be defcribed about the fauare ABC Das itts chidentinthe freure A BCD. W herfore about a {quare cent is defcribed a cir ae ES cle: which was required to be done. | ¢q A Propofition added by Pelitarins. ot {quatre errenmfcribed about a circle, ts double to the fquare infersbed in the fame circle, Suppofe thatthe{quare ef BCD be cir- cumfcribed about the circle EF GH, whofe centre let be X. And let the. poyntes of the A \@ \auee B diameters E Giand / #7, and thefé right lines EF,FG,GH, and HB, there fhall be in{cri- hedinthe circle a fquare EFC H (by the fixt of this booke )sEhen Tsay; that ‘the fquare ef BCD, is double to the fquare EF GH. For fora(much as the fide ef 2B ofthe greater fguare,is(by the 34.0f the firft}equall to F 4, whichisthediameret of thelefiefquare:. but the {quare of F 4 is double; to the fquare whofe diameter it is, namely, to the fquare EE GH (bythe 47. of the'firtt) . Wherefore alfo the fquate,of ot B whichis24 BC Dis doubleto the {quare E F G7: which was re- guiréd to be proued. Thysimay-alfo bedemonftrated by the equalite of the triangles and {quares contaynedin the great {quares, SaT he 10. Probleme, ,.Tbe10.Propoftion. To make a triangle of two equall fides called Wfofceles which foall bane eyther of the angles at the bafe double to the-o- ther angle. T ake A Propohiion added by Be litavius. rie. due ’ : : 4 . if i y : Pol ns : } | : uly t : ; ii aay © RIAL ah a | } : , : || B i f { | t i} | 1} Th it | ; Tih Bit ie L | i | } i it | i ' j ao ‘ if | : 4 " } i \ Lil 2 Weed bh " ith Hite: | wis Wit y Vis Wl t ;") ‘ f cue 2) he i H " H 's t { i} | ; ne : ft i} He aE Se ar ahi in | ih Te ay BH : \ ; ‘ | . | i + nw i } | ae ay iat | itt Wh Hat } i 1 og i Lae I | i 1 ; +" . ; Bil | Fi lat} Hilt mn ai Miu W } het ie } : wink : mi f. Wil » ra hi py : oe Ht bo. i} a) : if } y 1 ig} 1 . t * ay 4 4 { | ité 7 Bi Me | 1a .) : Oh i BE fd: ; ey . nifA sai 7 ip { Hi oF {) es Sia! weet in) 2 SR it 3 F ‘ f 1" F . y e AN , >! it { : "i he ec ued | bh A 4 “ i = a ~ =, “2 => - on = = - > eae — = Confiruttion. Demonfiras $408. er thefpace.AB; defcribe (hy the 2. peticion). acircle BD E; and (bythe ».of- the fourth into.the arcle B DE apply aright line BD equall tothe right hne AC which is not greater then the diameter of the circle B DE. And draw lines from Ato D and from D to C. And by the 5. of 3 fourth ) about the triangle AC 'D deferibe a civtle ACDE And fora/much as the rect anele figure contained "yn+ der the lines A Band BC is equall to the [quare which is made of the line A C:(For that is by Juppofition) But the line A Cis equall vnto the line BD. Wherfore that which is contayned dnderthe lines A Band BC 1s equall to the {quare whichis made. of the line B D. And forafmuch as without the circle AC D Fis taken a poynt B,and from Bynto the circle ACD F are drawen two right lines BC A,and BD; in fuch fort that the one of them cutteth the circle,and the.ather endeth at the circumfes rence and that which 1s contained Ynder the lines A Band BC 4s eguall to the Jquare which ismade of the lime B'D,therfore (by the 17. of the third ) the line B DO toncheth the arce AC DF. Andfora/muchas the line BD.toucheth in the point D and from D where the touche is 1s drawen aright line DC, theres fore (by the 32. of the fame) the angléB DC is equall ynto the angle D AC, which is in y alternate feoment of y circle. And forafmuch as y angle BD Cis ea gualipntay mighD) AC put the angle DA common vnto thé bothwherfore J whole angle BD A is equal to thefe two angles C D'A,ex D AC, But nto angles CD A,r D ACis equall the outward angle BC D by the 32.0f the 1.) Wherforey angle BDA is equal'ynto y angle BC D.But 5 angle BD Ais (by 2 5.0f the firft) equal ynto the angle C BD for (by thers. eee of > first) the fide A D is equall ynto the fide A B: wherfore (by the 1. common fentence ) theangle DB A is equall ynto the aigle BC Do Wherefore thefe three an gles BD AD B A,anid BC D-are equallthe one tothe, other. And forafmuch as the angle D B Cis equall ynto the angle BC D.,the fide therfore BD is equall ynto the fide DC. But the line BD is by fuppofition equall bnto- the lyne CA. Where yy 7 of Enclides Elementes. Fol.117. wherforethe line AC alfois equal ynto the lyneC D. wherfore alfo (by the §: of the first) the angle D Ais equall ynto the angle D ACW: herfore the ane ¢ks CDA ind D A Care double tothe angleC AD. But the angle BCD is equall pnto the angles CD A ind D AC. Wherfore the angle BCD is double tothe angle A D.But the angle BC D is equall to either of thefe angles BD Ate DB AW herfore either of thefe angles BD Aix DB Ais double to the angle D ABW herfore there is made 4 triangle of two equall fids A BD hauiig either of y.augles at the bafe DB ,double to.the angle remayning: Which was res quired to be done. : : | Ds Here Campane addeth,that the two dircles MC D'and BD E, docutthe one theo- ther: andthaethe circle ef CD cutteth of front thecircle BD Ean arke equall to the arke B'D; and thatthecircle BDE chtteth of from the citclee#CD an arke equall to the arke DC. — The firft partis manifeft . Forifthe leffe do not cut the greater but touch ‘itasin the poynt D . Then (by the rivofthe third) thecentreof either of them fhall beanone line namely, in the line AD: forthatinitis the centre ofthe greater circle,and in the {elfe fame is the poynt of the touch. Wherefore (oy the 31. of the third ) the angle -4CD isaright angie. And therfore(by the 13 ofthe firft)the angle D CB isaright angle ; and fo fhalltheangle 4B Dbea right angle(for.itis equall to the angle D ce, as it hath bene proned ) which (by the 32. of the firft) is impoffible . W herefore they fhall cut the one the other asia : the poyntes D. & E. Now I fay, Sa ds that the arke E D of the greater eircle,is equall'to the arke DB: and that the arke. FD of the jeffe circle, isequall to thearke OPraw thee Hight lines E74, ~ EC:&'E D-Now the(by the 27- | of the third) the foureangies |, PECCEA, DAC, WADE, |~ ~ are equall : for that the arkes | | Ced and CD are equall (by z the 28.of the fame). Wherfore “\ the whole arigle AED is dous"<\ bletothe angle B AD,& ther=_ fore is equall to/cithersof the” . ahgles 4B D and 4 D B. And’ forafmuchastheangle AED is equall to the-angle woe (bythe 5. ofthefirit ) for HORSE PS et A eS She lines of D andeve ae. (Sfit 387 0 erorpns hs Gsisd ; drawen from the centre, therefore the two 4rigles at the poitites E and D of the trian> gle AED, fhallbeequall to the’ two-angles at the 'poyntes Dand B ‘of the triangle "4D B:and therefore the anpleremayning of the wheat the popat o4% thal be equall to the angle remayning of the other at the {amie point (by the 3 lof thedirft),. Where fore(bythé 26. of'the thitd the arkeE D of thie preatercircle; 18 equalito the arke D-B: ‘And bythe famie'the arke FD’ of the feile’cirele, 18 equal tathearke DC which was che wiangle A BD. Chainely/eythel of whofe angler the bal isdouble to the at BS ar the toppe).the angleac the toppe, asin thys cxample,the ariglé‘at the poynt - “Sere pehrayins noteth, thatin euery fach Lforteles wrist a thys placeds is one third pattofanightangle, and, moreouct one fift part ofa third ofa tight angle : that is, two fift partes of one tight angle : and to be briefe, one fift part o Gg.je two Certaine ads ditions of C Amp Ante The first part demon stratede The fecond part demote , firatede i ee eo Re f Propofition ; added by Pe- tarilius. “= oF ewo tight aneles.And either ofthe angles at thebafe.is two fift partes of two right angles, of foure'fift partes of one right angle. Which fhall manifeftly appeare, if we denide tw’o right angles into fue partes . For then in thys kinde of triangle,the angle at the toppe {hall be one fift part, and eyther. of the two angles at the bafe fhall be two fift partes. ) Thys alfo is to be noted, thatthe line ef C is the fide of an equilater Pentagon to béin{cribed in thecircle ACD. For by the latter conftruGion itis. manifeft,rhat the three arkes AC,C D,and D E, of the leffe circle are equall . And forafmuch as by the faine itis manifeft that the two lines «4D and 4 E-are equall thearke alfo-4E thal? be equall tothe arke 4D (by the 20.0f the third) . Wherefore their halfes. alfo aree- quall .Iftherefore the arke AE be (by the 30. of the third) deuided into two equall partes, the whole circiiference AC DE A thall be devidedinto fiue equall arkes. And forafmiuch as the lines fubtending the fayd equallarkes are (by the 29.0f the fame) es guall.therefore euery one of the fayd. fides fhall be the fide of an equilater Pentagons which was required to be proved . Andthe fameline 4C fhall be the fide ofan equila- ter ten angled: figure to be infcribed inthe circle BD £E: the demonftration wherof T omitte, for thatit is demonftrated by Propofitions following. gq A Propofition added by Pelitarins. Upon a right lie gewen being finite, to defcribe an eguilater and equiangle Pentagon figure, Suppofethat the right line genen be 4B, vpon whichitis required to defcribe an equilater and equiangle Pentagon. Vpon the line 4B defcribe ( by the 23, and.32, of the firft) an Iofceles triangle 4B C equiangle to the Ifofceles triangle defcribed by the former Propofition : namely, letthe angles C_4A B and CBA, atthe bafe 4B, be equallro the twoangles 4 BD and ADB inthe former conitruGion.: fo thateyther of them fhall be two fift partes of two right angles,and the angle at thetoppe, namely, . the angle C, fhall beone fift part . Then deuide the angle GC: into two equall partes by *4 drawing the right line, CD. And vpon the line 4 C, and yntothe poynt-4,defcribe the “) angle CAD equall to theangle e4 CD, by drawing theline 4 D,whichline 4 Diet coucurre with the line C.D, inthe poynt D: and that within the triangle et BC, for theline CD being produced,fhall fall vpon the bafe 4B, " geria ah CORON ES abc andthe line.4 D.vpon the fide BC.Anddrawaline from, .. . kh. ED thepoynt D to the poynt 8 . And forafmuch as in the trianete «7 CD thetwoangles 4 and C areequall,ther-_: ee fore (by the 6. of the firft) the two fides e4 Dand€ D are. . Se NS equall’, Againe forafnuch as thetwo fides CB and CD, _/ \ | /; ofthe triangle CB D, are equallto the two fides C_A and CD of thetriangle «4 CD,and the angle C’of the one,: Tiki hey NPA is equall to the angle C of the other ( by conftruction). aie ah AS therefore (by the-4. of the firft) the bafe DB is equallto: .“-—"—. Ie | : Sey the bafe D 4, and {0 is equall to the line DC. Wherefore ‘Kee (byithe oj0fthe third) the poynt D thallbe the centre of a thevirole deferilsed,about,the triangle ef BC. Defcribe. .... sifa~ the: fayd.citcle andletitbe,4 BECF..;Now thenthean-... .-... aarerh gleid’D Bisdoable to the angle, ACD (by the.20,0f 0) i aero ngs thethird). Wherefore theangle,4‘D.B maketh two fift partes of tworight angles,thae is, onedift partioffourerightangles, And for afmuch asthe {pace about the cota is equall to foure right angles,then if the fayd {pace be deuided into. fue angles equ to-thd angle) fl BLA, nanely into fing fift partes ,by drawing the right lines D E & DF, which with thelines DA,D B,and D.C.will caule the fayd {pace to be deuided. into fine equall. partes, aid ifalfo there b¢ drawen theferightlines e4 F,F C,C E,and E Be there thall. be defcribed’ a te@itiné Pentagon figure e4 BE C-F, which {lial be eqnilas ter, by therule-of acitclt ‘and ofa citeumference,and helpe of Babs om = , Ce eo of Euclides Elementes. Fol.n8. he fiue angles : ‘anole(by the 4. and 5. ofthe fame) . Fort wor a “¢ rig re ae are equall partes:which was required to be done. ‘tarius.it will not be hard for vs, Gderwell thys demonttration of Pelitarius, it will not be bi ; ‘son avigheline peueh bo deferibe the reft of the figures whoft infcriptions here oy) after followe. (aT he u. Probleme. The 1. Propofttion. : Ina circle genen to defcribe a pentagon figure eq uilater. an equiangle. Se = > that the circle geuen be ABC D E. Itis required in the circle ane Verh D Eto inferibe afigure of fiue an gles of equall fides rie o es : Se yall an gles. T ake (by the propofitron going before ) an I ‘fofce es trian gle FGH haz A. ‘ning eyther of the angles at 7 bafe GH dous ble to the other angle,namely, mitotheangle g |, Fo And by ) 2. of y fourth in - os A BCDE m= feribe a triane YN Be AGD te. co. quiangle. ag A+ & al triangle GH So that let the an gle C AD be equall to the angle F ex the angle AC D nto the angle Gand likewife the angle CD A to the angle HW. met o. ther of thefe angles ACD, andCD As double to the angle € A D. ee (by the 9. of the first) either of thefe angles A C D,andC DA a ire ‘ Hi partes by the right lines E and DB: and draw right Saees "% ade B to C,from CtoD, from D to E,and from Eto A. And forafmuc fof of thefe angles AC @,andC D Ais double to the angle CAD: ana t are deuided into two equall partes by the right lines CE and D B; therfore the oe angles D AC, ACE,EC D,CDB;andBD A are equall the one to the o- ther. But equall angles (by the 26.ofthe third ) fubtend equall heceaae c Wherfore the fine circumferences A B;BC,CD,;DE, andEA ave me ! ‘ ~ one'to 9 other. And (hy ¥ 29- of 7 Jame ) nto equall circumferences sf [ubtede equal right lines: wherforey fine right lines AB,BC,CD Hive E . are ese ! ‘Jone to the other. Wherfore y figure A BCDE bawin g fine anglesisequt “- Now alfo Lfay that it is equian ole. For forafm Ge 4 the Gre si Te N fs Corftruttione Demonitra $i0tte fl ' * i Vind il ifs - i 1 \ile a | a 3 7 ie ¥ = a = —_ ae = ] i! , { i) | ; fis. . i | ni ibe ; { \ | ii { : i] { | i / ni | : : j | } ' i : iia | H { | ; ! iii) | } i] t : | i | : | I | | ’ | ‘ | i ! : i i i ‘ 1 ; \ | | } f ; a f | | ) | i a ; : : ily : 4 ¥ Rig. ; - ; > mI - n M4 +i ; ’ ta! bai : a i i ‘ ; ; 7! 4 4) MBAS: i . | " 1} ' ‘ \ f t \ ’ vy | ry | : Uy ane | a H : Mh y My my) | ~ ‘ } an i - i My } il ! he : 1 eee i Wry vs i ne | 7 he: , * F F | ; ) ated, f Ths Ling mt f oy ; ih. aah! - | | inte Be While a ; ae ut \i amv ith Dut WH ] ba iy qth ' "i i} ay iat | Hi wi +} hal Vi f ithe aE a i) ' : ‘|e ee Hi] : ye ii ! | : wy 4 4% } ee | a, 7 ' F hit) a } 6 tL hy j h ah ! ; : fi ' i 1 ‘ ail 4 i: ae ut ) hg ui] a ‘ 4 \ Hel tA i 14 \ ‘ | vot f , | at Ba | i! yay ie ee: ¥ Ty, { He y i f ie. eee aa ; i | iva H pit a i} i) We Mie ay d it UI | \ a) ef ee iq i ; \y 4 Bi) <0 iw { oy , w Mit | if ie a7 j ‘4 , By | ey ; ys ee ! { i” ot } * All i at’ b f ‘ epee } fone Ne ’ ‘4 My : 4 4 mie ; tian } H } ae. nt ’ wr) hit } ia ny eae . \ ti) : y at r , op b ‘et Dial | |) we) | a | Wg i { ! | 3 : ee : ; i\ We Oa ¥ {i ee en) we an | i j . 4 i] wi) 4 4 ete i : +) ba | Wet : : ite! an : if i ; i EP . if ss equoll tothe circumference DE, put the circumference BCD common bn to them both. herfore the whole circumference A BCD is equall to the whole aarcumference-E DCB: and vpon the circumference ABC D confisteth the ancie AED , ee i | | re Ra Be ireumferenc VID “AWD EDCB, cone : gba epee Byas e. fifteth the an- gle BAB, Wherefore the . | angle BA Eis equall toy ane gleAED(by the 27. of the third) and by the fame reas jon enery one of thefe angles 7 7 4 BC,and BC D,and CD E,is equal to enery one of thefeanglesB AE.and AED, Wherfore the fine angled figure ABCD EJs equian ib, and it is pros ued ,that is alfo equilater.Wherfore in a circle genen ts defcribed a figure of fine angles equilater and equiangle: which-was required to bé done. ysis kas : . ae et tee I P - " i ny renee ares: - Ss ss 2 - s ee ~ - . i —~ ——— oe eR = a Sa ¥ oy OD —— ~~ ; > a -*, . = 2 ee < v > —— = —— = --- _— Se A738 ene —, > = om - 2s me a ee $a ee eee —— - _ a - ro eee See i tm oem eS eS . a —— 7 =~ —— : er - -+ <= = £ ~S—. oan —_ ae _ 2 wa = = —— SS — ~ = oe eee Aga = . ~— ss - — — = 2+ y Another way to do the fame after Pelitarins. 4 aw + ee An ether we Suppofe that the Ifofceles triangle defcribed by the former Propofition,be DE F, $s doe fa ‘etl fo that let eyther of the angles E and F bedoubleto the angle D : Andlet the cirele pte Ate uen be 4B C: thecé- ? after Pelita- tre wheroflet be K. +. - ™M beep And vpon’ the centre K defcribe the angle BKC equall toone of the angles E or F of the triangle DEF. And draw a\right line ‘ from B to C..ThenI , fay,that the line BC 1s the. fide of the Pen- tagon figure to be in- {cribed in. the circle ABC.Deuide the an- gle B KCinto twoe- quall partes by draw- ing the diameter et - KL. And draw thefe rightlines BA& HC. . Now théitis manifeft, (by the 20, of the .. third) that the angle ~~” : BDI Ath | 4 y Lina es | BR AW ‘| | Pei ay | 4 Py \ aAe | ‘ - eo, of Euchdes Elementes. Fol.t9 BK Cisdoubletothewhole angle Bef CiWherefore the whole angle BAC is equall to the angle D, ynto which angle, the angle BKC allo is double. And forafmuch as inthe triangle 43 Kthetwoangles A and B are (bythe 5. of the firft) equal! (for thelines K.4 an@KB are drawen from the centre) therefore (by the 32. of the fame) the outward angle BK Lisdouble to eyther ofthe inward angles K4£ and K BA. And by thé fame reafon,the angle C K L isdouble to eyther of the angles K AC and KC AWherfore fora{much as the two angles atthe poynt K are equall, the two an- gles Aand B of the triangle 4B\K; are equall to the two angles 4 and C of the trian- te AC K,theone to the other: and therefore (by the 26. ofthe firlt) thetwo bafes <4 Band AC are equall. Wherefore ABC isan Ifofceles triangle . And forafgiuch asthe wholeangle BAC is equall to the angle D, the two angles remayning ef 5 C and ACB, thall (by the 32.0f the firft) be equall to the two angles remayning PL& F. Wherefore the triangle 4 BC is equiangle to the triangle DEF. Andnow- you may procedein the denionftration as you did in the former, imagining firft the lines BG and CH tobe drawen. Here itis a pleafant thing to beholde the varietie of triangles : for in the trian- ele ABC either of the angles atthe point A is one fift part ofa right angle. Whet- by is produced the fide ofa ten angled figure to be inferibed in the {elffame circle: Which is manifeft if we imagine the lines BL and LC:to be drawens Forthe arke B C is deuided into two equall partes in the poynt L (by the 26.0f the third). So then by the infcription ofan equilater triangle, is knowen how to. infcribe an Hexagon figure, namely, by deuiding ech of thearkes fubtended vndeér the fides of the triangle into two equall partes . And fo alwayes by the fimple number of the fide,is knowen the double thereof : as by afquare is knowen an eight angled figure : and by aneight angled figure a fixtene angled figure. And {0 continually in the reft. Pelitarius teacheth yet an other way how to infcribe a Pentagon. Take the fame cir~ cle that-was before,namely, e4 BC, and the fametrianglealfo DE F. And (bythe 17. éfthe third ) draw theline e/e4 A touching the circle inthe poynt’ 4. And ypon theline ef Af and tothe poynt 4,deferibe( by the 23 ,ofthe firk ) the angle 27 4AB equall ro one of thefe two angles Eor E(eyther of which, as itis manifeft,is lefle then aright angle ) by drawing the right line e472 : which tet cut the circumference inthe poynt 8. Agayne, vpon the line AA and tothe poynt in it 4, defcribe the angle NA G-equall tothe angle MAB, by drawing the right line AC: whichilet:cut the ‘circumfererice in the poynt C, And draw a right line from B toc. Thent fay, that BC is the fide of a Pentagon figure to be ‘afcribed in the circle 4 BC. Which is manifett, sf we deuide the arke 7.2 into two equal! partes in the poynt H, and draw thefe'right lines A-HandBA;andif alfo we deuide the arke 4C into-two. equall: partes:in the oynt G, and draw thefe right lines AG and CG ..For.taking-the quadrangle figure “ABCG, itis mamifeft (by the 32. of the third) that the angle ABC is equallto the al- teriate angle 4 Cyand therfore is equal to the angle £ ‘Likewifetaking the quadran- gle figure 4C 2H, the angle ACB hall be equall tothe alternate angle 44:4.B, and therefore isequalltothe angle F . Wherefore(by the 32. of the firft)as before, the tri- angle 4B C isequiangle tothe triangle DEF: And now inay you procede in the de~ monftration as you didia the formers® / Ae OSL SPSS Sah SS ST her. Probleme. The 12, Propofition. am | Gy.iy. ; Pe | ~ A bout acircle geuen, to defcribe an equilater and aquian- antsy Suppofe oa Jn other Yay alfo after PelitartHs. KID en . } al i \ Hh ‘ "~~ | : | } | | i : } | ' i J {i Conftractione Demon fra- Lion The fourth Booke tke Vppofe that the-circle ceuen be ABC DE. Itis required about the cire Sayicle 4 BCD E to defcribe a figure of, fie angles confifting of equal fides | and of equall angles ..T’ake the pointes of the angles of a fine angled figure deferibed (by the 1x. of the fourth) fo that by the propofttion goyng bee fore, let.y circumfertces A B,BC,CD,D E, and EA be equall the one to the oa ther. And by the pointes A,B,C, D,E, draw( by the 17. of the third )right lines touching the circle, and G let the fame beG HHH ... | K,KL,LM, andM o> fo E " oa eS G. And{ by the x of the third )take the centre of yin the circle es let the fame a | be F. And drawe right \> mM lines from F to B, from : F to K , from F to (, / f | a ‘ jrom to, and from \ coy ti FtoD. And foralmuch zB pes niew ah asthe right bne KL, ~~ \ po toucheth the circle AB se we 3 F K CD Einthe pointe (, and from the centre F dnto the point ( where the touche is is drawen a } . right line F (therefore (by the 18. of the third )F (is a perpendicular line ‘one to L.Wherfore erther of the an igtes Which are at the point.(is aright an ele, and by the fame reafon the angles which ave at the pointes D and B are right angles, dnd forafmuch as the an igle F (Kas a right angle, therfore the /quare whichis made of F K1s( by the 47. of the first.) eqnall to the fquares whith are made of F Cand CK: And by the ‘fame'reafon alfo the [auare-which is niade of EK isequall to the | quares which are made of FB and-B K. Wherefore.the fquares which are made of F Cand CK are equall to the {quares -whicharé made of EBand BK of which the /quare whichis made of F (is equall to the Jquare which is made of FB. Wherfore the fquare which is made of ( Keis-e quall to the fquare-which ismade of BK W berfore the line Bx is equal onto the line CK. And forafmuch as FB is equal bato F Cand F.K is common. ta them both therfore thefe two BF and F K ave equall to thefetwo C F er:F Ki And the bafeB K is equall ynto the bafe CK. Wherfore( by the 8.of the fir St) the angle BF K isequall vnto theangle K EC : and the angle BK F to the angle F KC. Wherefore the angle BE C is double to the an gle K FC. And the angle B K Cis doubleto the angle F K C, And by the fame reafon the angle C F D's double to the angle F Land theangleD L Cis double to the aiigle F L (. And forafmuch as the circumference B (is equall bnto the creme pee iar ? Cc L of Euclides Elementes. Fal.tr0, CD, therfore (hy the 27. of the third ) the angle B F' (1s equal to the angle C E®@. And the angle BEC is double to the angle K.F Cand the angle DFC is double to the angle Ls F C, wherfore the angle K F Cis equallynto the angle L FC. Now then there ave two triangles F K Cand F L C ,hauing two angles e quall to two.angles and one fide equall to one fide namely,F C;which is common to them both. Wherfore( by the 26. 0f the third,the other fides remaynmg are e+ guall ynto the fides remaynur sand the angle remayning nto the angle remay- ning Wherfore theright line K Cas-equall to theright lineC L,, andthe angle EK C6 theangle F LC. And forafnuch as KC is equall to € Ls therefore KL is double to KC ,and by the fame reafon alfo may it be prowed that 1 K. is double to BK. And fora/much as it is proued that ‘BK, 1s equall ynto KC, and KL is double to KC, and HK double to BK; therfore H¥ is equall vito KL. In like fort may we proue -y enery one of thefe nes F1G,6 M,cz M. Lis equall vnto either of thefe lines. K and KL. W herefore the fine angled figure G AK LM is of equall fides. I fay alfo that it ts of equall-angles. For forajmuch as theangle FISC 1s equall’ynto the angle ELC, and it 1s proued that the angle FAKL is double’to the angle F KC, and the angle KLM is double tothe angle F LC, therefore the angle AKL is equall to the angle K L M. In like [ort may it be proued that enery one of thefe angles KHG, HG M,andG ML is equall to either of thefe angles HKL,and SLM. Ww heres fore the fine angles GHK,AKLKLM,LMG, and MG H_¢ are equal the one to the other.Wherfore the fine angled figure GHK L Mis equiangle, and it isalfo proued that it is equilater and it is defcribed about the circle AB CD E: which was required to be done. q An other way to do the fame after Pelitarius by parallel ines. _ Suppofe that thecircle ge. nen be AB C,whofecétre let Bethe poynt F: and init ( by the former: Propofition ) in- {cribe ‘an equilater and equi- anole Pentagon ABCD E: : by whofe fue angles drawe from the centre beyond the circirference, fine lynes, FG, * ¢ FH,PK,FE,and FM. Andit.\ igmanifeft, that the fiue an-- “\ les at the cétre Fare equall, | len as the fine fides of the triangles within are equall, and alfo their bafes. It is ma- nifeft alfo , thatthe. fiue an- gles of the ‘Pentagon which are at the circumference, are deuided into, ten equall. an- eles (by the 4. of the firft) : Now then betwene the two 4 PLS RS okt lines FG and FH, draw the K w a Gg. iil}. line An other Way todo the fame after Pelita- tus. ; Dem Oi tra 7 EAC MH. Tbe fourth Booke ime GH parailel to the fide A B,and touching the circle ABC (which is done by a Pro- pofition added by Pelitarsus after the 17.0f the third). And fo likewyfe draw thefe lines HK,KL, and LM, parallel to ech of thefe fides B C,CD,and DE, and touching the circle . And for afmuch as the lines F G and FH fall ypon the two parallellines AB and G.H,the two angles FG H,& FH G, are equall to the two angles FAB and FBA, the one to the other (by the 29. ofthe firft), Wherefore ( by the fixt of the fame) the two lines F Gand FH are cquall. And by the fame reafon, the two angles FHK & F K H, are equall to the twoangles FGH ahd FHG the oneto the other: and theline EK is cquall to the line FH, and therefore is equallto the line FG. And forafmuch as the angles at the poynt F are equall, therefore(by the 4.ofthe firft)the bafe H K is equalt co thebafe GH. Inlikefortmay we pfoue, thatthe three lines F K, FL, andF M, are equall to the twolines FG and FH, Andalfothatthe two bales KL and L M, are e- quall to the two bafes GH and HK: and that the angles which they make with the lines FK, FL, and F M, ate equall the oneto the othet . Now then draw the fift line MG: which hall be equall to the foure former lines (by. the 4, of the firft) for thatas we haue ‘praned , the two lines FG & FMjare equal, & the angle GF M is equal to cuery oneof the angles at the poyntF. Thys line alfo MG toucheththe circle, ForH¢ vnto the point where the line LM touchcth the circle whi- che let-be N, drawe the lyne #N,Anditis manifett(by thé 18,of the third). that either. of the'angles at the poynt N, isa right angle’, Wherefore forafmuch.as.the-angle L of. the triangle FLN, is equall tothe angle M of the trian- gle FMN, &theangle N of . os He et) Beer the one,is equall to the angle £ Ste ait of os ek N of the other: and the lyne K ae L .. FN Is cOmon to thé both, theline NL thall (by the 26.0f the firft) be equall to theline NM.And fois the line ML denided equally inthe poynt.N. And forafmuch asthe three fides of the triangle FG P-are equall to the three fides of the triangle F M P,the angle P of the one fhal be equall to the angle P of the other(by the 8.of the firft), Wher- fore cither of théis aright angle by the 13.0f the fame).And forafmuch as the two an- gles F M Pand FP M of the triangle F M P,are equall to thetwo anglesF MN & FNM of the triangle F M N,and the fide F Mis common to them both, therefore theline FP isequall to the line FN. Butthe line F N is drawett from the centre to the circumfe- rence. Wherefore alfo the line F Pis drawen from the centre to the circu mference.And forafmuch as the line MG is perpendicular to the line F P, therefore (by the Corolla- ry of the 16. of the third) it toucheththe circle. Wherfore the Pentagon GHKLM circum{ctibed about the circle is equilater : itisalfo equiangle, asitiseafie to proue by the equalitie of thehalfes : which was requiredtobedone, = $a The 13, Probleme, The 13. Propofition. An equilater and equiangle pentagon figure beyng penen, ta defcribe initacircle, oa — Suppofe of Euclides Elementes. Fol.12. Ba ppofey Hequilater cr eqniangle Pttags figure gene be ABCD E.It : is required in the Jaid fine angled figure ABC D.E to defcribe a circle. ese Deuide( by the the 9.0f the first Jerther of thefe angles B C Dyand CD E intotwo equall partes by thefe right lines C F and FD and from the point F where thefe right ines CEand DF meete.Draw theferightlines F BLE A, FE. And forafmuchasB Cis equal pntoC D and C F is'common tothem both. Wherefore thefe tio A lines BC and CF are equal to thefe two lines. @ Cand C F: and the angle BC F is equall to theangle D C F.Wher fore (by the 4. of the first the bafe BE 1s e- quall tothe bafe DF, and thetriangle BOF is equallto thé triangle , DCF, and the an: gles remayning are/ez quall ‘ynto the angles remaynyng the one to the other , bnder which are [ubtended e- = ae quall fides. Wherefore the angle CBF iis equal to the angle C DF. And fora/much as the angle CD. Eis double to the angle CDF. But the angle C ®D Eis equallto the angle. AB C,and theangle CD F is equall to the angle C BE. wherefore the angle B Ais double. totheangle CBF... Wherefore, the angle AB F is equall tothe angleF BCA herefovethe angle A BCuis deute ded into two equall partes by the right line BF In ‘Dyke forte alfo mayat be proned that either of thefe angles B AE, and AED ave denided imto'two e- quall partes. by either of thefelmes F Aand EF. Drawe (by the 12- of the fir ft.) from the pointe E to.the right lines AB; BC, CD, and EA perpendi- culardines EGF AF K yE Land. FM. And forafmuch as the angle J.C F soyriall tothe angle KeCF; andthe rightangle Ft € a equall.tathe right angle F K C: now then there are two triangles F FIC, and F K €hawne two angles equal! to two angles the one to the other, and one fide equall to one fide : for E Cis-commondnto.themboth and isfubtended ynder one of the equal angles. Wherfore( by thé 26.% f the first) the fides remajning areeqrall pnto the fides xemayning W- bers oxé the per pendiculer FEZis equall ynto the perpendicus ler P Ke And in like fort alfo may it be proned that enery one of thefe lines FL, F M,and F Gis equall to chery one-af thefelines F FLandFK:W. hévfore thefe fine right ines FG,F HFK , FL, and FM are equall the one to the other. there Demonfirae 130 Ne — ¥ = = ~ a ne ee ______ - = _— — = ee — ~ oe > 5. re we +2; > > - ~* Pw. . <-—-* J 2 m: ~—t = — * ~ — ~ senaaianeaieeedtien - SPF. SEER OME Se +S Ee ES ee Oe eee ae I a ee eee . : a Say). -ocba eae barry Se EE =e J - —-- + —————— SS arene = —— = —————— SS —» ee - = ee —_ ss ze ae. a — “= = ” “ = — = ss > - = ~ ~ — —— —_——————— — ————o — a ~——— * “hese ve es ad x > ? —_ = Iie A = by tate s Se: WP Se = — — a SS a a 2 ae Po BS} Altes = = ee aa = hr ie SSS — _ Demonfirae tion leading to an abfure dities % Corollary. tines AB,BC,CD,D T he fourth Booke Wherfore making the centre F and the fpaceF G,or F Hor F K or FL, or F M. Deferibe a circle and it will pafSe by the poyntesG,H ,K ,L,M. And [hall touche the right ines A BLBC,C D,DE,and EA (by the correllary of the 16 of the third ) for the angles which are at the pointes G,H,K ,L,M, are right angles.For if it do not touche. them buticut,then from the ende of the diameter of the circle {hall be drawen a right line making two right an gles and falling with in the circle : whiche ts “1 (by the 16.o0f » third) | proued to be impofsible | Wherefore makyng F thecentre, andyfpace | one of thefelynes, FG, », FHF K ,F L,FM, defcribe a circle and it fhall not cut the right E,and E A.Wherefore = &' (by the corollary of the 16. of the third )it fhall touche them as it is maz nife/t in the circle GFA K_L M. Wherefore in © the equilater and equt- | S angle pentagon figure geuen,is defcribed acircle:which was required to be done, K D By this propofition and the former, it is manifeft that perpendicular lines drawn froin the middle poyntes of the fides ofan equilater and equiangle Pentagon fi- gurée.and produced, fhall paffe by the centre of the circle,in which the fayd Penta- gon figure is infcribed, :and fhallalfo deuide the oppofite angles equally; as here, A Kis-one right line,and deuideth the fide C D and the angle\A equally. And fo of the refi which is thus proued. The {pace about the centre F is equall to foure nightangles; which are deuided into ten equall angles by ten right lines metyn together in the point F.Wherfore the fiue angles AF M)M FE,EF L,L FD,an DF K are equall to tworight angles, Wherfore (by the 14.0f the firft) thelynes A F and F K make one right lyne. Thelyke proofe alfo will ferue touching the reft of es lynes, And this is alwayes true inall equilater figures which confift of vne- uenfides; > yea : v jaeThe 14--Probleme, The 14. Propoftion eAbout a pentagon or fieure of fine angles geuen beyng equi later and equiangle,to de/cribe acircle. . Supe of. Euclides Elementes. eee gat ES ax wl“ ppofe that y Pentagon or figure of fiue anoles. geticn being of equal A Ma fides'and of equal dies (or ee t 3 vehi sie he ya t Mr Pentagon ABC DE, to defcribe a circle. Denide( by the 9 vof the first ether of thefe angles BCD; GC D E intotwoequall partes. Conftruttion, by either of theferight lines CF and DF. And from thepoynt F where thofe right lines meete draw bnto the pointes B,A,E, thefe right lines F BLE APE. Demanjiats And tn like fort ( by the Pree iis pofition going before ) may it be proued., that euery one of thefe angles CBA, BAE, and AED is denided into two equall partes , by-thefe right ines FB, FA, @ FE. And for afmuch as the angle BCD 4s equall to the angle CDE 37 the halfe of the ans gle BC Dis the angle FC D, and likewife the halfe of the angleCD Eisy angle CDF. Wherforethe angle FC D is equall to the angle FOC. Whereforey fide FC 1s equall tothe fide Fk D.Enkke fort al- : fomayit be proued that enery one of thefe lines F BF A,and FE is equall to ee uery one of thefe lines FC and FD .Wherefore thefe fine right lines F A,F B; ECF D,and. FE: are equal the one to the other Wherefore making the centre Band the fpaceF A jor: EB or F Cor E Dior F E.Deferibe.acircle and it will paffe by the pointes A,B;C/D5E; and {hall be defcribed about the fine angled rune d BODE which ts equiangle andequilater. Let this. circle be defcris bed andlet the fame be ABC DE. Wherefore about the Pentagon geuen bes tie both ejuiangle and equilater j1s defcribed acircle : which was required to be Woe st OMS SSS PATS SD olor DIAGN S84 | oo oD be 15. Probleme... Che vs. Propofition. ~ * aRogerativele gene Matar te ain me feat cele setae Re gWilatcr And CPMAMBLey soa. hot hang “ Tea ps See SoC Hoth ORIN ey OS IS « | vS 7 NGM ppoferhat the circle venen be A BEDE Fo itis required inthe tire Ni? Cle. genen' ABCD Exodeferibeaficure of fixe angles of equall jides “ andofequallantles: Draw the diameter of the circle ABCD E F,and let 28) Lhe fourth Booke.. : Conftrnction. let the Jame be AD. And ( by the first of the third take the centre of the circle | and let the fame be G. And making the centre D, and the fpace DG, defcribe mV | ¢by the third petition) acircle CGE. 1: and drawing right linesfrom E toG, sett and from G to C extend them tothe pointes B and F of the circumference of the right lines AB, BCLCD, circle geuen. And draw thefe 3. —. DE,EF,andF A. Then A ay | I fay, that ABCDEF is Ta | an Flexagon figure of ts COA quall fides and of equall ane Demonfirae gles. For fora/much as the ston" point G is the centre of the circle ABCD EF, there: fore (by the 15 definition of the first) the line G Eis es | guall pnto the line GD. A+ | gaine forafmuch as the point a hi Dis the centre of the circle ; a, CGE FX( therfore by the ‘i felje fame jibeline D Eis e | guall ynto the line DG-And | it is proued that the line GE is equall Ynto the lne Gs Doi herfore thelineG E is equall nto the lunkek Df by ‘ , wtp fe.d ave equalltthe one.to the other, But wndenegrall tircumferences are fubtens é : a x ee of Evclides Elementes. Fol.123. ded eqnall right lines (by the 205.0 \f the fame. ) Wherefore thefe fixe right lynes #B:BC COD EEF yand F A are equall the one to the other. Wherfore the Hexacon ABCD EF is equilater. I fay alfo that itis eqniangle. For forafe much as the circumference A£4s.equall puto the circumference E D_adde. the circumference AB CD common to them both. Wherefore the: wholecircuinfes rence. ABCD is equall to, the whole arcumference EDC B A. And Dppon the circumference F ABC D confisteth theangle F ED: and vppon the.cirz cumference E D CB Aconfifteth the angle AF E.W herefore the angle APE és equall to theangle DE Fn like fort alfomay st be proned that the ref? of the angles of the Hexagon ABC DEF, thats eneryone of thefe angles iA B,,ABC,BCD,andC D Eis equallto enery one of thefe angles APE; and FE D.wherfore the Hexagon figure A BCD E Fis equiangle, and it is pro- ned that it is alfo equilater and st is defcrtbedin the arcleA BCD EF, whers fore in the circle ceuen A BCDE Fis defcribed a figure of fixe angles of equal jides and of equall angles. Which was required to be done’. ~ q An other way to do the fame after Orontius. ant Xs ol 7 | es An ofl, Suppofe that the circle geuen be ABC DEF in which firftletthere be deferibed 5, pbs an equilater and equiangle triangle ACE (bythe fecond of thys booke). Wherefore after Orbsinse the arkes ABC,CDE,EF A ate (by the 28.of the third )equall the one to the other, Deuide every orice of thofe three arkes into two equall partes(by the 30,0f the fame)in the pointes B,D,& F, And draw thefe right ee lines AB,BC,CD,DE,E F,and F 4. Now. then by the 3.definition ofthis booke there fhall be defcribed in the circle geué an Hex- agon figure 4 BCD EF, which muftnedes be equilater: for that euery one of the arkes which fubtend the fides thereof areequall the one tothe other. I fay alfo that itis e¢- quiangle . For euery angle of the Hexagoft figure is fet vpon equall arkes ,namely,vpon foure fuch partes of the circiference wher- of the whole circuinference cotayneth fixe, Wherfore the angles of the Hexagon figure are equall the-6ne to the other (by the 37. ofthe third).Wherefore in the circle geven eft BCDE F ts infcribed an equilater and equiangle Hexagon figure : which was ree quired to be done. q An other wayto do the fame after Pelitarins. Suppofe that the circle'in which is to be infcribed andequilater & equiangle Hexa- Ay osher gon figure be .4 BCD E, whofe centre let be F. And from the centre draw the femidia- ‘Way efter meter F 4. And fromthe poyntA apply (by the firft of thys booke)theline A4Bequall Peltarims. to the femidiameter. Which Hay-is the fide of an equilater and equaness Hexagon fi- gure to be infcribedin the circle -4BCDE. Draw &tight line ftom F toB. And for afpauch as the line 4B isequall to theline F A, & itis alfo equall to = line FB, a —— | . Hh.j. ore The fourth Booke fore the thidngle <7 F Bis equilater, and’by the sfofthefirt equiangl¢. Now then vp- onthe. centre’F defcribe the argle BFC equall to the angle e4 FB, or to the angle PS Atwhich isallonc) by draving the right line F GC. And draw aline from B ro C.And forafmuch as theangle «47 F 3 is the third part oftwo Mishtaneles by the’ s.and'3 2. ¢fthe Arlt; the anele BFC allophallbe the third part.of tvorightangles » Where-. fore cither of the two angles renaining F BCand FCB, for afmuch as they ate equall by the 5,of the firft, fhall be two third ‘partes: of two right angles (by the 32, of the fame), Or (by,the.4.ef the frit) forafmuch as the an- gle BF Cisequail tothe angle IZ e74,and the twofides FB and FC are eqtialfto the tvo fides_4 Band B F,the bale BC fhalbbe equal! tothe bafé BF, and therefore. isequall to the lite FC, Wherelore the triangle FB Cis equilater andequidngle . Laftl. make the angle CF D equall to'eyther of the angles atthe poynt F, by drawing - theline F‘D.. And drawalinetroniCto'D. Now then by the former-reafoh thetriande CD thall be equilater andequiangle,. And for af- much as the three angles at the soint F are equall to two right angles ( for ech of then is the third part oftwo right angles) therefore ( by the 14.0f the firft)e4D is‘one right line : and for that caufeis the dameter of the circle . Wherefore if the other femicircle ef F Dbe denided into fo, miny equall partes as the femicirclee 4B C Dis deuided into, it {hall comprehend fominy'equail lines fubtended ynto it. Wherefore the line A B is the fide of an equilater Eexagon figure to be infcribed in the circle : which Hex- agon figure alfo thall be equiangle : For the halfe of the whole angle B is equall to the half of the whole angle Cs whch was tequired to be done. -Now.then if we draw fromthe centre F a perpendicular line ynto 4 D, which let be F Esand draw alfo thefe rigtt lines BE and'C E : there hall be deferibed 4 triangle BEC whole angle E whichis asthe toppe fhall be the 6, part of tworight angles by the 20.of the third. For the anzle B F Cis double vnto it. And either of the two angles at the bafe,namely,the angles! BC, and EC Bis dupla fefquialter to the angle E:that is cyther of them-contayneth tie angle E twife and halfe theangle E . And by chis rea- fon was found our the fide of a1 Hexagon figure. Sp Correlary. Hereby it is mant- ‘e nz fest,that the fideo |. f an Hexagon figure v gin arn ; defcribedinacircl ’ ¥ ; De is equall to a right . line drawen fro the cetre of the fatd-cir- _cleynto the circum- LO ES ference. And ify » othe pointes A; B; ~ C;D; BF y B62 of Euclides Elementet. Fool.12.4.. »\s ramen right lines touching the circle, then fhall there be ° “ delcvibed about the circle an Hexagon figure equilater and \o eguiangle , which may be, demonftrated by. that which hath «: \ibenefpoken of the deferibing of a Pentagon about a circle. ~~ & F BS aye DoNSLOS circumference i Bbeing the cs fi part of a: circle [hall contain thre, ‘wherefore the refidue BC fhall containe two. Deuide (by the 39. of the first) the arke BC into two equall partes in the point E.wher fore either of Gg E Cis the fiftene part of the circle ABC D.lf therfore there be drawn right : —— HAiby dines. a C owtrultiote DemonS nag: thee ee en ferencesBE, ... } |) hi) ail) I] Ai 11 | | - i } t : i bie Hit | i i ite ' a Ith i He 1h ar SS ———s — eee os = ae ————— woe Lhefourth Booke tutes.froip Bt E aud from-E.to C andthen beginning atthe point B. or at the * point C there be appliedinto the circle ABC D right lines eguall ‘ynto EB or = Se ee Te ae —— ie > ~ -- - ++ = : 3; a a a FREE os . ——— ma rs Sa i ee a — “ " _ — — - - Fe = " ~ =< —— = = — -—- 4 = — ————— ——— —_—-— ~ ~ ; - ——— " ‘ ] EC.and fo continying- till ye come to the point Cif you began at B, or top point Bifyou besan at C anid there phall be deferibed in the circle ABE D a figure of fiftene-an: cles equilater-and eguiangle: which was required tobe done: And in | like fortas.in a pentagon if by the pointes where the circle is deuided ;be drawen right lines touching the circlein the [aid pointes there fhall be defcribed about § circle a figure of fiftene an 7 atcha’ eH equiangle. And in like fort by} felfe Jame obferuations that were in ~entagons,wemay in afigure of fiftene angles genen being equilater and equiangle either infcribe or circumfcribe a circle. a) { SE 4 | / ah 1 i 1B! bi (| ' { ; nl " sy (Be f f ey } 4 y vie | HI } f Sta ie | % Hi} { bat) | | | iia bar | t } ! t \ j An ddiises . Y An_addition of Fluffates to finde out infinite figures of of Finfsatese many angles. "A Poligoncn If into a circte from one poynt be applyedthe ides of two * Poligonon figures : the ex- figureisafi- . wolfe ofthe greater arke abouethe-leffe, fhall comprehend an arke contayning fo many Bure confifting “Tides of the Poligonan ‘figure to be infcribed by box many Unities the. denomination of of many fides. the Polrzonon froureofsbe leffe fide excedeth the denomipation of the Poligonon feure of the-eréater fide: and the number of thefides of the Polizonon figure'to be infcribed | 8 produced of the multiplication of thedenominations of the forefayd Poligonon fieures PUES the One iptd the other. i bydrisl hen too cee aS Asforexgmple. Suppofethatinto the circle ARE beapplyed the fide ofan equi= laterand equiangle Hexagon figure (by thers -Oftthys booke) which let be-e4B: and. likewife the fide ofa Pentagon (by the 11.o0f this booke)which let be 4C: and thefide of a fquare (by the 6.0f thys booke) which let be 4D: and the fide of an eguilater tri- angle (bythe 2.ofthis booke) which let bee4 E, Then I fay that the exceffe ofthe arké e4 Daboue the arke 4 B, which excefle is the arke B D,contayneth fo many: fides of the Poligononfigure to be infcribed ,ofhow many vnities the denominatorofthe Hexe agon 4 B, which is fixe,excedeth the denominator of the {quare .4 D, which is foure. And forafinuch as that exceffe is two yni2 ‘ Py ties therforein B D there fhall be two fides, * A And thedenominator of the Poligonon fi- a~<> gure which is to be infcribed (hall be pro- duced of the multiplication of the deno- minators of the forefayd Poligonon fi- sures,namely, of the multiplication of 6. into 4.which maketh 24. which number is‘the denominator of the Poligonon fi- _ gure, whofetwo fides fhall fubtend the arke BD . For offuch equall partes wherof the whole circumference cotayneth 24,0f fuch partes E fay, the circumference 4B con- tayneth 4, and the circumference e4 D contayneth.6 . Wherefore if from ef D Ds which fabtendeth 6. partes be taken away Soo" 4, which e4 B fubtendeth, there thall re- : a 3. mayne vnto BD two of fiich partes of —. - | | which the whole contayneth 24. Wherfore | ofan Hexagon and a fquareig madea Poligonon figure of 24. fides. Likewyfe of the . Hexagon ef Band of the Pentagon eC thall'be made a Poligonon figure "s hi Fa . cs, oo ms | of Cuclides Elementes, Fol.zs. fides; one of whofe fides thall fabtend the arke BC. For the denomination of 43 whichis 6. excedeth the -denomination of «4C whichis 5. onely by ynitie. So alfo forafmuch as the denomination of «4B which is 6. excedeth the de- — nomination of 4 E which is 3. by 3.therefore the arke 8 £ {hall * gontayne 3. fides of a Poligonon figure of 18. fides, And obferuing thys felfe fame methode and Order, a man may finde out infinite fides of a Poligonon figure . (-°.) $5 The end of the fourth booke of Euclides Elementes. Y : 4 gre 9 . — m r al ne 4 ,s > . bi a —_-= - — = = 2 any = = —— —ens = peat — oe Pee ee a = =: es ———— a SR SS ae re qT he fifth bookeofEu-_.. clides Elementes. HiS rrirrusooxs of Euchdeis of very great commoditie and vie in all Geometry, and muchidili- The argument DY - ; | gence ought to be beftowed therin.It ought of all o- Vs _ vB) 4 ther to be throughly and moft perfectly and readily l\artes:as Mufike, Astron omy ,Per{pectine, Arithmetique, \}/ the arte ofaccomptes and reckoning,with other {uch A> like. This booke therefore is as it were a chiefe trea- fure,and a peculiar iuell much to be accompted of. it entreateth of proportion and Analogie, or proportionalitie, which peftayneth not onely vnto lines, figures, and bodies in Geometry : but alfo vnto foundes & voyces, of which Mufike entreateth, as witnefleth Boetius and others which write of Mulike. Alfo the wholearte of Aftronomy teacheth to meafure proportions of tymes andmouinges. Archimides and Iordan with other,writin g of waightes, athrme, that there is proportion betwene cf and waight, and alfo betwene place & place. Ye fee therefore how large is the vfe of this fift booke. Wherfore the definitions alfo thereofare common, although hereof Euchde they be accom- f modate and applied onely to Geometry. The firft author of this booke was as it The frit in: seated” of many,one Eudoxus who was Platos {choler,but it was afterward fta- beck Be. med and putin order by Eaclide. doxis. ee Sap ‘Definitions. Tie firft defi sisted eA parte 1s 4 lee magnitude in re[pett of a greater magni. tude,when the leffe meafureth the greater. As in the other bookes before, foin this, the author firftfetteth orderly the definitions and declarations of fuch termes and wordes which are neceffarily re- quired to the entreatie of the fubieét and matter therof, which is proportion and compariion of proportions or proportionalitie. And firft he fheweth what a parte is.Hiere is to be confidered that all the definitions of this fifth booke be general to Geometry and Arithmetique,andare true in both artes, euenas proportion and proportionalitie are common to them both, and chiefly appertayne to number, neither can they aptly be applied to matter of Geometry, burin refpect of number and by number. Yet in this booke,and in thefe definitions here fet, Euclide emeth to fpeake of them onely Geometrically, as they are applied to quantitie continu- allas to lines, fixperficieces,and bodies: for that he yet continueth in Geometry. I wil notwithftanding for facilitieand farther helpe of the reader,declare thé both A parttaken by example in number,and alfo in lynes. | two maner of For the clearer vnderftandyng of a parte, it is to be noted ,thata partis taken in The ff way, ‘he Mathematical Sciences rwo maner of wayes . One way a partis a ar ape of Euchides Elementes. Fol.126. titie in refpect of a greater, whether it meafuire the greater or no. The fecond way, a part is onely that lefle quantitie inrefpett ofthe gteater, which meafureth the greater. A lefle quantitic is fayd to meafure or numbera greater quantitie, when it,beyng oftentymes taken, maketh precifely the greater quantitie without more or lefle,or beyng as oftentymes taken from the greater as it may,there remayneth nothyng. As fuppofe the line AB to contayne 3. and the lyne C D to contayne 9. thé doth the line AB meafure theline C D:for that ifit be také certayne times,c iia cadihioie “) Oona ee. namely, 3. tymes,it maketh precifely the A 5 a lyne CD, that is 9. without more or | leffe. Agayne if the fayd lefle lyne AB be taken from the greater C D, as oftemas it may be,namely,3. tymes,there fhall remayne nothing of the greater. So the nil- ber 3.1s fayde to meafure 12. for that beyng taken certayne tymes, namely, foure tymes,it maketh inft 12.the greater quantitie: andalfo beyng raken from r2.as of. ten as it may,namely,4. tymes,there fhall remaynenothyng.And in this meaning and fignification doth Ewclide vndoubtedly here in this definea part : {aving, that itis a lefie magnitude in comparifon of a greater, when the lefle meafureth the ereater.As the lyne AB before fet, contayning 3. is alefle quantitic in compa rifon of the lyne C D which containeth 9. and alfo meafureth it.For it beyng cer- tayne tymes taken,namely,3. tymes,precifely maketh it, or tacen from it as often as itmay,there remayneth nothyng. Wherfore by this definition the lyne A Bisa part of the lyne C D. Likewife in numbers, the number 5. is a part of the number 15. for itis a lefle number or quantitie compared to the greater, and alfo it meafir- reth the greater : for beyng taken certayne tymes,namely, 3. tymes, it maketh 15. And this kynde of partis called commonly pars metiens or men[urans,that is,amea furyng part : fome call it pars meultiplicativa: and of the barbarous itis called pars aliquota,that is an aliquote part,And this kynde of parte is commonly vied in A- rithinetique. The other kinde ofa part,is any leffe quantitie in comparifon ofa greater, whe- ther it be in number or magnitude,and whether it meafure orno. As fuppofe the line A B to be 17,and let it be deuided into two partes in the poynt C, namely,in- to the line A C, & the line C B, and let the, , rh AR. REARS, NGS SS lyne AC the greater part containe 12.and let theline BC the leffe part contayne 5, Now eyther of thefe lines by this definition is a part of the whole lyne A B.For eyther of them ts a leffe magnitude or quatity in coparif6 of the whole lyneA B: burneither of the meafureth the wholeline A B:for the leffe lyne C B contayning 5. taken as oft€ as ye lift, will neuer make precifely A B which contayneth 17. Ifye take it 3, tymes it maketh only 15.0 lacketh it 2.0f17.which is to litle.Ifye take i 4.times,fo maketh it 20.thé are there thre to much,fo it neuer maketh precifely 17.but either to much orto litle. Likewife the other part AC meafureth not the whole lyne AB: for take once,it maketh but 12.which is leffe then 17. and taken twife, it maketh 22. which are more then 17. by 5. So it neuer precifely maketh by takyng therof the whole A B,buteither more or leffe. And this kynde of part they commonly call pars constituens,or componens: Becaufe that it with fome other part or partes,maketh the whole,As thelyneC B together with the line A C maketh the wholelyne AB. Of the barbarous itis called pars aliquanta.In this fignificationitis taken in Bar/z- wm in the beginnyng of his booke,in the definition ofa part,when befaith + Exery | Hh. iid. lefSe The fecond way. How aleffe quantity 1s faydte mea- {ite a greeters ig what fro~ nificated hue clide bere ta< keth & dite Drs metiens ov menlurans. Pars multipls Catlva, Pars aliquota, T 4s binde of part comonly Go fedin Arith- metigue. Fhe other kinde of parte Pars conflity- CNS,0F Com pow CIS 6 Pars aliquan- td; | The fecond a C/iNttiOR Numbers ve~ ry neceffary for the ynder- landing of this booke and the other bookes follo- Win?» The third de« finitione what propor- £1011 80 Thinges com= pared together this WAYESe Example of shes definition in magnt- suade Se T he fifth Booke: oe number conipared to a greater, is [ayd tobee part of the greater, whether the lefve mea- 4 ‘ure the greater ,or mcalurest not. Multiplex is a greater magnitudes refpett of the lefe,when the lefse meafureth the greater. As the line C_D before fet in the firftexample,is multiplex to the lyne A B.For that C Dalyne contayning 9.is the greater magnitude, and.is compared to the lefle,namely,to the lyne A B contayning3. and alfo the lefle lyne A.B meafureth the greater line C D : for taken 3. tymes, it maketh it, 2s was aboue fayde. So ¢ Ons pene; in numbers 12. is multiplex to 3: for 12 is 3 the greater number, and is compared to A ee? the lefle,namely,to 3.which 3.alfo meafu- reth it: for 3 taken 4 tymes maketh 12. By this worde multiplex which is a tetme proper to Arithmetike and number,itis eafy to confider that there can be. no ex- act knowledge of proportion and proportionalitic,and fo of this fifth bookewyth all the other bookes followyng, without the ayde and knowledge of numbers. ? Proportion 1s a certaine refpette of two magnitudes of one kinde, according to quantitie. | “ Euchdeas in the firlt definition,fo in this & the other following,and likewife in all his Propofitions of this booke, mentioneth onely magnitudes, and geucth his examples and demontftrations of lines : for that iaere inthe 4. bookes before he hath entreated of lines & figures, and fo c6tinucth in his fixth booke following after this,comparing figure to figure, and fides of figures to fides of figures, with- out mention of number at all . Notwithftanding as itis fayd they are. generall to all kinde of quantitie, both difcrete and continuall , namely , faites and magnitude : and neede for the young reader and ‘ftudientin thefe artes to be de- clared in both. For, the opening of them in numbers(in which they are firft and naturally founde) geueth a great and marueilous light to their declaration in magnitudes. Proportion (fayth he) is acertaine behauiour,that is, a certaine refpect or comparifon of two quantities of ome kinde : as of one line to an other, and one figure generally to an other, and onenumbef to an other, as touching quantitie, that is to fay, thatthe quantitie compared, is to that wherunto itis compared, eyther equall’, or greater, or leffe then it . For after thefe three maners may thinges be compared the one to the other.But quantities of diuers kindes can not be-compared together. A fuperficies can notbe compared toa line : nor number toa body : nora body toa line or number: for that they are not of one kinde. For example of this definition,take two quantities, namely, two lines AB and. CD, and compare the one to the other, namély,A B to C D according to fome certaine tefpect of greatnes, A or leflenes, or equalitié. namely, in this example, ler AB be greater then’ C D, & containe it twife. Now chys comparifon,relation,or refpect of AB to C D, eae 3 and generally of any one quantitie to any other,is called proportion . Likewife ts B. ENS Mi) Pe itaf - of Euclides Elemente. Fol.12.7. . ~ as in the third:example.The likeis italfo in nunabers- | ; * Dupla fe[guse ferlias Tripla [e[qus- gqvAarla. Multiplex fw perpartient ° Dupla [uperti- partsens. Dupla [upertrs partiens . eee y ripla f Hper@ guadriparticns. Tripta feper- gusintipartsens, ee rate’ -Thefifeh Booke proportion tharnuinber hatirto number) the-proportion betwene them is irratioe nall.confufed,vnknowen,vneertaine; and furd ;'\Andsthis kinde:of proportionas found onelyin magnitudes, asin lines and figures (and notinnumbers) of which he of purpoie en treateth in his tenth booke.Wherfore I wil here omit to fpeake of it,and remit it to his due places And fomewhat will I now fay for the elucidation of the firft kinde. onion armie o OME bik 2: | | Proportion rationall is denidedinto two kindes.into. proportion 2 ~equalitie and into proportion of inequalitie. Proportion of equalitieis; when one quanti: tie is referred-to-arr other equall vnto itfelfe:asifye compare § to 5, or 7 to73; Xe fo of other.And this proportion hath great vfe in therule of Gof. For in itall the rules ofequatiotis tendeto.none other endebutto finde outand bring forth aniis ber equall to, thé numberfuppofed,which isto put the proportion of equalities: Propoftionof inequalitie is, when one vnequall quantity is compared: toamos ther,as the'greater to the leflejas'8. to 4: org.to 3>0r the lefle to the greateras-4s to 8:0r 3.0 9 oO} bOI (i) sanseriroes Prdportionof the greater to theleffe hath fiuekindes,namely, Multiplex, Supers particnlarss uperpartiens,M ultiplex ‘fuperperticular,and M ultiplex fuperpartiens. Niuleplex,s whemthe antecedent: containerh unit felfethe confequent cers tayneétimes without more orlefle:.as twiée, thrice, foure tymes, and fo farther. And this proportion hathynderit infinite kindes.For if the antecedent contayne the ¢onfequent inftly twile,itis:called duplaproportion,as 4: to 2. If thrice tripla; as. 9.to 3.1f4,tymes quadrupla as 12. to 3.1f5.tymes quintuplaas 15.to 3. And fo infinitely after thefamemanen.: » 31978 ite deyerhuy ) Superperticular is, whé the antecedét containeth the confeqirentonly:once,& moreouerfome One part therof;as-an halfé,a third,orfourth;&e. This kinde alfo hath vnderit infinite kindes.Forifthe antecedent containe the confequent Once and.an halfejtherof itis called Se/quialtera,as:6,to 4:if once and a third part Sefqute tertia;as 4.to 3sif once andafourth part Se{quiquarta, as5. to 4.And foinlikemas nér infinitely. 1D 3a | y Superpartiens is, whéthe antecedent cétaineth the confequent onely once, moreoner more partes theri:one of the fame,as two thirdes, three fourthes, foure Efthes and fo forth. This.alfo:hath infinitekindes vnderit.. .Forif the antecedent containe adboue the confequent two partes;iris-called Superbipartiens, as 7. to.5.1f 3. partes Supertreparticns aS TtOcds If'4. partes! Superquadripartiens; as 9. to 5. Tf 5. partes Superquintiparticns as Ut0 6. And fo forthinfinitely. = Multiplex Sup erperticularis when the antecedent containeth the confequent more then once; and: mofeouer onely one parte ofthe fame. This. kinde likewife hath inSnite kindes vnder it.Forifthe antecedent containe the confequent twile and halfe therofitis called»duplaSe/quialtera;as 5, to 2.1ftwifeandia third Dupla Sefquitertia as'7.00 3.1f thrice andan halfe Tripla fe(quialtera as 7.t0.2.1ffoure times and an halfe OvadruplaSe{quialtera,as 9.to2. And io goyngon infinitely... > MultiptexSuperpartients: when the antecedent contayneth the confequent more then once,and alfo more partes then. one of the confequent. And thiskinde alfo hath infinite kindes vnderit«For1fthe antecedent containe the confequent twife,and two partes oueryitisicalled duplas uperbipartiens as 8. to 3.1f twice and three partes,dupla Supertripartiens as 11.00 4. If thrice and two partcs, it is named Tripla Superbipartiens a8 11, to 3. If three tymes and foure partes Tripla Swper- uadripartiens as 31. to 9.Andto forth infinitely. | Here is to be noted that the denomination: of the proportion betwene any two. numbers, is had by deuiding of the greater by the lefle. For the sl or a, icist number . peel, ~ > Ss > = ™ of Cnclittes Blenvontes. Fol.t28. number produced’ofthat diuiiien is cues the dehorhifiation UF the proportion. Whichin the firft kinde of proportion,namely,multiplex,is euer a whole number, and inallorherkindes of proportion itis a brokenmumber. 7 io poAsafye wilhknow:the denomination of the proportion betwene 9 and 3De- vide by 3:fothalky chatieinithe quotient 3,whichis. awhole number, amdis the denoniination ofthe proportion and fheweth that the proportiombetweneigsX 3018 Etipla.So theproportion hetwene.12,and 3. 1s: quadrupla, for that 120 beyng deuidedi by zethe quotientss 14: and foof others thekindedfmultiples:And 2i- though in this kinde the quotient be euer a whole number,yet properly itis refers red to vnitie,and fo is reprefented in maner of a broken number asi-and.it for we nitie is the denominaton to a whole number, 7 Likewife the denomination of the proportion betwene 4 and Pis'1. *. for that . ; - » . . = . 4 i ¥ “ 4 deuided by 3. hath in the quotient 1 — one and a third part,of which third part, : 5 : bs ee " itis called fe(quitercia; fo. the proportion betwene,7,and 6.is.1-—oneand.a fit, of which fixyparvitis called fefyaifexta,and.fo of other of thatkinde: Alfo betwene and ‘the denomination ofthe proportion is 24. one and twofifthes, which des ; . . ~ ~~. ee _“ * 5 : j > nomination Cofifteth of rwo parts namely ofthe munerator and denominator’ of the quotienrof 2and s:6f which two fifthes it is called /upérbipartiens quintas:for 2 the numerator fheweth the denomination of the number of the’pattes,and §. the denominator,fheweth the denominati6, what parts they are,& fo of others. Alfo fie denomination betwene s5.and.2,4s'2:-* two anda halfe,: which confifteth of a whole number and.a’broken, of 2.the whole number itis dupla,and_ ofthe halfe, it is called /efqusaltera, {0 1s the proportion dupla fe[quialtera. - Agayne,the denomination of the proportion betwene 11, and 3. is3 *. three and.two thirdes,confifting alfo of a whole number anda broken, of 3. the whole gumbre itis called tripla,and of — the broken number, itis called Superbipartiens .— ihe ie : - . : eB Pause . : . _~ , tertias fo the proportion is tripla {uperbipartiens tertias » Thus much hethertd tou- ching proportion of the greater quantitie tothe leffe. Proportion of the leffe quantitie to the greater +hathas many kindes, as that of the greater to the lefle,which kindes are inthefame order : and hane alfo the felfe {ame names,but that to the names afore put ye muftadde here this word fab. As comparing the greater to the leffe, it was called! multiplex; [uperparticular, fuperpar tient, multiplex {uperparticular,and multiplex fuperpartient, now comparing the lefie wantitie ro the greater, itis.called /ubmultiplex; {ubfuperparticular, [ubfuperpartient, Aibcwnls ple feiiamaasnaie, and_/ubmultiplex fuperpartient .. And{o in like manerto allthe inferior kindes ofall fortes of proportion ye hall adde that worde /ud..The éxamples: of the former ferue alfo here; onely: tranfpofing the;termes of the pros portion making the antecedentconfequent; and the conféquent the antecedent: AS 4.to:22is-dupla proportion: fo 2.to 4zisfubdupla As 9.to 3.is tripla: {0 is 3.to gfubmipla;And.as.9.to 6: is fefquialterayfo 6. to.gsis fubfefquialtera. As 7. to 5. isfuperbipartiens quintas,{o is 5.to.7.{ubfipérbipartiens quintas.As 5.to 2.is dupla fefquialreta; fo is.2.t0 §.{ubdupla fefquialtera, And alfo as 8.t0 3.is dupla furperbi- particns tertias; fois 3.t0:8 -{ubdupla fuperbipartiens tertias. And fo may ye pro- tede infinitdlyinall others; Thus much thoughtI good in this place for the eafe of the beginnento be added:touching proportion. = ‘Pre- How tokvow the denonii- nation of any proportion. Proportion of the lefSe in the greater. Submulte- plex. Subfuperpar- ticular. | Sub/uperpers tient.oe* Fhe fourth é: fibteOne Example of ehis definction in magnt- vudes. Example thereofin auimabers. Note. The fifth defi- $3L20Re An example of this acfint- tion in mag- Ritudel. FV hy Exctide BH defining of Proportion vied mutlis- pricatione 8: Zhe fifth Booke. oP roportionalitie,ss.a fimilitude of proportions.. As in proportion are compared together two'quantities,and proportion is no- thing els but therefpectand comparifon of the one to the other, and thefe quanti- lies are che termes of the proportion’: fo in proportionallitie are compared toge- thertwo proportions. And proportionallitie is nothing els,but the refpe& & com- parifon of the one ofthem tothe other.And thefe two proportions are the termes of this proportionallitie. He calleth irthe fimilitude,thatis,the likenes or idempti- tie of proport- qesdinutrsiod vy s ip 31 | ons:Asifyewil cOpare the pro- portion of ,the line A contay- hyng 2. to the — : | ine B contayning 1,to the proportion: of the line C ‘contaynirig ‘6, to. rhe line D contayning 3 ,cither proportion is:dupla. This likenes, idemptitiejor equallitie of proportionas called proportionallitie. So innumber 9.t03.and 21.to 7. either pro» portion is tripla:, Where note that proportions. compared together, are fayd to be likethe onc to the other; but magnitudes compared together, are faid to be equal A C : . the one.to. the other, to Phofe magnitudes are fayd to bane proportion the: one tothe other which being multiplied may exceede the one the others’ Before he fhewed and defined,what proportion was,now by this definition he declareth betwene what magnitudes proportion falleth.faying: That thofe quanti« fies are {aid to haneproportion the one to the other, which being multiplyed,may excédethe one the other.As for tharthe | ) line A being multiplied by what foeuer muitiplication or nibet, as taken twife, thrifejor forre, fiue; or morte tines, or onceand haifejor once and a third, & fo A 3 ofany-other part, or partes, may excede EAT O and become greater then the line B» or contrarwile,then thefe two lines. arefaid to-haue proportion the one to the other. And:fo-ye may {ee that betwene any two.quatities of one kinde, there is.a propor> tion. For the onetemayning vamultiplied,& the other being certaine times mul- tiplied {hall be greater then it. As 3.to 24.hath a proportion, for leaning 24.vamul- tipliedsand multiplying 3.by 9, yethall produce 27 : which is. greater then 24,and excedethir. Hereisto be noted,that Euclide in defining-what quantities haue pro- portionjwas compelled to vfe multiplication, or els:fhould not his: definition be generallo either kinde of proportion : namely,to rationall and irrationall:to fuch proportion I fay which may be exprefled by number,and to fuchas cannot be ex preffed by any determinate number,burremaineth furd and innominable . In rati+ onall quantiues which haue onecommon meafure,the excefle of the one aboue the other is knowen,and by itis knowen the proportion, which may be expreffed by fome determinate number. But in irrationall quantities which haue no comon meafure,itisnotfo. For in them the excefle of the one to the otheris euer vn- knowen, st teeeeeeenenn agpemeeene een ee) of Euclides Elementes. Fol.129. knowen,& therefore is furd,and innominable.As betwene the fide of a{quare and the diameter therof,there is vndoubtedly a proportion , for.that-the fide certaine times multiplied may excede thediameter , Likewile betwene the diameter ofa circle and the circumference therof there is certainlie,by this definition, a propor- tion.for that the diameter certaine times:nultiplied may excede the circumference of the-circle : although neither of thefe proportions can be.named & exprefled by puniber . For this,caufé therefore vied Euchde thisananer of defining: by shultipl- cation. iL] ; M2 Hu Pons} 391303 | eMagnitudes are fayd to be in one‘or the felfe fame propor Th: ixth de _ tonsthefur/t tothe Second, and.the.third to.tbe fourth, when frien ’ a the equsinultiplices of the firit and of the third beyng compa> red with the equimultiplices of the second ana of the fourth, +» according to any multiplication ; either together exceede the one the other, or together are equall the one to the other, or togetber are leffe the one then other. ee In the definition laft going before, he fhewed what magnitudes haue propor- tion the one tothe other, & now-this diffinition fheweth what magnitudes are in onéand the felfe fame proportion,and how to know whether they bein one and An exemple the felf fame proportion,or not.Itis plaine that euery proportio hath two termes, ofthis defini {o that when ye compare proportion to proportion, ye mutt of neceflitie haue 4, 07 ss magnte termes, thatis,an antecedent and a confequent,to cither of the proportiés.As fup- tudete pofe A,B,C,D, to be foure magnitudes, the firft,B the fecond,C the third, and c pein ncaa D the fourth inow ifye take the equimultiplices of A and C the firlt & the third, that is,if ye multiply A and C by one and the felfe fame number, as let the multi- plex of A be E,and let the equimultiplex of C be F. Likewifealfo if ye take the equimultiplices of B and D, the fecond and the fourth, that is ifye multiply them by any one number,whether it beby that nuinber wherby ye multiplied A&C, or by any other number greater orleffe, as lerthe multiplex,of B be G, and the e- quimultiplex' of D be H: now itthe equimultiplices of A and C be both greatet hen the eqnimultiplices of B and D, tharis ifthe multiplex of A be greater then the ninttates of B, and the multiplex of C be greater then the multiplex of D, or if they be both leffe then they : or both equal! to them, then are the magnitudes A,B and C{Diifi fie and the felfe fame proportion. an Likewife in'umbers 8, to 6. hath a proportion,al{o 4. to 3. hath a proportion: GES ipaeea ante 28 18 gins now = —— ———— - - -- = = - = -- —- _ eto — = > _ = a —" = za = a ~~ 7 — — c = - _ i - ~ - ~ > —-~ = = -. 7 ~ > “Qatar x “a aa _—— = s- = - - . . a _ = Anexample in numbers. z se nei =~ _— = ri oe SS errno ae yA Wh oie Thy: ee =. \ : et a= oes — = = : te A J , ae ete ——__. _ _ -- = tre . en, — ———— - SS Me he _— ~ s . se oe ee —— = ————— — — — —_ — ~ z ; _ - — ee a -- ~ ~ —— —— - —_ . ee — oe ee a -< - 3 x = ey » —_ —- Paz _ — = “ via = > -— ——— — — a — = Ss ae a 7 et — - re ,, 7, =>. 29 wes > ae. > («= ey = 3 x. bn = ~ = Ps 7s a > : = _—* ro - —— i = pa SG ae 7% ee ae » anaes ss =e ee &. i Se om ye ---- hoes et ae ee “ ~ — = Sie ie? ~ 5 _ rs ~ lt es == Sus sas - a. —-- ons — —_ . 4 e — i ——- z a oe comes — —- —— r3 - z s = <= 7 “Sate: = — = = mens Roce Te Ses - _ =-> - - += = x ; - = —— ————_—_ =~ aren ee = - . — ,- res nae . > ee > = = = noses = aap ol Pas =, ee eS Fh : a st ye did, the tiple of 6 is18. and the triple win other ex. awiple in wunbers. An other example in mumbers, St A te A _€quall to the mulriplices of 6 and 3. the fecond and the fo urth, nam fe , . | Lhe fifth Booke \ ‘nowto feewhether they be in on’ and the felfe faine proportion ornot, fet than atvorderas athe examiple here written; 8 the firtt, 6 the fecond; 4'the third and. a the fourth, Now-take the equitnuleiplices of 8 and 4. the firft, and the third, tharisymultiply'theni by oneandthefelfe ovo: SHS IS MUI Sir bos sou ‘fame number, fuppofeitbe by 32foithe trie 5 SUETIOD Tage CRS \pleof8 is24. Sethe triple of gris tz: likes fs ga Saito terres Third “wilerake the equimultiplicesof 6 and 3. 09/077 i002? ink FPP4 the fecond and the fourth, multi lyeng Second 6 92° 0¥ —— 7 r Ae them likewife by one and the felfe fame nunaber.{uppole itbe\alfoby.g4 excedlcth the triple of Sythe econd,namiely,i8 « likewifethe triple of 4the.third numbers namely, 12. excédeth the triple of 3.. the fourth,namely,9. Wherefore by the fist part of this definition the humbers 8 to 6/and 2 to 3: arein onéand the felfe fame proportio, Décauletharthe equimultiplites of 8 and-4. the firlt 8 the third,do'both exceede the equimultiplices of 6 and 3, the fecond and the POUND. 2 Po ge) ©) Againe,tike the fame nambér and try the fame after this maner: Take the equimultiplices of 8. and 4. the firft and the thir smultiplieng eche by 3.as before ye did, fo fhall ye haue 24 for the tri-- wth ) | ple of 8.and 12 forthe triple of 4,as ee Gee 24 | page y Had before. Then take the equimulti- ah Banc Pek on de Chied pices of 6.and 3. the fecond and. the oe | _—_—_— 7. ' fourth multipliyng them by fome one Second “6 | Fourth oy HUMmber,but nor by 34s before ye did: - ba Fxg vont bur by 4. fo forthe quadruple ofé the 24 3 ou Hotes: econd tumber, fhall yehane2g. and Ty | orthe quadruple of 3 the fourth number,ye fhall haue 12. And now ye fee that the equimultiplices of 8 and 4.the firft and the third namely, 24 and12. are both ely, to24 and 12. Wherfore the numbers geuen,are by the fecond part of this definition in one and the felfe {ame proportion, becaufe the €quimultiplices of 8 and 4 the firftand the third,are both equall to the equimultiplices of 6and 3. the feconde and the fourth, 4 | Agayne to fhew the fame,and for the fulnes of the diffinition, take the fame numbers $,6,4,3. and take the equimultiplices of 8 and 4. the firftand the thirde, multiplieng eche by 2. fo haue ye 16 for the duple of 8, the firft number,and 8 for the duple of4 the third number : then takeal{o the equimultiplices of 6 and 3.the fEcond and the fourth,m ultipliyng eche by 3. fo haue ye.18 for the triple of 6 the lecond, and forthe triple of 3. the =... ree = fourth nuniber: And now ye fee that , aw er hi the’ equimulti lices’ of 8 and 4. the pati a PASE cage firftand the cite. namely, 16.and 8 mh | = | as i z are both leffe then the equimultipli- Second 6 | Fourth 3 ces of 6 and 3 thefecond & the fourth tne Fee Peary Be namely 18 and.9. Fori6areleffethen. “* esuny, 185andS arelefle then 9. Wherefore. aostean ea bY the third parrofthis diffinition,the numbers rs ETT DY 1 are ser St ~~ ? Te ~ ropofed are in oneand: the felfe oes : Fp fame proportion, for that the equimultiplices of § and 4-the firtt and the. third are both leffe then. the equimultiplices of ¢and 3 the fecond and the fourth. oe , Further coo pm De of Euchdes Elementes. Fol.1204 Farther in this diffinition; this: particle (according to any multiplication) is moftdiligently to bé:confidered, which fignifieth by any multiplication ‘indiffe- rently whatfoeuer.:For whenfocuer the quantities ‘be in one'and the felfe fame proportion,then by any multiplication whatfoeuer,the equimultiplices of the firft and the third, {hall exceede the equimultiplices of the fecond and the fourth, or hall’ be equall vnto them, orleflethemthem. Yerit may {6 happen by fome onc multiplication,that the equimultiplices of the firftand the third,do exceede the e- quimultiplices of the fecond and the fourth,and yet the quantities genen thal not be in one and the felfe fame proportion. As in this example here fet, where the e- quithultiplices df 6and §, the firt and-thethirdey namely,'18. and 15. doo both exccede the equimultiplices of 4 and 8 is 9 5 the fe send Mice | —— | Rae: 3. the fecond and the fourth, namely, Bie ey Third ? 8 and 6. yetare not the numbers ge- ven in one and the felfe fame propor- tion. For 6 hath not that proportion to 4. which §. hath to 3. In this exam- ple 6 and 5 the firft and the third were multiplied by 3. which made their equimul- tiplices8-and 15. which exceede the equimiultiplices of 4 and 3, the fecond and the fourth beyng multiplied by 2. namely,8 and 6: but ifye fhall multiply 6 and 5 the firftand the thirde by 2. ye fhall produce 12 and x0 for their equimultiph- ces, and'then if-ye multiply 4 and t2 10 3. the fecond and the fourth by 3.'fo Fivit oe fhall ye produce for their equimalti- plices1zand 9. Now ye fee that by this multiplication the equimultipli- ces of the firftand the thirde doo not 7 bothex¢eede the equimultiplices of the fecond and fourth : for 12 the multiplex of 6doth not exceede r2'the multiplex of 4. and therfore the numbers or quant- ties. are notin one and the felfe fame proportion, for that it holdeth not in all mul- tiplications whatfocuer. + Andbecaufe this diffinition requirethall marier of multiplicatids to bring forth the excefles,cqualities,and wantes of the antecedents aboue,to,or vnder the con- fequénts,to auoide the tedioufnes and infinite labour therof,I hauc fet forth a rule much to be made ofand eftemed,wherby ye may in any rational! proportion pro- duce equimultiplices of the firft and the third equallto the equimultiplices of the fecond. and the fourth. The rule is this, take two numbers whatfoeuer in that pro portion in which your quantities are,& by the number which is an tecedent mul- tiply the confequents of your proportions,namely,the fecond and the fourth sand by the number which is the confequent multiply the antecedentes of your pro- portions,namely,the firft and the third:then neceffarily fhalbe produced the equi- multiplices of the firft and the third:equall to the equimultiplices of the fecond & the fourth. As by example,take 6 to.2. and 3 to 1.which are in one & thefelte {ame eo & taking thefe two niibers 9 & 3.which are in the fame proportio, now y-9 the antecedent multiply the confequéts 2 & 1. andfo fhal ye haue 18 & 9 for the equimultiplices of the fecond & the fourth,then by 3 the confequent multiply che antedéts 6 & 3,fo hal ye haue 18 & 9 for the equimultiplices of the firft & the third, which are equal to the former equimultiplices of the fecod & fourth. Wher- of itfoloweth thatifye multiply 18 & 9 the equimultiplices of the firft and the third by any niiber greater thé 3.wherby they were now multiplied,they fhal both ever excecde the equimultiplices of the fecond & the fourth:& ifye multiply thé H.Aj. by Second. .4 Fourth = 3 (se ee ee —_--- § 6 Second 4 Fourth 3 pan ee (ee a T. 9 Note this pare ticle according to any mults- plea ti0tte An example where the equimultipli- ces of the f:rft andtbirdex= ceeds the egut- multiplices of the fecond and fourth, and yee the quantities genen arenot an one and the Selfe fame pro- portion. A rule to pro~ duce equimul- tiplices of the firké and third equall to the equimultiplt= ces of the fe~ conde and fourth. Example there of A. so - —= = deere SS : = DEB ite Se b - i) rt | } , ' : | : ; f } ! j } i ) : Lill ] | in| | Wa 4 4) i ml | i) | ue | ian} | ! } | hi } ' i i : ‘yen 4 ty if } Hin ; 1 i] - | ia \ | a { ’ } t i { 4 { t | ' | i [ : i 7 q eid H \ | | : Wy ; ' i] 5 ! 2 i f i | : ' : f | : i ' ' - } { | \ ; ; ( ; moe : ‘ , : rl ' i 1 } \ \ ie Ne ul | i : : } } | | \ 4 j } : ' i : j ) i ; J] | i ) f 1) } i y i : ‘ H j i - 4h \; Bal j i i : ie hal + " - ; 7 RY |), qi i, 14 \ : | ‘5 J ) ha " ht - 1 | iit ‘ ‘ : | “ y | i On ee La ‘ | ‘ i vee y ; ‘ 4 Th ) o)) ) }) > " i itt \ ia a ae aie ae} 1 ] haa ay >) ogy: pa otf Tal aD | yi | mY y : | af j t Ny : } 4 | ' t ~~ 4 |! BE | ‘ : iii 1 14 ie] at { Th i \ an ‘ } ' f : ae i) ie { im OF { ‘ ty ey 1) ay { Pi) : f i : i (7 if Me a } : E t | ?- Tbe fenenth definition gor 3 4 Propartiona- bity of tivo fortes,conti- nual and dif- continual « “An example of continual proportionalt- tie in nume berse 16,3.4.2.1 In coniinnall proportionals- te the quan~- Sities cannot be of one kinde. Difcontinuall prope rtrona= bitte. Example of difcontinual proportionalt- ty in numbers. dn difeotennal proportionalt- tre the propos tions may be of diners kindes, Lhe fifth Booke by any numbetleffe then 3. they thall ever bothavantof them. So that whatloe- uer multiplication it be, they {hall euer both exceede,be equal,or want aboue, to, orfrom13.andg. the equimultiplices of the fecond and fourth, Magniindes which are in one and the felfe fame proportion, are called Proportional. As ifthe lyne A,haue the fame proportion to the line B, that the lyneC hath to the lyne D, then are the {aid foure magnitudes A,B, C,D, called proportionall, A +———+——_.. c..,-~ Alfo in numbess.for that.9. to 3-hath that {ame proportié thattz hath to 4: therefore thefe foure niibers 9.3.12.4. are {aid to be p:oportionall. Here is to be noted that this likenes or idemptitie of proportio whica is called.as- before was {aid proportionalitie,is of rwo fortes: the one is. continual,the other is difcontinuall, Continual proportionalitie is, when the magnitudes {et in lyke proportion, iare fo toyned together, that the fecond which 1s confequent to the firft,is antecedentto the third,and the fourth which is contequent to the third,is antecedent to the fift,and fo continually forth. So eue- ry quanutic orterme in this proportionalitieis both antecedent and confequent (confequent inrefpeé of that which went before, & antecedent in refpect of that which followeta)exceptthe firft, which is onely antecedent to that which follow- eth,and thelaftwhich is onely confequent to that which went before. Take an.ex- ample in thefe rumbers,16,8.4,2.1. In what proportion 16, is to 8, in the fame is $.to4,in the {ane alfois 4. to 2, and likewife 2.to 1. For they all are in duple pro- portion:16. thefirft is antecedent to 8,and 8. is confequentvnto it : and he felfe fame 3. is antecedent to4.; which 4. beyng-confequent to 8. is antecedent to 2, which 2. likewif: is confequent to 4.and. antecedent to 1:which becaufeheis the laft,is onely confequent,and antecedentto hon e,as 16. becaufe it was the firftswas antecedent oney,and confequentto none, Alo in this proportionalitie all the magnitudes mult of neceffitie be ofone kyndeé,. by reafon of the continuation of the proportionsin this proportionalitie, becaufe there is no proportion betwene quantities of divers kyndes.. Difcontinuall proportionalitie is, when the magni- tudes which arefetin lyke proportion,are not continually fet,as before they were, hauyng one terme referred both to that which went before,and to that which fo- loweth,but haue their termes diftin@ and fevered afonder : as the firft is antece- dent to the fecond,fo is the third antecedent to the fourth. Example in numbers, as 81s to 4.f0 is6.to 3, for either proportion is duple. Where ye fee,how ech pro- portion hath hys owne antecedent and confequent diftin@ from the antecedent and confequentof the other,and no one numberis antecedentand confequentin diners refpectes And by reafon of the difcontinnaunce of the propottions in this proportionalitie,the quantities compared,may be of diuers kyndes, becaufe the con{equentin the firft proportion is northe antecedent in the fecond proportion. So that ye may compare fuuperficies to fuperficies,or bod y to body in the felfefame proportion thatye do lyne to lyne. d et ee oy a Ir ay sg ee en Whew of Enchides Elementes. Fol.131. When the equemultiplices being taken, the multiplex of the T he eight firft excedeth the multiplex of the fecond, @ themultiplex of “fn the third,excedeth not the multiplex of the fourth : then hath the firft to thefecond a greater proportion,then kath the third to the fourth. In the fixt definition was declared what magnitudes are {aid to. be in oneand the fame proportion :, now he fhewetlrin this.definition what magnitudes are faid to be ina greater. proportion . And here is fuppofed the fame order of multiplicati- on,that there in that definition was vied: namely, that the firft and the third be e- qually multipliedsthat is,by one dethe felfe fame niber ; and-alfo thatthe fecond and the fourth be equally multiplied by 4 the fame or fome other number: and then ifthe multiplex of the firft, excede the wnultiplex of the fecond:& the mul tiplex of the third. ; excede not themul- i tiplex of the fourth,the firft hath a grea- ter. proportion to the fecond, then hath | | the third to the fourth. As fuppofethat | | there be foure quantities,A,B,C;D:-of a which let A be the firft,Bithe fecond, G /# the third,& D the fourth, And let:A the, | es firft cotaine 6.and ler B thefecond.con... taine 2.& C the third 4. & D the fourth f 3:,Now.takethe equimultiplices of A....:} 8 and @,the firft & thethird,whichletbe .. - oF EandF,fo that how multiplex E, is. to.) 4 2 A fo multiplex let be to C: names. .. + ly.for example fake let. either of them... E,, A BG be triple : fo hawe you 18. for the | multiplex of A, and -12.for the mul- ee ae Soe : tiplex of CG. Likewife take theequimultiplices of B & D,the fecond & the fourth, multiplying them alfo by one and thefelffame number,as by 4 : fohaue ye for the multiplex of Bthe fecond 8, namely zthe line.G, and for the mtltiplex.of D the fourth r2,;namely,the lineH; Now becaufe the lineE multiplex to the firft, name- ly,18 ,excedeth the line G fnultiplex tothe fecond,namely,8,; And the line F mul- tiplex to the thirdjnamely,12, excedeth not the line H multiple: to the fourth, namely,12 (for that they are equall)the proportio of A to B the fit to the fecond, is greater then, the propertionofG to D-the third t6 the fourth,So likewife in ni- bers: take 11.to 2.,& 7; to 3-and mul- See eee yk eee tiply 11.67. (the firft, and the third) ~ ig by 2,d0.fhall ye haue.22, forthe multi; .b. epg Fin 3 aes Sanoaiiehat end 144 for the, multic... 2. waste Baer. : ee plexofthe third: and multiply 2.and | Second, 2. | Fouth "3 3.theiccond and the fourth by 6; fo. Pere 18.). fhall the multiplex.of thefecondber. . > and che multiplex ofthe fourth be 18: ’ ASS "Now An example of thes definis tion 17 mag nitedes An example in numbers, Wotes at Pd of A multiplication,that whé the equinul- TL he fifth Booke Now yefeé 22. the multiplex of the firft,excedeth 12,the multiplex of the fecond. But14.the multiplex of the third, excedeth not 18. the multiplex of the fourth: Wherefore the proportion of 11.to 2.thefirltto the fecond,is greater then the pro- portion of 7.to 3,the third to the fourth, And {fo ofall other quantities and-num- bers,which are not in one and the felfe fame proportion, ye may know when the firft to the fecond hath a greater proportion then the third to the fourth. q An other example. This example haue I fet to declare | that although the proportion’ of the 16 i8 firftto the fecond be greater then the Frt SL oh — proportion of the third to the fourth, re chee yet the multiplex of the firft excedeth Second 4. | Fourth 9g or —~— notthe multiplex of the fecod. Wher- | rH a fore it is diligently to be noted, thatit : is {ufficient to fhew that the proporti- on of the firft to the fecond is greater thé the proportion of the third to the fourth, ifthe want or lacke of the multiplex of the firft from the multiplex of the fecond, be leffe then the want or lacke of the multiplex of the third to the multiplex of the fourth. As in this example 16. the multiplex of 8. the firft, wanteth a séihe multiplex of 4.thefecond,foure: wheras 18.the multiplex of 9,the third, wateth of 45,the multiplex of 9 the fourth,27 . And fo ofall others wheras(the proportions being diuers) the equimultiplices of the firftand the third are both leffé , then the equimultiplices of the fecond and the fourth. Likewife if the equimultiplices of the firft and the third do both excede the equimultiplices of the fecond & the firft, thé fhall the exceffe of the multiplex of the firftaboue the multiplex of the fecond, be gréater thé the excefle of the multiplex of the third,abotie the multiplex of the fourth “As in'thefe numbers here fet,the eqtimultiplices of 6. and 4. the firtt and the third,namely,1z.and 8.do both excede the equimultiplices of 2. and 3. the fe- cond and the fourth namely,4.and 6. But r2.the multiplex of the firft excedeth 4. the multiplex of the fecond by 4,and 8.the multiplex of the thyrd excedeth 6.the multiplexofthe fourth by 2. but 8.is . | more then 2. Howbeitthisisgeneral- — (" 12 § ly’ certain thar when foeuer thie pro- Fir SON Bhd sa portion of the firft to the fecéd 1s grea-~ | ter then the proportion of the third to © Second 2 Fourth 3 the fourth, there may be found fome es ge a tiplicés of the firft and_the third fhall ; eet ag be compared to the equimultiplices ofthe fecond and the fourth, the multiplex of the firft fhalf excede the multiplex of the fecond;& the multiplex of the third fhall not excede the multiplex of the fourth,according to the plaine wordes of the de- a finition... Brae. res 4 aah Inlikemaner when you haue taken the equimultiplices ofthe firft & the third, and alfo the equimultiplices ofthe fecond and the fourth, if the multiplex. of the firft excede not the multiplex of the fecond,and the multiplex of the third excede the multiplex of the fourth} then hath the firft to the fecond.a lefle proportion, then hath the third to the fourth. As in the example before, if ye chaunge the termes,and make C the firft,D the fecond,A the third,and B the fourth:then fhalt BS a ee F,namely, of Euclides Elementes. Fol.132. F, namely, 12.the multiplex of the firftnotexcede H, namely, 12.the multiplex of the fecond: but E,namely,18.the multiplex of the third‘excedeth G,namely,8 the multiplex of the fourth . Wherefore the proportion of C to D, the firtt to the fe- cond, is leffe then the proportion of A to B,the third tothe fourth. Euen fo in numbers. As in this ex- ample, 5.to 4.and 7 alt . eri ( rs ar s.and 7.the firftand the third eche a Sey 4, Fe ie 3, ye fhall for the multiplex of 5. the beg Schoh et ot firtt haue rs: and for the multiplex of Second 4 Fourth 3 7.the third fhall ye haue 21: againeif | ve J oe | ye multiply 4.and 3. thefecond & the fourth by 6,for the multiplex of 4.the fecond ye fhall haue 24,and for the multiplex of 3.the fourth , ye fhall haue 18 . So yefee that 15.the multiplex of the firft,is leffe then 24;the multiplex of the fecond: and 21.the multiplex of the third is greater then 18.the multiplex of the fourth. Wherefore the proportion of5.to 4.the firft to the fecond is lefle then the propor- tion of 7.to 3.the third to the fourth. Proportionallitie confifteth at the left in three termes. Before it was fayd,that proportionalitie is a likencfle of an idemptitie of pro- portions. Wherfore of neceflitie in proportionalitic, there muft be two proporti- ons,.and euery proportion hath two termes, namely, his antecedent and confe- quent. Therfore in euery proportionalitie there are foure termes.But for that fom- tyme,one terme {upplieth by divers relations,the roume oftwo, forin refpe to the firftitis confequent,and in refpect to that which followeth, itis antecedent: therfore three termes at leaft and not vader may fufficein proportionalitie,which three arein power foure,and occupy the rome of foure,asis Gayd.As fuppofe that A hath to.B that proportion, that By... | hath to.C; then are thefe thre quan-4 og: ho a oie tities A, B,C, fetin the left number B of proportionality. Likewife in num- C bers,as 8.4, 2.and9, 6.4. | Su DOig = When there ave three magnitudes in proportion,the firft /hall he ynto the third in double proportion that it is to the Jecond. » But when there are foure magnitudes in proportion the furst _. fhall be vnto the fourth in treble proportion that it 1s to the Je- » ond: And fo alwaies in order one more,asthe proportion fhalt Peeetended, . This definitionisialfo vaderftand in coftinuall proportionalitie.As if the thre magnitudes A, By bee proportional : then fhall the proportion of A is an Ii, 111}. Tl The ninth des finition, An example of this defint- tion in maz~ mitudes. Example in numbers. The tenth definition. Example of this definition in magnt- tudes. T be fifth Booke firftto.C the thirde; bee double to the proportion). 4 thr battnieleseccecs Sagal Wl ny whichis betwene A& B EE TERE the firft and the feconde, . thatistheproportid of A ¢ to Btaken trwile, or added to itfelf (which is all one) fhall make the proportié of A to C.For the eafier vnderftidyng of this & the pra ctife therofjitfhall be much neceflary fomwhat to inftrué the rude beginner how proportions may be added-one to an other. Which is done by this rule. Multiply the antecedent of the one proportion by the antecedét of the other, ————f ce) _- a ees and the number produced fhall be the antecedent of the proportion which-con- ons to propor. tayneth them both.Likewyfe multiply the confequent of theone proportion by te the confequent of the other,and the number produced fhall be confequent of the proportion which fhall contayne them both. Example _. (An example. Ifye will adde the proportion which is betwene 4and 2. (which shercof, is dupla) to the proportion which is betwene 9 and 3. (which is tripla) multiply 9. the antecedent of the firlt proportion by 4. the antecedent ofthe fecond proportion, and ye fhall produce 36. which referue and kepe 36 for the antecedent of the proportion which yefeeke for. Likewife 9 4 multiply 3 the confequent of the firft proportion,by 2 the confe- 3 2 quent of the fecond,fo fhall ye have 6.which 6. {hall be confequent 6 to the former antecedét, namely, t0'36¥fo fhal the proportié which is betwene36 and 6, namely, fextuplajcontayne in it the two proportions geueri) namely,tuipla,and dupla.And by thismeanes are they added together, & brotight into one.And by this may-ye adde all other kyndes of proportions whatfoenet they be Now for thatthe diffinition fayth,that if thete be three quantitiés in’pro+ portio,that is,what proportié the firft hath to the fecéd;the fame hath the fécond to the third,which forexample let be triple,as in thefe niibets,27. 9, 3.addé triple i to triple by the.ruleabouefaid. And forafmuch as itis eafiér to workeifi Guall ong 70943" bers thenin great,reduce thefe proportids to theyr leaft denomination:So 27, to . sréduced to the left termes in that proportion,is as inuch 4s 3. to t. Likewife 9'to Pas 3 reduced to theyrleft termes are alfo as much as 3 to 1, noW adde together thefé sai Hg two triple proportions thus reduced, multiphyng 3 by 3y the one antecedent by ‘* ‘the otner,fo thall ye produce 9 for a new antecedent, thén mtiltiply 1 by r,the one conlequent by the other, fo thall you produce 1. which let be confequent to 9. your antecedent,fo the proportion betwene 9 and 1. which is noncuple contay- neth both the two triple proportions. And becaufe they were equal the one.to the other, itis duple to eché ofthein\ Ye fee allo that the proportion of 27'to¥. the firlt to the third, is alfo noncuple. Wherfore according to the definition, the proporti- on ofthe Art to the'third,is double tothe proportion of the firit tothefecond, as 9totebeyne noncuples isdouble 3 tox. which is triple, ‘becaufe it contayneth ir ~-So ifthere be4. quantitiés in continual proportion,the proportion.of the firft tothe fourth: thall be triple ro the proportion whichis betweéne the-firt\and the fecond,thatis. it thal contayne it three tymes.As for exampld, Take 4-numbers in + 2 2 continual proportion 8.4.2.1. Ye fee that the proportio of 8 to 1. the firft to the # it 1 fourth,is odtupla : the proportion of 8 to 4. the firft to the fecond is dupla, now Ags! treble dupla proportion, thatisyadde 3-dupla proportions together, by ahe tule before geuen,as ye fee in the example. Mustiply all the antécedentés ane Zs ey ¢ ~~. : ra = ~N : a ant i waggstids en aes ————— ~~ ~~ _- = a ~~ < — = —_ > —_ oe = So ee a ree = r = > w =, en Ses y= a er - = F dat if Ath be i an { | " ha) Dl ie iat i | v = of Euchdes Elementes. Fol.133. the antecedent of the'firft proportion, by 2.theantecedent ofthefecond,fo haue ye 4:which-4.multiply by athe antecedent of the third proportid, fo thal ye haue 8 foraniew antecedent. : In lyke maner multiply all the confequentes together, 1. the confequent of the firft proportion by 1. the confequent of the fecond propor- tid,fothatye haue rwhich+.multiply agayne by 1. the cdfequent of the third pro- portion,fofhallye haue agayne 1: which 1. let be confequentto your former ante- cedent 8: fo haue ye 8 to 1.which is o¢tupla,which was alfo the proportion of the Getto the fourth which'o@uplais alfo brought fourth of the addition of thre du- pla proportions together, and contayneth it three tymes,wherefore octuplais tri- pla to dupla,and therfore as the diffinition fayth: the proportion of the firftto the fourth # tripla to the proportion of thefirftto the fecond: And fo confequently forth 48'fong as the proportionalitie continueth accordyng to the fentence of the diffinition, the termes of the proportions exceding thenumber of proportions by one.As ifye haue§. termes in proportion, the proportié of the firft to the fifth fhal be quadrupla to the proportion of the firft to the fecond,and if there be 6.termes, it fhall be quintupla and fo in order. e Magnitudes of lke proportion, are fayd to be an tecedents to antecedentes and confequentes to confequentes. For that before it was fayd, that proportion wasa relation ora refpect of one quantitie to an other,now fheweth he what magnitudes are fayd to be of like pro- portion,namely,thele whofe antecedents hauc like refpeét to their confequentes, and whofe confequents receyue like refpectes of their antecedéts. As putting 4.magnitudes A,B,C D.IfA antecedent to B, be dou--B -—___—. sy wn ble'to Band C antecedent to D, be double alfo to D3théhaue the two antecedenites like refpectes to their confequents.Likewife ifB the confequent be half of A,and alfo D the confequent be halfe of C,then the two confequentes Band D receiue of their antecedentes like refpectes and relations.And by this dif- finition,are thefe magnitudes A,B,C,D, oflike proportion. Alfo in numbers,9.3. 6. 2:becaufe 9 the antecedentis triple to 3. his confe- quent,and the antecedent 6.is alfo triple to 2 his confequent : the two antecedéts 9 and 6 haue like refpectes to their confequentes, 9. ©. and becaufethat 3 the confequentis the fubtriple or third partof 3. 2. 9his antecedent,and likewife 2 the confequent is the fubtriple or third part of 6. his antecedent,the two confequentes 3 and 2 receiue alfo lyke re- {pedtes of their antecedentes, and therfore are numbers of like proportion. ————— in cmeniaitit — -neiie Proportion alternate, or proportion by permutations, when the antecedent is compared to the antecedent, and the confe- quent to the confequent. The vnderftanding of this definition & ofall the definitions following, depen- deth of the definition going before,and vie it for a generall fi oe: to tT ens sue The eleuenth definition. Example of this definttion in magnituds. Example in numbers. The twelfth definstion. & xample of this definition $2 magnituds. E xample in numbers. T hethirtenth definition. Example of this definition $n WsAguituds Examp le in numoers. T he fourtenth definition, Exansple of this definition in magnituds The fifth Booke hauefoure quantities in proportion.Suppofe foure magnitudes A,B,C:D,to be in proportion,namely,as A is to B, fo let C be to D. Nowif ye compare A the ante- cedent of the firft proportion to C the antecedent of the fecond as to-his confe- quent, & likewife if ye compare B the cofequent of the firftpro- A portion asan antecedent to Dg... the confequent of thefecond as to his coniequent ; then fhall ye haue the magnitudesin this fort : as A to Cante- cedent to antecedent,foB to D,confequent to confequent,& this is called permu- tate proportion or alternate. In numbersas 12. to 6/0. 8.to 4. either is dupla.. Wherefore by permutation of ...Duple Duple proportion, as 12.to 8.antecedentto antecedent,fo is 6. j Be i 28 on to 4.confequent to confequent,for either is fefquialtera. 47° 7, 2.4 Sefquialter. P ———+ Conuer/e proportion, or propo tion by. conuerfion isswhen the confequent 1s taken as the antecedent , and fo ts compared to the antecedent as to the confequent. Suppofe as before foure magnitudes in proportion,A,B,C,D,as A to By fo C to D : if ye referre B the confequent of the firtt proportion,as antecedent,to A the antecedent of the firft,as to his confequent : and likewife if yereferre D the confe- quét of the fecond proportion as an- 4 c —- tecedétto C the antecedét of the fe cond proporti6, as to his c6fequent: thé {hall ye haue the magnitudes in thys order.As B to A cofequent to antecedét: fo Dto C confequent to antecedét.And thys is called. céuerfe proportion.Soalfo in numbers, 9.to 3, as 6.to 2, eyther is tripla;wher- . ——— fore comparing 3.to 9,theconfequentofthefirtto Triple Triple hys antecedent 9, and alfo 2.the confequent of the = —*4 a fecond to hys antecedent 6, by conuerfe proporti- 9-3 ! 6.2 on itcommeth to.paffe as 3.t0 9,f02.t06: Fore “7 oi pe: ther is fubtripla. ge Subtriple. Subtriple. a ee "DD jnceminial Proportion compofed, or compofition of proportion is, when the antecedent and the confequent are both as one compared ato the confequent. Suppofe that in the former foure A - ee in proportid,A,B,C,D ; TTT ty eee as is to.B,fo is C to D: if yeadde * a, Remreresse es A and B the antecedent and the confequent of the firft proportion together,andcompare. A.B, . 3G oe them fo added as one antecedent to B the ¢ | confequent of the firft. proportion as to hys 33 confequent:and likewife if ye adde together of Enclides Elementes. Fol.134., “Cand Dtheantecedentand the confequentiof thefecond proportion: aid {o ad- ded,compare them as one antecedent to D the confequént of the fecond propor- tion,as to his confequent:then fhall ye haue’the magnitudes in this otdéi: As A B to B,fo C Dto D, for either of them is tripla.And this is called compoled propor- tion, or compofition of proportion. And foalfoin numbers. As 8.to 4, {0 6.to “4? S.arid 4 the ahtécedetitand confequent’ df the fir pré- “portion added togetherinake 12: which 12/as antecedent &d- 84 ‘pare to.g.the conlequent ofthe fitkt proportionias to his con- . “O43 { fe adde wcetl eecab SRoleeAre) Mad) equent : fo adde together 6.and. 3, the antccedent'and con eee fequent of the fecond proportion,they make 9:!' which 9!as | antecedent compareto3.the confequent of the {eednd proportion.as ro hie caH- > 33% fequent : fofhall ye hauebycompofition of proportion;as 12.0 4,fo 9.10 2/for tL ther of thenvistripla. | | Proportion denided, or diuifion of proportio is, when the ex. ceffe wherein the antecedent excedeth the-confequent,is com: “ “pared to the confequent. Thys definition is the conuerfe of the definition going next before < in it was vied compofition,and in thys 1s vfed diuifion..:. As before fonow {uppofe foure magnitudes in proportion. AB. the firft, B the fecond, C D the third, and. D the fourth; as A BtoB: fo C:D.to D: at 3 A.B, theantecedent. ofthe firlt pro- Se ortion.excedeth B the.confequent C D of the firft proportion by. the cael, Usaeees : tude A, wherfre Ais the exceffe of the antecedent A B.aboue the confequent B: fo likewife C D the antecedent of the fecond proportion, excedeth D the confe- quent of the fame proportion, by the quantitie C , wherefore C is the excefle of B «the antecedent C D aboue the confequent D.. Now ifye compare A the excefle of AB the firft-antecedent, aboue the confequent B,as antecedentto B the con- fequent; as to-his confequent : alfo if ye compare.D. the excefle of the fecond an- tecedent € D, aboue the confequent D, as antecedent to D the conféquent,as to his confequent : then fhall your magnitudes bein this. order-As'A to B fois C to D : whichis called dinilion of praportion,or proportion deuided. And {foin numbers, as 9.to 6,fo 12.to 8,¢ither proportion is fefqutaltera : the exceflé of 9. the antecedent of the fitft 9. 6: 12. 8 -proportion aboue.6.the confeqnent of the {emeis 3.: the ex- As Oy. 8 cefic of 12.the antecedent of the fecand proportion aboue 8, the confequent of the faine,is.4: then if yé.coinpare 3.thé ex- cefle of 9.the firft antecedent aboue the confequent, as antecedent to 6,the. con{eé- quent,as to hys confequent:and alfo ifye compare .4 the excefle of 12-the fecond antecedent aboue the confequent, as antecedent, to $.the confequent,as to hys confequent,ye fhall hauc your numbers after this maner by diuifion of proporti- ‘01,45 3.t0 6:10 4.to 8 : for either proportion is fubdupla. Conuér fio of proportion (which of the elders is commonly cat: led enerfe-proportionjor enerfro of proportion 1s ,whe the anz | 7 feceaent Example sis numbers, ~~ Lhe fiftene acfinition. This is the e3 uerfe of the former defini- F10% 6 Example in magnttuites Exaraple im numbers. The fxtens definitzone : T he fifth Booke | tecedent is compared to the.exceffe, wherein the antecedent excedeth the confequent.. ee | igi | | An example ~~ Foure magnitudes. fuppofed as, before, AB the firlt;B the fecond,C D the ofthis definte third and D the fourth. As ABto A,fo.C D to C: AB the antecedent of the firlt te i, proportion excedeth B.the confequent of the fame, by the magnitude A ,wherc- : fore A is the exceffe of the antece-.. Ans: Bobi dent A Baboue the confequentB:. 5. ot {o alfo the inagnitude,C is theex- ‘ rae ) ; 7 cefle of C D the antecedent of the fecond proportion aboue D the confequent of the fame: now if yereferre AB the antecedent of the firft proportion,as antece- dent,to A the exceffe therofaboue the confequent B, as to his confequent : ifye compare alfo C D. the. antecedent ’of the fecond proportion as antecedent to C the exceffe therof aboue the confequent D , as to his confequent : then fhall your magnitudes:come to thys order. ‘Ais AB to A, {o\C D to'C, and ‘thys isicalled conuerfion of pfoportion, and of fome euerfion of proportion.Likewyfein num- An example bers, as 9.to 6, {0 12.to 8. eyther roportion is fefquialtera: su numbers. the €xcefle of 9.the antecedent of thefirltproportionaboue 9.6: 12.8 6. the confequent of the fame is 3: the exceffe of 12. the ante- rage © cedent of the fecond proportion aboue 8.the confequent of oiBit ist the {ame,is 4: now cOpare the antecedent of thefirft propor- i tion 9.as antecedét to 3. the exceffe therof aboue 6. the confequét, as t6 his con+ fequent, likewife compare 12. the antecedent ofthe fecond proportion as antece- dent to 4.the exceffe therofaboue 8.the confequent,as to his confequent: fo fhall your numbers be in thys order by conuerfion of proportion : as 9.to 3: {0 12.t0 4: for either proportionis triple. i te Proportion of | equalitie 1s,when there are taken , number of dfitun ~-“Magnitudesin one order,and alfoas many other magnitudes - inan other order,comparing two to two beyng in the Jame pro portion,it commeth to paffesthat as in the first order of mag nitudes the first 1s to the laft,/o in the fecond order of magnt, . tudes is.the firft to the laft. Or otherwife it is 4 compart= Jon of extremes together the middle magnitudes being taken away. ‘ Tothe declaration of thys definition are required two orders of. magnitudes equall in number, and in lyke proportion : Asif there be taken in fome detet- erate, minate number ccrtayne magnitudes, namely, foure, A,B,C,;D_. And.alfo in ‘this define : ve f ashe mage the lame number be taken other quantities, namely, foure, EF, G,H : a weitudes. cake the equal] proportions by two andtwo : as A to B, fo Eto: Fras = of Euplides eae: Fol.135. 49 F to nda dse@fosB, fobGo2s) is 02 fh 2...8 0. £0.) SNovw)according! to: thé firft0 aifosaly 30" 10) definition , itoAnche firth magnitude of the firlt ondet be-to:D the laft mags nitude:of the dame orderA.as:E the firft magnitude ofthe fécond order is to H the laft magnitude the fame, ise, itis Called proportion, of equa- tie, Or = proportion ; By. the Poe Sar as sik ie \ is all one in fubftance with the firft, ye.deaue the meane magnitudes in eyther order, namely, B, C,,0n,the one fide , ‘anid F,G..on the other fide, and onely compare the extremes of wy —~ ers ey! Nala rane ech fide together s which by thys de- : | finition fhall be in lyke proportion , as SA is.to.D, fo is E to H. Euen foin humbers, take for example thefe two otders, 27.9.12.24,15,d0d 9.3.4.8. 5) there areiineche orderas> yeeeyfiue_ numbers, then fe thatall.the proportions taken: whi examble by, two & two.be hey betwéne,, . OT, in tionberi,. 27. 9, numbers. of the firkt or- 2h5) 9 | der,and betweri¢ g,and Z5nUM=--» 2°. 13 bers ofthe fécond order, there-/.2 2° ind is oneand'the felf fame propors ; 24 8 tid,namely,tripla : alfo betwene 15 5 9. ands12, numbers.of the firft : r order and 3: and 4, nuinbers of it econ Sides | is slike’ pro- b portion;namely, fubfefquitertia proportion Sfo betwene 12: : and 24, numbers ofthe firft order, and 4.and 8,numbers of ) obs of hs ‘| thefecond ordér, 1S alt lyke propertion, nainely, fubdupla? fy | Laft ofall,betwene 24.and 1550 umbers of the firftrowe,and “+ -® Ps HH betwene 3 and 5, numbers of the fecond. towe,the. proportion is one, namely, fi- pertriparticns quits ¢ Whercfore by. this defhnition, Soe out. all the meant fivitibers Ofeche Hae ye: iiay Compare together onely the extremes,and conclude that as 27.0f the firft row jis to 15. the laft of the fame row, fois. 9.the firlt of the fecond rowe to 5,the laft of the fmerowe: forthe proportion of ech is faperque dripartiens quin tas. Here is tbe confi dered, thatitis not of neceffitie that all the proportions. in eche rowe of numbers be fee in like order, as in the-one-fo in the other: butit fhall befufiicient thatthe proportions be the fame and in equall numberineche rowe Whether it bein the felfe fame order, or in contrary, orinuerted, order, it nah nomen Ag Hevehele nuinb rs, 12. 6..2.in the firtt fow,and3 24. 8. Ae. in thefecond . As ¥3%s't6 6 the firft eo tlie fecond of the firft row, fo, t tht i " { mn) ht A WU TEP — ' 7 ” - ‘| ' u 1} : 1 nie!) Ee ai 4) ; ala } : 4 H) ee) Nit i] } { 1a it) WAR Wal ant | by HH it i : aD 5 us {| ae alll 4 ) | t ‘ UE | ‘ inl d f Pans H hid 1 ! } iWeia | : ae } f ‘ ne iy | iP; rf) } : { ; aL Bee Rot H A} | tae i Mi) . | DETAR | Silt Hit ea i\ y DBE i] 1h A ae } ay ii “1g t yi) |) ae - \e i ‘ i ha 7 a id {| | ml hie , i} fyi a ht! 1h { j it on ' |} silt i a} Wah) ‘ (a! |i i Sn j aT { * i] i a | I if | ut |] de tH | ] hy the - ANE: |) | Ha dae! Hay | {hm | \ | wi my a. tt WE | 0) iim | : : { { aah, | Cap) eh iit ay 4 : be tal We | : aR) n | eet! tS ‘a ibis } } " i 4 ; , | , r ne |i wu y t i . | t : Wels j ir Wil 7 tag! | oR! Bj | i , } : 1 | 5 ei 4 ie | 5 j 1) oe i ir ; et : Hi i i : } ? s a. |} : t iid i) 17 yin ah i If , H ; a Hh ae me \ 1 ( Fe - { uP | 2a : ; Ny : | - ta : } x ! hh uv \ ¢ ini Ni) me | ie |). Y a UE t 4 Hh Ne ; iy hal G : ' | | i th i i th 1% if 1 . ae , : Pia Ne Hl : ae ; a ‘ ‘ / + \ ' ; ‘Bh } elt ae i) mh | t i) ' iy 7 { | . oe ' of Th i ¢ y] | 4} te y : ,| ! | ul) i et) ae ime ite Ul ' t 7 : . t } opr ' , i : } iH ‘ Li it i F im | if | | ‘ aT ; i) ey ‘(in { i rt . Hh thy t ; } ! ' : i | t a — — =< ~~ Notes “Ato {2.24.5 is 8.fo 4. thi fecofid td the third of the {cond row: either 1s duple | hs oe pos -. Bi as 'S. to? eas fecoud to the third j in the fict Boa dirt: | ee ee Oi Eko Order: e= 4S The tight- senth defins- £30Mt. fin example of this definse tion in map wisndes. Example iv wherberse The xintenth An example of this defini- sion 34 magnt. gades. Example wn murnvers. wLbe fifth Booke- order : fo is 24. to 8. the firft to the fecoridcdn, thiedecond: order. Where ye fee that the proportions are not placed in onéand thefelfe fame ordersand courlé, and yet notwith{tanding ye may conclude by equalitie: of proportion, : leauing the meanes 6.and 8: as 12. to a.the firfhto the laf ofthe firfbors ‘12°24 der, fo 24.to 4. the firftand laft of the fecond orders: And.fo of others) ic22>g0 what foeuer and how foeuer they be placed . : eAn ordinate proportionality is, whem as the antecedent is ta the confequent, fois the antecedent tothe confequent, and as the confequent 1s to another,fois the confequent to an other, For the declaration of this definition are alfo required two orders of magnit tudes . Suppofe in'the firft order,that the antecedent -A,to his confequent B,haué the fame proportion that the antecedent D, hath to his c6fequent E inthe fecond order: and make the confequentB : antecedét to fome other quantine, A as to C . Alfo make the confequent Eantecedét to an other quatitie, as to F, fothatthere be thefame pro- C , portion of B to C, which is of Eto | toF. And thys difpofition of proportions is called ordinate proporti- onalitie . Likewife inmnumbers; 18.9.3. and 6.3.2. As 18.to 9.antece-- 1 dent to confequent, fo is 6. to 3. antecedent to confequent : eitheris dupla proportion: andas 9. the confequentis:to an other, namely, to the number 3, fois the confequent 3. to an other, namely, to vnity. And this ordinate proportionalitie may be extéded as farreas ye lift, as ye may {ee in the example of numbers in the definition next before. , bee re re ee et D “ete Yep pimereags. 4 ——— ° r w ©. o¢ e An inordinate proportionality is) when as the antecedent vs tothe confequent,fots the antecedent to the confequent: and as the confequent ts toan other, fo is an other to the ante- cedent. This definition alfo as the other before, requireth two orders of magnitudes... Stippofe in the firlt order that the antecedét A be to the cofequét B,as the antece; dé Cy in the fecond 2 | et Order ts to the ¢onfe+ qient D; & ler B the © confequét of the firft ® proportid be to fome other.namely, to the ) sugincde Fs (Ome pe OE Wy. other, namely, the magnitude Fis to the antecedent C of the. fecond. proportids this kinde of proportionalitic is called inordinate or perturbare. te (ODS) pec peleeay Také alfoan example in numbers, as 9 to 6,the antecedentto 9. 3.8.) théconfequent,{o is 3 to 2 the antecedent to the confequent,ei- = 6 Bois ther proportid is fefquialtetaand as 6 theconfequentofthefirft 3° 6 | : : : proportion, of Euchdes Elementes. Fol.26. proportion,!s to an other, namely,to the number 3401 is another nainely, the hum- ber 6. to 3. the antecedent of the fecond proportion, for eytheri iS qupla pr propot- tion. Anextended biopoebaly: 15, wen as FY she ed anes is to the confequent,/ots the sudecedent tothe confequent,.and as the con/eg uentis toan other sf is the confeiuent toan other. ef pertue bate propor dinates is; ry rhe Whe 1 Pita: wie bes ps ing ig compared to three others magniti ee st.cometh ta pale, Te thatasi inthe farit mag onitudes. the poor see te is tobe con. ee hin nent, foin the fecondi 1s the antecedent-torbe confequent, eo i ayrin theprst magnitudes the confequent is toan other i nig nitude, Join the fecond magnitudes 15.4n other magnitude to the antecedent, , Thefe two lat definitions hie put by. Saratiees Cie a one with sthe other two lait before feti Wherfore it isnot lyke that they were Written and fet ‘here by Euclide,for that they feeme not neceflarysbut rather fuperflaous, neither are: they found in the Greeke examples commonly fet forth in print, nor mentioned Of a- siy\that hath written‘commentaries vpon Exclideolde ornew : Not of Campane, s ceubelins;Pellitarius, Or ontius; norFlafates: vebieshoves itis.not of neceffitie to ade Vnto them any explanation or example either in magnitudes or in numbers, The examples of the two laft definitions sie a am likewite ferue for them alfo. A , eri% = : A aT he i. aoe: ce The Propo it10n. 7 there bea Sumber t magnitudes how many foeuer eque- > multiplicesto.a like number of magnitudes ech toech : how ~\naultiphexon magnitude is to one ft multiples are all the magnitudes toall, tA B aa D C saabnateblirs tou-like Sber of mage \! nitudes E.and Fech toech. Then d fay, that how maltix AEE plex A Bus.to E, fo multiplices are A Band DC tok aN hand FE, For for afmuch as how multiplex. AB ts to E, fo \ mag isD C.toF therefore how many magnitudes "2 therearein AB equall ynte Efo man ny are hare ein DE | K ky. equal ihe 26. definitzctte ? e The 21 .efi nde f20%, Thefe tivo laff definstions not found in the greeke e%- aim plerse CoujivaGion. Demoufire- chon, Demouflra- tion" The fifth Booke equall pnto F . Denide' AB into the inagnitades A which are equall pnto E,thatis into AG andG'B: and likewife D C into the magnitudes which are ee guall ynto F that is into D Hand HC,Now then the multitude of thefe D Hex HC,is equall ynto ‘ the viultitude of thefe AG iG B.And forafmuch ~~ as A Gasequall punto Band DH nto F: therfore AG and D Hare equall ynto E and F: and by the fame reafon forafmuch as GBs equall mmto E,and HC yntoF, GB alfoand\ HC areequalltntoE 2 and F. Wherefore how many magnitudes there are | in AB equall ynto E,fo many are therein AB and DC equall ynto Ew F: Wherefore how multiplex A Bis to E, fomultiplices are A 4 and DC toE and F. If therefore there be a number of magnitudes how many ‘[oener equemultiplte ces to.a like number of magnitudes ech to ech aw multiplex one-magnitude is to one fo multiplices are all the magnitudes to all: “which was required to be proued. $a The 2.Theoreme. The 2. Propofition. If the firft be equemultiplex to the fecond as the third is to the fourth and if the fifth alfo be equemultiplex to the fecond as the jixt is to the fourth:then [hall the firft and the fifth compo- fed together be equemultiplex to the fecond, as the third and the fxt compofed together ts tothe fourth... :. iat V ppofe that there be fixe quantities, of which let AB be the firSt,C the sea: /econd,D E the third,F the fourth BG the fifth,ix EF the fixt:and a Juppofe that the firft, AB, be equemultiplex nto the fecond,C, as the third,'D E ,ts to the fourth,F: and let the fift, BG, be equemultiplex bnto the Jecond,C jas the (ixt,B i,t to the fourth ,F. Then I fay, that the first and the fifth compofed together which let be A Gs equemultiplex onto the fecond Cas the third and fixt compofed together, which let be D Hi, is to the fourth,F.For forafmach as AB is equemultiplex toC, as D Eis 3 to F, therefore how many magnitudes there arein “| B | : gon | . 4 A © equall ynto C, fo many magnitudes are there in ze D Evequall ynto F : and by the fame reafon how mas ny there avein BG equall ynto C, fomany alfoare ‘ therein E. 4 equall’ynto F .. Wherefore bow many there areinthe whole AG equall ynto C fomany are there in the-whole DH equall ynto F.. Wherefore how multiplex AG is vnto C, fomultiplexis DH pnto F. Wherefore the firft and the fifth compofed 4 e together, Q -— of Euclides Elementes. Fol.137. tovether, namely, AG is equemultiplex nto the fecond C,as the third and <> . the fixte compofed together, namely, DH, is to the fourth F. If therfore the firft be equemultiplex tothe fecond as the third is to the fourth , and if the fifth alfo be equemultiplex to the fecond as the fixt is tothe fourth: then fhallthe firft ex the fifth copofed together ,be equemultiplex to the fecond, as the third and the fixt compofed togetherts ta the fourth : ‘which was required to be proned. ST he 3. Tbeoreme. The 3. Propofition. Af the firft be equemultiplex to the fecond,as the third is to the fourth,and if there be taken equemultiplices to the firf ¢> to the third: they hall be equemultiplices to them which were firft taken, the one to.the fecond, the other to the fourth. V ppofe that there be foure magnitudes of which let A be the first ,B the eve? fecond,C thethird,and D the fourth. And let the fir/t,A ,be equemulti- she? plex tothe fecond,Byas,§ third,C,1s to the fourth, D. And ynto A and Coph rnétion, _ Otakeequemultiphees, which let be EF andG H , fo that how multiplex E F ist0-A, fo multiplex let HG betol. Eben I fay that E F is equemultiplex yne to Bas G His nto D.For forafmuch as E F is equemultiplex pnto A,as G FZ is vnto C, therefore bow many magnitudes - there are in EF equall nto A, fomany magnitudes alfo are there in GH equall = rato C. Let EF be deuided into the mags nitudes that are equall pnto A, that is, into se ON Demonitra- K | b40fle en : : E. G D And by the fame reafon KF ts equemultie plex vnto B,as LH istoD. Nowthen pamn,\0 ot J there are fixe magnitudes whereof E K is the first: By fecond: G L5 third: Dz fourth: KE the fifth: L H the fixt. And forafmuch as the firft EK is equemule tiplex to the fecond B, as the third G Lis to the fourth D: and the fift K Fis equemultiplex tothe fecond Bas theifixt Ls Histo the fourth D: therefore( by the magnitudes equall Ynto C, that is into GL and LH. Now then the multitude of thefe magnitudes EK and K F ts equall pitoy multitude of thefe magnitudes GL and L L.. And forafmuch as Ais equemuls tiplex to Bas CistoD: but E Kis equall ~wnto A,andG L tntoC, therefore E K is equemultiplex.vnto'B, as GL is ynto D, = . A. § EK and KF. And lkewife G H into C.. kt. the _ the fecond of the fift ) the firft ex the fift compofed together naniely,E F is equer Lhe fifth Booke multiplex pnto the fecond B, as the third and the jixt compofed together namee 4y,@H is tothe fourth D . If therefore the firft be equemultiplex to the fecond, as the third ts to the fourth , and if there be taken. the equemultiplices to the firft and to the third they fhall be equemultiplices to them which were firft taken, the one to.the fecond ther other tothe fourth : which was required to be proued.. The 4. Fheoreme. The 4.. Propofition. Ff the fir[t be vnto the fecond in the fame proportion that the thirds to the fourth’: then alfo the equemultiplices of the Sirft and of the third, onto the equemultiphes of the fecond and of the fourth ,accordyng to any multiplication,fhall haue the fame proportion beyng compared together. Demonstra- E20%e ; :3 therfore( by the 3: of the | any other equemultiplie ces, thatis, Mand N. And forafmich as E is equemultiplex ynto A as Fis pnto Cand bnto FE and F be taken the e» guemultiplices K er L, fifth) K is equemultis plex to A, as Lis toC: and by the fame reafon alfo M is equemultiplex ynto Bas Nis toD. And feing that as A is to B,fois Cto D, and of Lteg he ae A and C aretaken eques: © ps ‘45. RG pony multiplices Kand Ly . Yop a Boa At | and likewife of B CF. De ones pk SS ms te ghey are taken other equemultiplices namely,M and.N,, therfore if. Koa in | | atfo I, i DH NP of Euclides Elementes. Fol. 138. alfoextedeth N ?andif it be equall, itis equal, andsf it be lefSe it 1s lefSe (by the conuer/e of the 6. definition of the fifth ). And K_ and L are equemultiplices to F.and F: and Mand N are other equemultiplices toG and 1. Wherefore as EistoG,fo is Fto Hd by the faid fixt definition. If therfore the firft be ynto the Jecond in the fame proportion that the thirdis to the fourth: then alfo the eque- multiplices of the firft and of the third ,ynto the equemultiplices of the fecond of the fourth according to any multiplication {hall haue the fame proportion bee yne compared together: which was required to be proued. en Afumpt. Wherfore feng it hath bene proued that if K exceede M, L alfo excedeth Nand if it be equallit is equal: and if it be lefSe, it is lefSe: it is manifef? that if Mexceede KN alfoexcedeth L: and if it be equatlit is e- quall : and if tt be lefse tt ts lefse : and by this reafon as Gis to E, fois F. to F. ss . oS ee Corellary, » -on Hereby, itis manifett chat if there be foure magnitudes proporuonal,they {hal alfo. by conuerfion be proportionall : that is, if the firft be vnto the fecond, as the thide-isto the fourth ; then by conuerlion as the feconde is to the firft,'{o is the Fourth to the third?” The 5,Theoreme.. The 5.Prapofition. Ife a magnitiid e be equemuttiplex fo a magnitude, as a parte taken away of the one, isto 2 part taken away from the other: the refidue alfo of the ane,to the refdue of the other, /hal be e- guemultipler,as the wholeds to the whole. ANC V ppofethatthe whole magnitude AB hepato the whole magnitude C Bice D equemultiplexas the part taken away of the one, namely, AE, is to ”™ the part taken away of the other,namely,C F, Then I fay that the refie dueof the one namely ,E'B, 1s to the re/idue of the other namely, D F equemul: tiplex as the whole AB is to the wholeC D. How multiplex’ A Eis to CF, fo multiplex make E'B to CG. And forafmuch as( by y first of the fifth) A Eis to GF equemultiplex,as A Bis.toG F-but AE is toC F equemulti plex, as A Bas toC'D. Wherfore A Bis equemultiplex to either 4 of thefe G FandC ®. Wherfore G F is equall yntoO D.-T ake a- 6 way C E-which is common to them both. Wherfore that which re mayneth namely © Cis eqnall "puto that which remayneth names * 4 ly,D F. And forajmuch as AE is to F equemultiplex as E Biss P+ toG C but G.Cis equall pnto DF, heretal A Eis toC F eque- multiplex as EB isto F D. But A E 1s put tobe equemultiplex tos CF, as ABistoCD,wherfore E Bis toF D equemultiplex , as ABistaC Dowherfore thevrefidueE Bis tothe refidue F D ec guemultiplex,as the whole AB is to the-wholeC D. Iftherforea pg 5 K king. magni A Leinma, or an alfurmste A Corollary. ¢€ onuerfe pre- pertzon. fo Conftruttion, Demon ftre- £3092. - —— =... = i —s > eo — a Fa ge Rete ac | a a Oe a ha se me = -— a == 269 a VK Hi : 1 tH i ™ | | |] M th } Hh. - aL ae | i" } a 1 Hl! | AH bf i} int ‘ 1 ' at | mh ye + hi H) ii } ; } ; aah Hh a an 1G ee ' | Ba A | il BRT} Ai) aie bh m1 ee i! by EM EL) 2} 1) ae i | i} A! Ht : 4 | : i j | \ | » Is ty Wat) \ Th ~ | - i iy WW j 1) ay th NY | { AY i] 1 1} AL ny j Ps bi | Pay 4 ie, ij H : i) ii | f | ee ; ti) ee 4 4 1 in J iif | ee t | | t Bi t 4 the } hi} : 4 BAILS ff ; Py i a if i i. ft) 1 ae 4 1 In te uu u's 7 wil tt ‘ea } i : The. Hide} ) (| a I a 1 hi q Nl f _.! Be ie ir q | se 5 i 4 hy in aif } : | {| ae hie t 4 f | r f | ‘ HI | ; hie | ae Pn |) . ' | : if 4) i} : ii z| | \s - | it i} i ) |) ni ne)! >), iF f° ap || oy A Th j Lie We Liv ) ‘| ( | 1 ; he iii - . i 7 HH] d wi a i) i iV \\ ' ; *) aR I i} ! Th iy iy | aw The fifth Booke magnitude be equemultiplex to a magnitude as a part taken away of the one is to a parte taken away of the other : the refidue of the one alfo to the refidue of the other fhalbe equemultiplex as the whole ss to the ‘whole; which was required to be proued. SyThe 6.Theoreme. The. ‘Propofition. If two magnitaaes be equemultiplices to two magnitudes,¢o any partes taken away of them alfo,be eqnemultiplices to the fame magnitudes:the refidues alfo of them fhal vnto the Jame magnitudes be either equall,or equemultiplices. = ppofe that there be two magnitudes A Band CD equemultiplices if, | to two magnitudes Eand F,and let the partes také away of the mage | ,. |nitudes A BandCD, namely, AG andC H be equemultiplices to Tos cafe its : * the ame magnitudes E and F Then I fay that the refidues G Band rit pita D, are vnto the felfe fame magnitudes E and F either equall, or els equee multiplices. apa ecu Ree The firfte Suppofe firft that GB be equall vnto E. I’ hen Lfay that Viativattion TE Dis equall‘ynto F. VntoF put an eqhall magnitude C K. yan Demoniire. 24 forafmuch as AGis equemultiplexvnto E,as C Fiis.. tiene ‘vnto F: but G Bis equall vntoE, sae he 7. Theor eee, | : The 7: Propofition. Equal magnitudes bane to one ‘hem the felfe fam om agnit ude, of Enclidés Elemenites. Fol.1296 one and thefame proportion. And:oneand the Jame magni tude hath ta equall magnitudes one and the Jelfe Jame pro- portion. Se Whintle fad Te ; : | bk : ARV ppofe that A and B be equall magnitudes, and take-any other.mage , - CO} nitude namely ,C. T hen L fay,thateither.of thee A andB. bane mn See thefe A and 'B one and the fame propartion, T ake the equemuttiplices of A and B,andletthefame | be Dand E: and likewife of C, take any othen multyplex, — | and let the fame be F . Now forafmuch as Das ynto Lez | guemultiplex as Ets to B but Ais equall ynta B,therfore, ..\) (by the firft common fentence ) D 1s equall yntoB .. And | of C there is taken any other multiplex FW. hereforetf D. 7 exceede FE alfo excedeth F:and if it be equall.at 1s. equal, and if it be leffe it is lefSe. But Dand E are the equenmls | tiphices of Aex B, ana Fis of C an other multiplex:W here? foréas Ais toC, foi Btol. | | i I fay moreoner that Chath toeither of. thefe. A and'B ~ one and the fame proportion. For the fame order of con firuétio remaining, we may like fort prove, 3 Dis equal . wnto Ete there ts taken an other. multiplex.to C, names B Ly, E.. Wherefore if F exceede D,it alfoexcedeth Es and if it be equallit is equall: and if it be le[Je sts leffe . But F is multiplex toC: and D. oz E are other equemultiplie ces. to A and B.Wherfore as C is to.A,fors C toB .Where fore equal magnitudes haue to one and: the fame mea onitude, one and the fame proportion : and oue and the fame magnitude hath to equall magnitudes one and the felfe fame proportion : which was required to be demonftrated. Sa The 8.Theoreme. The 8.Propoftion. Unequall magnitudes beyng taken the greater hath to one and the fame magnitude agreater proportion then hath the leffe. And that one and the fame magnitude hath to the lefea greater proportion then it bath to the greater. Raat al /ppofe that A B and C be pnequall magnitudes , of which let AB be RASS 2 Sgr e : Wes the greater and C 9 lefSe. And let there be an other magnitude “whate ba \5 hee (RP : Hg ii-,| foemer namely, D. Then I fay that AB hath yntoD a greater pro Ei portion then hathC to D: and alfo that D hath toC a greater propore tion, N YA a 2 AY bs : f rt4anwe-e , fA ay EQ ye !toC one and the fame proportions and,that:C, alfo hath.ta, eithe; of Tie fir bare ef this Propo- fetion demon-~ firated, The fecond part demon firated. ahi \Fbeffth Boike ciousthen tt-bath to\d BE or forafmuch apd B asgreater.then C,let there be tas ken amagnitude ejuall ynto Cnamey BE. ORE OR to cost eae ee Bois Voie . & The fir part Now then the'tefse of thefe two magnitudes A Eand EB being inultiplied of this Propo. Will at the length be creater then D. RY c fitton demon First let AE be lefs e then EB: firated. = RA. , . 7 e j “% | = , The fir/t cafe till that ye produced d¢ breitter a ofthefame, then D3 dnd-lee'thue multiplex be F 2. ‘<“B ‘ i e7 fo forth encreafing by one till fuch “s 7; be | a) tyme as the multiplex of Do taken jo ag pasa | . be first greater then K. that is, that YB bhee bs | | ; e multiplex which amongft althe mul} 8] . | | | | ; tiplices of D doth firft exceede Kee to Opa Fo et | | let the fame be N which here is quae sees ss FRCD\ Dor ole . dru ble to D jand the fof ) rultiple breater then K. N OW forafmuen as Kiss yz \ firft multiplex lefSe then Ni therfore Kis not lefse then M. And for that F Gis “tod E equemultiplexas6 Fis to EB, therfore ( by the first of the pfth ) FFL is to.A B equemultiplex as F G isto 4 E: bat F G i equemultiplex to A Eas Ke is to C.Wherfore F Hand -K are equemultiplices to A'B and C. ‘Agane forafa much as Fl 1s equemultiplex to E B as K is toC; but EB is equal pnto'C; therfore G Fis alfo equall bnto K. But Kis not lefse then M: ~wherfore ney ) ther alfois G H lofse then M.' But FG is greater then D.Wherefore the whole ; FFLis greater then both thefeD and M. But both thefe DandM ire equal a pte Ns for Mis triple to D and M and D to ether are quadruple to D., and | _ Nalfo ts quadruple to D: “wherefore both thefe M and D are eqnalynteN} but F lis greater then Mand D,wherfore F H excedeth N., (thatis, § mul- , tiplex of the firft namely of A Bexcedeththe multiplex of the fecond, namely, of D.) But K excedeth not N (that is the multiplex of the third nanely,of C., excedeth not the multiplex of the fourth namely of the fame D:) anaF Hand K are eqnemaltiplices to AB; and C, and'N is a certayne- other multiplex ta D:Wherfore (bythe 8. definition ).AB hath to Da greater proportion ther hathCto D. ee head I fay morecuer that D hath to € 2 greater proportion then D hatr to A B. partofthe -Lorthe fame order of conflruction ftill remayning,we may in like fort proue that propofition de- NT 1s oreater then K and, that it1s not greater then FH, And Nis nultiplere eeclesiacsan 10 D,and F Hand K ave certayne other equeniultiplices to A BandC Where fore D hath taCa greater proportion.then D hath tod B. , a But of Euclides Elementes. Fol.134., But new fuppofe that A E be greater then EB. Nowthen E B beyn 19 the lefSe,and bung oftentimes multiplied, will at the length be greater then D.Let at be fo muliphed and let the multiplex of E B, namely, G HA, be greater then D. And bw multiplex G His toE B jo multipler let F G be to AE and K to F C. And by the-former reafon may “Wwe proue thatP Hand K are equemulti= | : | | 2S plices to A Band CLikewsfealfoletN be multiplecof Der alfolet it be 5 firft multiplex which ts greater then FG: wherfore arayn F Gis not lefse then M. ButG E11 greater then D: wherefore the whole F H,excedeth D ee M, that is ,N: but K excedeth not N. For FG whichis eriater then GH, that is, then K exceedeih not N. And fo followyng | the fame orter we did before, we fhall | | performe tle demonstration. Wherefore dnequall magnitudes beyng taken, the. K H C BD LAN greater hath to one and the fame magni o) Je mpc cer iret oe pen enc netted tude a greaier proportion then hath the lefs e:and that one and the fame magnte tude hath ti the lefSe a greater proportion then it hath tothe greater: which was required tobe proned. ) q For that Orontius feemeth to demonftrate this more plainly therefore I thought it mot-amifje here to fet it. Suppofe tiat there be two vnequall magnitudes, of which let «4B be the greater, and C the lefe : and let there be a certaine other magnitudej;namely,D.Then I {ay firft, that 4B hth to Da greater proportion then hath Cto D. For forafinuch as by fup- polition 4/4 is greater then the magnitudeC : therefore the magnitude 4B contay- neth the fane magnitude C, and an other magnitude tefides. Let E B be equall vnto Cand let ef E be the part remayning of H the fame magnitude. Now 4 Eand EB are ES ojos Rupa pd enon eyther vneqtallorequall the one tothe o='* * ) i c ther. Firft et them be vnequall: and let >: As FE yu B B pete einsil e/ E be liefl: then EB. And vatoe# E the 1 leffe take any multiplex whatfoener,fo that’ oo 2 Roy pole Hee it be preatei thé the magnitude D: andlet® thefame be FG, And how multiplex FG ?> Desi istoe E, fomultiplexlet GH beto EBy >) yy and KtoC., Agayne take the duple of D, saciatalatt inniia aie which let beZ, and then the triple , and lee NS the fame bee. And-foforward,alwayes adding one: yntill there be produced fuch a multiplex:e-D which thall be-next grea- terthen G#( thatis, which iinagelt the multiplices of D, by the contin watt additi- : onof The fecond cafe of the Firlt differtce of the fark park, s i — - eae Wh | | nt dh ; i, | i” | | ' t ' f - ; : : | rin ! Halt : . it ‘ie Mie | any }! ah ' q i] iid ‘ ip iz a 7 i} i / | i | ae if i ' ah 7 W ’ Wah: | |) an } 1 We tig 4 Et, ; 4 , A ‘a = Hitt iu s/ } ity | ( 1 ie | oe ' ' A, {| j Yi it \ - Th ya 1) M "| t “a Ht fc t 4 Y » : | AL ny it~ 4 y a ] uM 1 3 i | , n° i 4 til ) |) {i> i} 7 | sd ' | { '; ij hee ay ‘e | ! at we | ; ae f if yy } | ae ‘ MY hi a wo ii s i 5 1 vu HE - l H 7 fh! iW 4 4) oe Ua a A WY q tl . nt i u MH Hu) f I | g a) i: a ; ' im fi § 4 ( ae q i . | Va an i Wy Wy ie iri iD if D ‘iy im) | } | d |) a4 par Oa | - Ee J co M | Vy : Bik ) i ayy)! : aay re |i) : § ai ‘ 3 ry f he Wo cnt 1 i 5 4 ,? 1| 4 H i } \ y } ; 5 } i] By : j } i } 4 ) f FF A] mit oe Be : 5 mn | : ! i At : iW : he} iH 7 f Ch te He ' im | Hy 1 cs ' ¥ it * * { iq 1 ; " I ‘ r an 1) 1 ae 2 f | ; ' ; ne . : \e { fe i, H } 4 f \ yj iH ; Wt +) { ih 7 ' 1 J { Ny ; , : e Hy ed i " 5 4 if r ? ; i . aA Py) i) At A : | f fu He i 4 » i H , b ;) ' } : * ta i if t b 1 ae \ + av Fhepfth Booke 36 onofong, doth irltbeginne toexceede, G/t).and.letthe fanie be-AGwhich letHe qua- s druple.to D: Now then the multiplex G Ais thenext multiplex leffe then Nand hes. Demoupieatié foreisnotieffe then 27, thatis, iseithereqnall ynto it or'steater thensit , And forat- ofthejame | Much asFiGisequemuluplexto 4,48 7471s to FB, therefore How mtiltiplex F'G fir diferte, isto dE, fo Multiplexis F Hto AB (by the firlt of rhe fift)... Buc how multiplex FG isto 4 E, fo multiplex is K to C, theretorehow multiplex F H ista AB, fo multiplex is KtoC,. Moreoucr forafmuch asG HanéK are equemultiplices vnto FRand C: and E Bis byconfiruction equall ynto C, theriore (by the common fentencé)G\A is equall vnto, XK. But G # is not lefie then 247, 4s hath before bene fhewed,and F G. was put to be greater then D. Wherefore the whole 7 Zis greaterthenthefetwo Dand 412. But D and M4 areequall vnto DH. For A_is quadruple to, And 47 being triple to D,doth together with D make quadruple vnto'D ‘AWherefore F. His greater then WN. Farther, Kis prouéd to be equall to GH. Wherefore\K is Jefe then N,. But F'H.and Karee- quemultiplices vnto 4B and C, ynto the firft magnitude,I fay,and the third: and N is a certaine other multiplex vnto D,whic reprefenteth the fecond & the fourth mag- nitude. And the multiplex of the firft. excedeth-the multiplex of the fecond: butthe multiplex of the third excedeth not the multiplex of the fourth . Wherefore e4 B the firft hath ynto D the fecond a greater p’Oportion, then hath C the thitd to D.the fourth (by the 8.definition of thys booke). L F000 19h 2 But it\_4 £ be greater then EB, let E34, the leffebé multiplied yntilhthere be pro- me, duced a multiplex greater then the magnitude D: whichletbeGA. And how multi- ¢ * + ATS nan = SO 8) Se See Pas. 7 ° a ie ee ae —<_ | — * 2 - aeoa + a — ‘ : = > —- - — aes z - ~ : — = - ~ =a ——— — — : > oa tte BR. ve Ss a fo . Fe Se «5. nee 2 < = ee . e ~ ~* a-+ ot " . a —— - ~ . - . 7 7 = aoa - : : ; 7 r ; = = “ : : ~ a = =—= Sr 4 =<—_ ae - . cami ~ ~ patil eS * . eer na z rt a --- > : = . ~ ——_ + # an ~~ eo > = Se : > -- am = ~ = ~ = —s ~ = a a = = = Sy orgies teal ~~ = 3 = eH Se — ee . os 3 - = — - —— ——— ae . 7 = . = — =. = = — + > os ——s 53 7 ar —. ee ~ _——— ape ee en gta noe — mee eee = am oe ~— sa _ —— - - — ~ ~ a = ———— —e! z - - _— - “ — - pent * * aon eel on _ “ = =~ = Se ee ad ——— = > —— ———- — - Sf == ear ee a ~ - — = ~ — — — = ————— ——— —— = 2 s > _ -- ba lt ” = ~ a F = —— > = a — ans a a * = — ~ — - = —= — : a Sa = = ae = ae . = ae = = = == <== == = - - — = ~ <= = - 5 = > - . suse tlc —_—— panenpnclethlatiiglegtita - staee. --- ~ -- ~--—- — : -~ - “ — ie ‘ ——_ ~ == = cages, plex GA is toE B, fo multiplex let F G beto 4 E,and ‘K alfo toC. Thentake vnto D . fucha multiplex as is next ereater then Fey andagaine let the fame be DN which: let be quadruple to D. And in like fortas be) O40 egy : fore may-we proue, that the whole F#7'‘s ge ae in vnto A BS equemultiplex as GA isto Es? BAST gui93 whi’ H and alfothat FA & K are equemultiplices «5. NE REO Ee tate Ve ynto.42 &.C;4nd finally.that G His.equal gt Ole E REALE 5 Sage vnto.X...Andforafmuch asthe multiplex.V. ee Se es isneve greater then F G:therefore F Gis not Pid Wipers 83 BAS leffe then AZ. But GA is greaterthen D ly Late ae: conftru@ion. Wherefore the whole F H's De ereater then Dand 47: andfoconfequen- . 9 . - on Hel ly is greater then WV: But Kexcedeth not J: MES LAAN jon Ww HH for K isequallto G HM for how multiphx \\ 2 Lat iawn: i} ii K isto EB theleffe, fo multiplexis F Gio ee ey | pfiothe greater > Butthole magnitudtse--- |) <>: tac ! i i which: dresequemultiplices’ ynto. ynequill 9d st0nt3 ; ao) mavnitudesare accorditig to the fame proportion vneqnall .. Wherefore Kis leffe then eal} # G;and therefore ismuchdetle-then QV. Wherefore. againe the multiplex of the firft Fi exceedeth the multiplex of the fecond: butthe multiplex of the third. excedeth not the toe multiplex of the fourth. Wherefore(by the 8:definition.of the fift) 4 B the firft,hathte wie a. D the {econd,a greater proportion, then hath Cthe third to D the fourth, D Third afr Butnow-if..4E£ beequall vnto E 8B, eyther <= 43 parce. . ofthem {hall be equall vnto C.Wherfore mtes;! i673 90 © goles rp either of thofe three:magnitudes take ecues Http multiplices greater then D. Sothatlet FC be: yo ol gy big iz Mel +e multiplexto 4 £,and GA vnto £B,and Gae) aaa ety roigniom yas eel gayne to C: which (by the 6.comon fentenge) | «gy. (i. shusiage 947 Sela agers ig on 3 fhall be equall the onetothe other. Let Ni ale - e+e wor bo As Stestte es fo be multiplex to D,and be next greater them’ og 9d AO Jaf sel intre) AY ola euery one of them,namely,letit be qnadriple }o seb eeeeesee yg 4 . O08 is toD . This coftruction finifhed,we may agaitt bye 01g it? 23 S03 OPS Ad Jolie prouethat F/and-K- are equemultiplicest£O; = ye taete tt 8c sit A Band C:and that F H the multiplex ofthe: pps oi) 2d 9" iy : snogeibbs firft magnitude exceedeth yz the multi plec.ol =: sere eter gett itt she fegond magnitudes. sind that Aeherme He Hegcorie daisy. .2i sada ) AOE Hea? 293 29 the pies ; r _ f was +. a of Enclides Elementes. Fol.4at. plex ofthethird excedeth not che multiplexof thefourth. Wherfore we may conclude that 48 hath vnto D a greater proportion,then hathC co D, Now alfo. ffay,that the felffame mabnitude D hath vnto the lefle magnitude Ca grea- ten proportion, the ithath to the greater4 BiAnd thismay plainly be gathered by the forefayd difcourfe,without chaunging the otdeér.of the magnitudes & of the equemul- tiplices| For feing that euery way it is before proued,that £ 4 excedeth N,and-K is ex- ceeded of the Re fame'Ws therefore cohuverfedly V excedethikX, butdoth notexcede FH. But N_ ismultiplemto\D chatis to the firftand third magnitude : and’ XK is mul- tiplex tothe fecond namely,to C; and £1 is multiplex to.the fourth, namely,to 42: Wherefore the multiplexof the firlt excedeth the multiplex of the fecond: but the mul- tiplex of the thitd excedeth not the multiplex of the fourth. Wherefore(by the 8. defi- nition of this fift boéke) Arche firft hath ynto€ the fecondsagreater proportionjthen hath ‘D’the third tore 4 ZB, the fourth: whichwasrequired to be proued. 635 Theo. Theoreme.., The 9..Propofition, eMagnitudes which haue.toane and, the fame magnitude one and the Jame\proportion’: areequall the one tothe other. And thofe magnitudes into wbome one and the fame mag- ' nitude bath one and the Jame proportionzare alfo equall. PN V ppofe that either of thefe two magnitudes. A.and B hane ROS to C one and the fame proportion, Then I fay that A is e —s * guallynt B. For if it be not, then either of thefe A andB fhould not hane toC one ex the fame'proportio ( by the 8.of9 fifth) but by fuppofition they haue, wherefore A is equall bnto B. Axgaine, fuppofe that the magnitude C haue to either of thefe magnitudes A and Bone.and the fame proportion, T-hen I fay that Ais equall pnto'B. For if it be not, C fhould not haue to either of thele A.and B one and the fame proportion. ( by theforimer propofie.. A tion )sbut by fuppofition tt hath, wherfore Ass. equall ynto BWher fore magnitudes which hane to one and the fame magnitude one and the fame proportion, are equall the one to the other. And thofe magnitudes ynto whome one and the fame magnitude hath one and the fame proportion are alfo equall:- which was required to be pros wed. hp T he 10: T beoreme. ; The to. P ropofition. QR - OF mapas compared to.one and the fame magnitude, that which bath the greater proportion; isthe greater « And that magnitude wherunto ont and the fame magnitude hath the greater proportion,isthelefe. . . | Lij. Suppofe The feeond pave cf ebas propofstion. The fir pars of thts Propo- fitton demon~ Arated. The fecond part proned. rb “> a all er i te. a ‘- -_ ents setts —_ ~ .. as at ai” ca %. = : _ ban eB atin, a re = ——S= = = The farit pare ef thts propof- sion proved, The fecond part demon- firated. LP hefifth Booke ay 4 ney Pepe that Ahane'to C a greater proportion then B hathto C220 hen T 2 Ne fay that Ais greater then D For if it be not then either A is equan puto rie DonlefSethenit, But Acannot be equal ynto B for then either of thefe Aand B fhould hane'yato C one and the fame proportion (By the Foo: ; of the fifth ):but by fappofition they haue not » wherfore Ass. note quall pnto'B: Nestheralforis\A lefSethen B, forthe fhoald Abane: 1oC-d'lefse proportion then hath B to C (oy the 8.of the fifth) but by fuppofitionst hath not. herfore.A.is.not lefse then B..And it is al/o.proned that itis not equall, wherfore Ais oreaterthen B. Agayne fuppofe that C haue to Bia grédter proportion then C hath to A.T hen I fay that Bis leffe.then A. For if it-be not,then is at esther equall yato it or els oreater, but’B cannot be eqnall bato A,for, then fhould Chaye to either of thefe.A and B one.and the Jame proportion (by the7.of the fifth Ybut* by ‘fuppofition it hath not, wherfore B 1s notequall yuto As Neither alfo\is\B greater then A, for then [bould C have to Ba leffe proportion.then ithath tor (by the 8.of the fifth ) but by fuppofition it hath not : ~wheree fore Bis notgreater then A: And it was proved that it is note qual pato A,sherfore B js lefse then A. Wherfore of magnitudes compared to one and the fame magnitude, that-which hath § Lrede ter proportionts the greater. And that magnitude ~wherunto one and the Jameinagnitnde- hath the ereaterproportion, is the lefse. ~ Which yas required to be proned. Coustrifion. Demonflra- tion Tie "i. F hebreme. | Phe i. Propofition: Proportions which are one and the felfe fame to any one pro- portion, are alfo the Jelfe fame the one tothe other. RSA phpofe that as Ais to B JoisCtoD,andas Cis to D ,jois Eto. F, SKM Then I fay that as A is to. B/o is Eto F. Take equemultiplices to A, SIC and E, which let be GK. And likewife to'B ,D and F také.ae , 4 iv . : 4 . \ fe " .~ ’ | ie 2 ° : analy * i e KVSy ‘ of Euchdes Elementes. Fol.t4.26 deth L: and Mn“ \; : ; Gy os ifit be equal: : ) ) tisveguall) wwx03xKs yi aspedtinmonns tay A ea and if it be | 4 | lefSe, it is leffe (by the ame cone uer fe Wher | ore if G exe | nm L then i K alfo exe , | cedethN, : and ifit be | | equalitises : | | quall and if | | it be leffe, 16) | | | | isteffe. Buts | Ger K are G A N eariemultiplr ~~ ee ae tes of Ate E.And L oN ave certaine other equemultiplices of Bee F.Whers fore ( by the 6. definition as Ais to B fois Eto F. Proportions therfore which ave one andthe felfe fame to any one proportion are alfo the felfe fame one to the. other: which was required to be proued, 9 Bie: ; | Gh D mM K nm MY eee SaThe 12: T beoreme. » “Fhe v2 Propofition, If there bea ntimber of magnitudes how many foeuet propor- —. tional: as one of the antecedentes 1s to one of the cofequentes, So are all the antecedentes to all the confequentes. Mae ppofe that there be anumber of magnitudes how many foeuer namely, aS, BC DIE SP in proportion : fothabas Ais to B, fo let 'C beto D, Nand E to F.T hen I fay,that a6:Ais to B;fois.A,CE,toB,D,F.T ake equemultiplices to.A,C,and E.. And let the fame beG, H, K . And likewife to BD and F take any other equeinultiphees which terbe.L,M,N. And becanfe thatas Aisto'B, fois C to D and Eto F. And to A,CjE are taken equemultt- plices GFA; K sand likewife to B,D, Fare taken certaine: other equemultiplices L,M,N, If therefare G exceede 12, H alfoexceedethM, and KN : and ifit be equall tt isequalland if it be leffe it is. leffe( hy the connerfe of the fixt defint LI. i. tion Conftrattion, Demonfira- tion® 7, ; : - P = —— —— =— 4: ata neat ia ey ie - . ~~ ~ —- = = es —— = —E en * nite ns ae owe — te = . ie ; orn " | = : : ae . fy | F - 2 t 3 ; : : i Be bBo ge ; | Zt Li WA ~ a , as 3 . ' : . 6 ; ' & ; te & Nt le iH | K | a AG (3238.22 F ALON magnitudes A ,C,E. For (by the first of the fift) ( if there be a number of mige nitudes.equemultiplices to alike number of magnitudes ech to.ech ee weadente anemagnitude gs to ne, female ameall magnitudes,to all), dnd by the Jame regfon.alfatz,and L,MN., are equemultiplices to the magnitude Bard to the magnitudes B,D,F : Wherefore as A isto B, [ois A,C,E, to B SOF (Ly the fixt definition of the fift ). If therefore there be a number of magnituces how many foeuer proportional: as one of the antecedentes is to one of the conye _quentes , foarealkthe antecedentes to all the confequentes-; ‘which-pas requind to be proued. ~voqr $a T he 13, Dheoreme, Thex3:'Propofition. Ff the frft haue vato the fecond the felf Jame proportion. that the third bath to the fourth,andif the third baue ‘ynto the .fourtha greacer proportio thé the fifth hath to the fixth:.the ‘ » fballthe; ft allo baue pnto the Jecond a greater proportion ahem bath he feo thefixth, Jaws 7 bik, Si Habe oF eatleninicsary EAN ppole that there be fixe magnitudes, of-whichlet be the frit, B ONG the fecond, C the hind, D the fourth, Ethe fifth, and F bee q » be Suppofe that.A the farSt hane vnto B the fecond the felf fame propo ua ton that C thethird hath to D the fourth. And let C the tbied i ‘nto D the fourth, agreater proportion. then hath E the fifth to F the faxth. 50% XG Then of Exnclides Elementes. Fol.143. T ven I fay that A the first hath to B the fecond a greater proportion ,then hath E the fifth to F the fixt.For forafmuch as Chath to'D agreater proportion then Conitruétion haib E to F,thérfore there are certaine equemultiplices to Cand E, and likewife any other equemultiplices ‘whatfoeuer to D and F which being compared togee thu the multiplex to C fhall exceede the multiplex to D but the multiplex to E foal not exccede the multiplex to F (by the conuerfe of the eight: definition of ; Pt ee ed eA A Ot ETE, —) ——oo et A OD eee em. q ' | | | “siete Peal oe A N | deebes “wake codes boot i . Gee Sisto B,fois to D:andto A and C are taken equemultiplices M andG.And DewmonStre- 810%. Wen phall-ehe firyt alfohane: putothe-fecond a. greater proportion then bath the Hib tathefreehdV hich was required to baprovediy 9) Vy | 3 <" , \ nt = c \\ al G a os ; erent webat ? \ LETS * ej 2 s F gy Anaddition af Campane. . , 21 + + tA 02 LL. i. if : ‘The fifth Booke An addition If there be foure quantities, and if thefirt bane vuto the ‘fecond a eréater proportion ef Campane thé hath the third to the fourth: then fhall there be [ome equemultiplices of the “hitund the third which beyng compared to fome equemultiplices of the fecond and the fourth, the mul. 1 tiplex of the first fhall be greater then the multiplex of the fecond, but the multiplex of the | third {ball not be greater then the multiplex of the fourth., rgsR '. Which is thus proued. Suppofe that ‘AB haneynto C a greater proportionthé hath D to E.And let AF betoC as Dis to E,Now then by this propofition & the téth,A Fis lcfle then AB. Let it be leffe then A B by the quantitie F B,And multiply F B yntil there be produced a quantitie greater then C, which let be GH: which alfo mutt be {uch a multiplex, as D be | yngfo oftentymes KK nun, mays OPO producea quanti- i G tienotlefethé E: 9 + whiche multiplex A F B D |} let be K. And Jet L biakinalhaibienictgaadid nacht: iis G be fo multiplex c r to AF,as G His to! =e oi FB,or K ro D.Now | then by the firftof ae eee ; aisha Be te Oe GP this booke LH is | equemultiplex to is ED eae ABas KistoD. | | AndletM be to E Oo rae ME. FET a STE Ee the firft multiplex i greaterthenK: & . | : . | | | let N beequemultiplexto Cas M oe E.Now then Nis the firft multiplextoC greater then LG : Forforthatas Dis to E,fois A Fto C,and Kis equemultiplex toD asGL i i ee therfore LH fhall be greater then’ N,And forafmuch as K islefle then M, therfore that which wasrequired to be proued,is manifeft, aa Although this propofition here put by Campanenedeth no demonftration for that itis but the.conuerfe of the 8, definition of this booke, yer thought Tit.noe worthy to be omitted;for that.it reacheth the way to finde outfuch equemultipli- ces,that the muluplex of the firft {hall excede the multiplex of the fecond, but the multiplex of the thitd fhall not exceede the multiplex of the fourth, 4. ee SeeT he 14.T beoreme. é i; he i4:: Propifition. : “t " Sibe fifi bane wenn Delica eee ale roportion that »: thethird hath vata the fourth:andsftk Bee icersoter then the third, the Jecondualfois greater then the fourths and ifit be equall itis equall:.and if it be leffe st is leffe. ee Suppofe of Euchides Elementes. row AV 4 LN SOX yy as MAAN) Ce Fol.14.4. |“ ppofe that there be foure magnitudes, of which let A be the firft,B therefore Bis grea= ter then D . And mn Icke fort may we prone that if A be equal yntoC,B fhallalfo be equall to D : and if Abe leffethen C,B fhall alfo be leffe then D.If therefore the firft haue ynto the fecond the felffame proportion that the third hath ynto the fourth’, and if the firft be greater then the third, the fecond alfois A greater then the fourth , and if it be equall it is equall,and if it be leffe itis leffe: which was required to be proned. Sa The 15. F heoreme. Lhe 15. Propofition. Like partes of multiplices,and alfo rheir multiplices compa- red together ,haue one and the famé proportion. a 7 AY ppofe that AB be equemultiplex to C, jas D Eis toF. Then I fay,that as Cis to F, WUlee,| [01s AB to DE . For forafmuch as how \ - , Multiplex AB is'to C , fo multiplex is DE to F; therfore how many magnitudes there are. in AB equallynto C, fomany are therein DE equall ‘nto F. Deuide AB into the magnitudes equall'ynto C that is into A0,G Land AB? and likewifeD E into the magnitudes equall pnto F , thats, into D K, KL and LE. Now then the multitude of thefe AG, GH, and HB ys equall to multitude of thefeD K , KL, and LE | And forafmuch as AG, GH, and” FAB) are equall-the one to 9 other: and likewifeD K , KL, andl E, are alfo equall the one toy other : there foreas AGis to DK fois CHikKl and B Ls) Demon tra- 640%. B C D A | | Conjiru ion. boone G | BK ] wi | Demon Stra- £20Me “= Lliiy. to L E, Demoftration # of alternate proportion. (onfirnttion. Demenfira- fon. wi The fifth Booke to L. E.wherefore( by the 12. of the fift )as one of » antecedentes is to one of the confequéntes fo are all the antecedentes to all the confequentes. Wherfore as AG isto DK, fois ABtoD E.But A Gis equall vnto Cand likewifeD K to F. Whereforeas C isto F, fois A BtoD E. Like partes therefore of multiplices and alfo their multiplices compared together shane one and the fame proportion that their equemultiplices hane: which was required to be demonftrated. Sa T he 16.T beoreme. Lhe 16. Propofition. If foure magnitudes be proportionall : then alternately alfo they are proportionall, | she sid Vppofey there be foure magnitudes proportionall, namely; A,B,C,O, i /othat as Ais toB, JolerC be toD,-I hen I fay, that. alternately alfo ~ they [hall be in proportion that is,as A is toC fois BtoM.T ake equee multiplices vnto Aer B,and let | | the fame be Ey F. And likewife to Cand D take any otherequer multiplices what foener , andlet the fame be rand H.Andforaf= mitch as bow multiplex E is to A, fomultiplex ts F to B, but lke partes of multiplices ¢ralfo their m1 ultiplices haue one and the felfe June. yo portion the.one. to. the on | ge = ther by the former Propofition).- |. Jj, | Wherefore as Aisto B fot Eto ' o coe ae F .But as Aisto B fois €to.D, |. | a wherefore ( by the 11. of the fift), B,.iA..\B. asCistoD fois Eto F.Agames 8 x | a forafmuch as G and FZ are. equemultiplices to C and D, but like partes of mitle tiplices and alfo their multiplices bane the.one to the other oneand the [hele proportion( by the 15. of the fift).W herefore as C isto.D, fois G to fA! at is to D, fos E to F Wherefore as Ets to F, fois G to H (by the 11, of the fift ). But if there be foure maguitudes in proportion; and if the firsk be greater then the third,the fecond al/o is.greater then.the fourth: and if it be equall itis easels and if it be “ff it ts lefse iv the 14.0f the fift ).If thevfore E-excede GF hall alfo exceede 1: and if it be equall, its equal; and if tt De leffesat is lefJe. But Fund Fare equemultiplices to AandB sand Gand Fare certaine other egug pe (by the b.definition of the fift as Ais toC, fos 0 maltiplices ta Cer Divers RIGS OI BtoD. If therefore there be fc ure magnitudes proporti onall , then. alt ernately — : = a ‘ * ot ipThe , * | I . Fe DH G eh SS — . - alfo they-are proportionalf sawhich wag required £9 be proweds. of Eucldes Elementes. Fol.14.5. SayT he.17.T heoreme. The vy. Propofition. If magnitudes compofed be proportionall, then alfa deuided they /hall be proportionall. aaa Y ppofe that the magnitudes compofed being proportional be A BBE, LESS" CD, DF) fothat as A Bis toB Eyfois&. Dito DF. T hent fay,that a" denided alfo they [hall be proportionall as AE isto BE, foisC F to DF. Fake equemultiplices pnto AB; EB; Ck, F D,andlety famebeG H, H K ,LM,andMN. And likewife to E'B, and FD, take any other eques multiplices what foeuer, and let the fame be KO ,and NP... And forafmuch as how multiplex G@ His to A E, fo multiplex is KH to E ®, therefore. how mule tiplex G Huistod E,fo mee isG K to,4 Boy the firft of the fifth ).But howynultiplex.G lis to A E, fomuyltiplexis L M toC F: Wherfore how mule tiplex G Kis to AB, fo multiplex is LM to C F (by the 11.0f5 fame). Againe forafmuch as bow multiplex LM is to CF fo multiplexis MN to) F therfore how multiplex. LM isto CF, fo multiplexis LN toCD ( bythe firit of the felf fame). But how multiplex L Mis to € F fo mul. tiplexisG K.to AB. Wherfore how multiplex G K_ isto AB, fo multiplex is LN to CD. Wherefore GK andLWN, are = sok to.A Ber D. ) Avaine foralmuch as how multiplex FL K_the firft is K | H A \¢ to EB the fecond, fo multiplex is MN the third to FD the fourth, And how multiplex KO the fift is to EB the fecand, fo multiplex is NP. the fixt to FD the fourth.Wherfore( by the fecond of the fame) how multiplex HO compofed of the firft and fift ts to EB, fo multiplex is MP compofed of the third and fixtto FD. And for that as A Bis to BE, fois CD to DF: and to AB gz CO aretaken equemultiplices G K andl. Nand likewifetoE Band F Dare tae. | ken certaine other equemultiplices, that is, FAO,and |. dp MP. If therefore G_K exceede HQ then LN alfo | exceedeth MP : and if it beequall st is equall:andif g 3 > \y it be leffe, it is leffe ( by the conuerfion of the fixt defi- nition of the fift).LetG K exccede HQ: Wherefore K H-common to them both being taken away ,the refidueG H, fhallexceede the refidue KO. But if GR exceede HO, then doth LN exceedeM PR: Whereforedet L, Nexcede M'P:and MN -which is comon to thé both being take away,7 refidue L M fhall éxceéde thevefidne NP wherefore if GH exceede KO, then fhall LM exe cede NP. And in like fort may -we proue, thatif GIL be equall onto KO, then Ly M fhall be equall vnto N P : and if it be leffe, it fhall be leffe: but G a an Demonratia of proportion by dinifion. Conflruttions Demonjfira- tion. = - —- —- —— : 7 -> 2 ha, a - Se toe eS = ee a em e ——— ae My " Wey . aH hh RL). Mill t Ht | | 2 \ | | ail ay | i th 1a mp Lt) | } : \ eb ii} 1) Pe Tn : \ Wa ¥ uy ie f! BY » M ' Ht ie & i \ ii ¥ bit! } It Diy i H! Ni | aa me ||} WD] ; mi} hy | i :f f ae tS U | : i! i } iF i Wee fy ail iui Hy ; WEY | | ith | Bi Nt y 1 Walt ra : itl My ri ‘ab. — } Ww : | it i] : | I ‘hy i Ae) ; | Hy) a WATE: ii HH. iit upab HE |), i | : itt ih Hh i i} a i Aaa. f i! i Waa Ma wie fle yA aB. || BY hil ie 4 ‘ab i OB ‘ |) ie || Pe i me ri ' M| My ||) 1 al P| i He i} : : } Vy Z 4 We | Se. Nie i A i! a t if i () ae FH \i Bei 1 Ih |! |) yi! i} neg hit ‘ iH § a y ij at * | a au if tt 4 | : i} i be ite ' | bi : eh rh: |! ! i] i} i i | Hi 4 v) : i} i f ji/ } 13 in - ] i ' 4 1 Bh | ie if } | ae | bi! |) ) | ie li, ul \ eg the} WM i). H at Hi my af i} Hy Wi ' } 4 it ihe he By r att { S| ‘ { p aft 4} ” 4 ; , 4 t | 5)! an i | Wi it ' ba t ‘ “ 4 re) a) if h | \ 1 ee a le Demonftrats- On 0 cal tion by com- polticne This propefi- gion is the con uerfe of the former. Demonftratio deaceng toan suzpofsibility. The fifth Booke and L Mave equemuttiplices to. 4 E and GF;,and: likewwife KO andN P are cer tayne other equemultiplices to E'B, and F D.Wherfore as A Eis to EB, fo CF t0 F D( by the fixt definition of thefift).1fcompofed magnitudes theres fore be proportionall,then alfo denided they fhall be proportional : “which was required to be demonftrated. SweT he 18. Ti) beoreme. The 18. Propofition. : Ifmacnitndes denided be proportional : then alfocompofed they fall be proportional. | Pies’ ppofe thatthe magnitudes deutded being proportional, be AE, EB, Go .c Fer ED fo that as AE isto EB, folet CF be to F D.T hen F fay, ==" that composed alfo they fhall be proportionall, that is, as AB is to BE, foissC Dto DF. For if AB benot bntoB Eas Dis toF D, | then foal AB be'vnto BE as. CDs either vnto a magnitude A : leffethen F Dor vnto a magnitude greater, Let it firft be vnto a a leffe,namely, to DG. And foralmuch as, as ABisto BE fois rk F’s | ) B D> CD to DG: the compofed magnitudes therefore are proportio- nall , wherefore deuided alfo they hall be proportional ( by 5 ¥7- of the firft). Wherefore as AE isto EB, foisCGtoG D . But by [uppofition as A EistoE B, foisCEtoF D. wherefore ( by the 11. of the fift )asCGistoG D, foisC Fto FD. Nowthen there are foure magnitudes, CG,G DCF and FD: of “which the firft CG is greater then the third CF . wherefore (by the 14. of the fift the fecond G D's greater then the fourth F D. But it is adfo put to be lefJe then it : which 1s impofsible. Wherfore it can not be that as AB is toBE) fois CD to a magnitude leffe then F D.. In like fort may we prone, that it can not be fo toa magni tude greater then F D.. For by the fame order of demonStration; it would follow that FD is greater then the Jayd greater magnitude: which is im ipofsible.W here foreit muff be to the felfe fame. If therefore magnitudes deuided be proportios nall,then alfo compofed they fhall be proportional : ‘which “was required to be prod, ©} | ieee sex » Se The 19: Theoreme. “The 19. Propofition. | Wf the whole be to the whole, as the part taken away is to the "part taken away: thenfhall therefdue be wnto the refidue, as thewhole is to the whole. 2 | wae Suppofe of Euchidés Elementes. Fol.14.6. OEE —- — — — —— ~ _ - — = = ——————= = am = i — _==> = ie 7 = — as — rs Gx V ppofe that asthe whole AB is tothe whole CD, fo At That which : HES iTS the part taken away AE. tothe part taken away . ithe emp of sbis ! i pC fay, that the 'refidue EB fhall be'bn-. | acai a Fo. the vefidne RDpas the “*~phole AEB is to the whole C D,; For: > ~ | wabiiees: i | for that as the whole A Bis to the whole C D, fois AE taC Fx | ' a Mere ‘I ye i penerally of al therfore alternately alfo ( by the 16, of the fift) as ABis to AE, : magnitudes. foROD t0-Ck tad forthat then macnifudes compofed are’ , 4 Propartional the famedeuded alfd-are.propor tionally by the 17. | of 3 fft kthereforess B Ets tad; fois D.F to FCW herfore alternately dfof by thet, of the fiftasB E18 toD 3 fois Bd to FC. Butas AE isto CF, fo(by fuppofition )is the*vbole AB to the whole CD. Wherefore therefidue EB hall beynto. ‘ the refidue F Das the-whole A Bis to the whole CD. If thers fore the whole be tothe whole, as the part taken away is tothe part taken away, then fhall the refidue be bnto the refidue as the whale is to the whole: which was required to be proued. Oy ee gi ee oS ee is Sea «, A Lemma or Affumpt. And forafmuch as by fuppofition as ABis toC D, fois AE toCF: andale 4 Lemma, ternately as ABisto AE, foisC DtoC F. Andnowit is proned, thatas. AB istoC D, fois E Bto FD. Wherefore againe alternately,as ABisto EB, fo isC Dito F D. Wherefore it followeth, that as A Bis to AE, fois CDtoCF: and againe, as the fame A Bis toEB, fois the fame DtoDF. | Sa Corollary. » And hereby. it is manifefl, that if magnitudes compofed be 4 "x \ proportionall,then alfo by conuerfion of Proportion which of connerfnof Jomeis called Proportion by Euerjion,and which is,as before °°" it was defined, whe the antecedent 1s compared to the excefe, ‘wherein thecantecedent exceedeth the confequent ) they fhall be proportional.’ song Ue The 20.T heoreme.. The 20.Propofition. Af there be three magnitudes in one orderand as many other magnitudes in an other order which being takentwo and two i Oe gpd ai eae parodia — : qualitie This propofi- Sion pert aie weth to Pro- portion of e~ quality tnore dinate propor- tionality. Tbe firft dif- ference. Dewmonitra- tone The fecond @ifferences The third differences * wreater ther C, and”. “Tbe ff Booke qualiticin the firft order the firkt be greater then the third, then in the fecond order the firft alfo al be greater then the third: and if it be equall it {hall be equall :.andifit be lefe st Shall be leffe. ye} SAS SIwWAM say N30 s the , % Se V ppofe that phere beshree magnitudes in one order, namely , A ,} BC 4 DOM let there be as many magnitudesin an other order which let be D, ERY E:, F, which being. taken. twovand two in ech order, let be in one and the Jame proportionsthat is,as Atte B,yoletD be toE andas Bis to C foktE betoFy 3: alydyo\ 415 ota At ee a And firft ofequas \\». | Wore hk. B -C\\.0\S Bo BAP Bape ditie let A be greater . thenC. IhenI fay, that D alfois oveater then F: and if it be es guall st is equall, and if it be leffe,it is leffe. Por forafimuch as Ais. : —+-— es me o£ « there isa tertaine os’ “Y°° ther. mavnitude., Op" namely ;B; but the 1 greater hath. to-one* + * and the fame magni tude a greater proe ! portion then hath the leffe( by the 8. of the fift).Wherefore A hath rnto B a greater proportion then Chath to Bs Butas A is to By fois D to E: and.as Cis toB, foisF to.E ¢ by the Corellary of the fourth y the fift). Wherefore D hath ‘puto Ea greater proportion, then F hathto E.. ‘But of magnitudes compared to one and the fame magnitude that-~which hath the greater proportion is the greds ter (by the 10. of the fift) Wherefore Dis oreater then F a \ ‘Bat now if Abe equall ytoC, Dalfofball be. equall:ynto F » Forthen A and U haue ynto B one and the fame proportion ( by the firft part of the fenenth of this booke). And for that as Ais toB, fois Dto E,andas Cisto B, fois F to E:: therefore D and F haue vuto E one and the fame, proportion . Wherefore by the firfrpart of the 9: of this booke D isequall nto F; ~~ ° But now [uppofe that A be lefSe then C.T hen alfo fhall D be leffe then F For by the, 8i0f this booke.C {hall bane nto Biagreater proportion then bath Ato B..Butas Ais toB, fois. Dito Eby fuppofition, and as.C isto B, fo bane we proued is Fto E. Wherefore F hath bnto E a greater proportion then hath D toE wherefore by the firft part of the 10: of this booke Fis greater then D. STIR Ay If therefore ie : | of Euclidés E:lementes. Folia I fe therefore there be threé magintudes in one Order dnd afmany ot her *hagrite sudes in. an other order which being taken two and two ineth.order, avest one and the fame proportion, and if of equalitie in the fir St order the firit-be greater then the third then in the fecond order alfo the first hall be greater then yahird; and if it be equall it [hall be equall : and sf it be leffe, it fhallbeleffe: which was required to be prowed. $a The 21.Theoreme. The 21. Propofition, If tiere be three magnitudes in one order,and as many other magnitudes tn an other order, which being taken tivo aad tig in eche order are inone and the fame proportion, and their proportion ts perturbate :if of equalitie in the firft order the rft bt greater then the third, thé in the fecond order the firft alfo ball be greater then the third: and ifit.be equall tt hau be equall : and if it be'leffe it halt be leffe. —s ane let there.be as many other magnitndes. in an other order, which let be DEF: which being taken two.xx two in ech order let be inoneand or the Jame proportionsand let their proportion be perturbate. So thatas Ais toB;folet EbetoF¢r as Bis to Cfo let D be to.Eyand of equalitie letA be greater then G. Fhen I fay that Dialfa 4s, greater then F ;.and if it beequall its equall xandsf tt be leffe ats leffey 9 4 mart wbirft let. Abe grease. Aa bee: AR zit ter.thenC. And forafe — | 3 | | much as A is greater | then qthereisacers taine other magnitude, namely, Betherfore(hy bbe 8 of fift ) A Gath |. a creater proportio bus |” f 5B: then Chanbte DB: | : — a , Pe ee ae (iy — i me mm Buia Ai to B for Bt. toF: and againe as Cis to B, fois Eto D (by , ~ Nee | / Pou I be 3 ay [ he | Hoe CoreHaryiof § fourth of the fiftWherfore E. 4.5 bath Into Bagrestip Se aalS ate ee proportion stheitsbath to.@.. But that ma nitude Soheranto one and the fame sragnitnde hath the greater proper tion 4 isthe loffe( by the. 19..0f. the ftp li here WIJ. ore (ts —+-_+—-& rs Oe it +—~_- ~ : ' 7 te S eens eat + : - ~*~ é . Whore FH "7 ~~ "ae be ‘ —_ oo : on. a wad oe ars Yy " eh. x f , 3 ; . - fre V ppofey there be threemagnitudes in. one order namely,d, BC, and Thixpropofes tion pertat~ neth to Pres portion of ¢a quality sre . perturvate proporttouge bity The firlt dif- ference. a a se ae + Theil Lhe fifth Booke > foreFisleffe then D. Wherefore Dis greater then-F.) 2s «eee gi, PARED FE Fe Po 3 SK Oye) SH ONO Sc aot, as guall ynto C , we. may | occas ayo proue that Dis es guall ynto EF’. For then ( by the firft part of the 7 of this booke) Aand C [hall bane bnto B one and the fame proportis on. And for that as Ais to B, fois Eto Fer as Cis to B, fois E toD: therefore E hath to ets ther of thefe D and F, one and y fame propors , 5 2° ¢ tion . Wherefore by the fecond part of the 9.of the fift D is equall pnto F. | Pie The third > Likewife, if A be leffethenC, D alfots leffe then F . For then C fhall hane differences "ynto ‘Ba greater proportion, then hath A to BY bythe 8.of the fift . Wherefore E alo hath ynto Da greater proportion then it hathto F . Wherefore by the fecond part of the v0. of this booke Dis leffe then F . If therefore there be three magnitudes in one order, er as many other magnitudes in an other order, which being taken two and two in ech order are in one or the fame proportion «> their -\ proportion is perturbate, and if of equalitie in the firft order the firft be greater then the third then in the fecond order the firSt alo fhall be greater then y.thira: and if it be equall it fhall be equall: and if it be leffe it fhall be leffe_: which was required to be proued. SaT he 22.T beoreme. The 22, Propofition. D> ome ee ee | D iw) yr Teeter ein to tae nats leant ti "" If there be a number of magnitudes,how many foever in one order ,and as many other magnitudes in an other order which being taken two and twoin ech order are in one and thefame - proportion,they fall alfo of equalitie be in one and the fame ~. proportion. “eee ae 77 V ppofe that there be a certaine number of magnitudes invone order |, : ee Proportion of TW 4 As for example : A,B,C, and let there be as many ather magnitudes equality in ore > 9% Sox! V2 ate propor Aa )y,,| 70 an other order, which let be D,E,F, which being taken two sonality. = arid two let bein one and the fame proportion . Sothatas AistoB, f ket Dbe to Ejand'as Bis to C,folet E be to. Then I fay; thar. of equalitie Jrgd 3 REY of Euclides Elementes. Fol.14.8 they fhall be in the fame proportion,that is, as Aista.C fois DtoF. T ake bnto A and D equemultiplices G ex H, and likewife to'B ex E take any other eques muftiplices whatfoeuer, namely, K and Land moreoueryntoC and F take any other equemultiplices alfo what foeuer namely, Mand N . And fora/much as, as A isto B, fois Dto E: and pnto A and D ure taken equemultiplices © and Fl raid likewife mnto Band E.are\takencertaine, other equemultiplices K_and Li therfore bint | ( b ry the 4.0f the fift e G.is to RHR. eR pays ees. the fame aes IS Wh PS A reafon.) as K isto M, fous L. to N . Sein therefore 9 there be in order three | | C Di El Fy SY ee ma lonitudes G, K, M. 2 G K er as many | | | other magnitudes in an other order, namely, H,L, N., which being compared m A 8B ‘to to twoare in one and the [ame proportion , therefore of equalitie( by. the 20+ of the fift if N exceede M ,then [hall A exceede G: and if it be equall it ‘hall be equall : and if it be leffe it fhall be le/fe . But G and H are equemultiplices nto A and D and M and N-are certaine other equemultiplices ynto C and F: there fore( by the 6. definition of the fift ) as Ais toC, fois DtoF. ’ Soalfo if there be more magnitudes then three mn either order, the firft of the one order fhall be to the laSt,as the firft of the other order is to the laft..As if. there were fourein one order, namely, A BCD, and other foure in the other order, ‘namely, E-F GH, we may with three magnitudes A,B,C, and E,F hond ‘thatas Ais toC, fois EtoG : And then leaning out in either order the fecond and taking the fourth, as leaning out Band F, and taking D and FA, we may proue by the/e three and three A,C,D,and E,G,H, that as Ais to D Jo is Eto H. And obferuing this order, thys demonStration “will ferne how many foewer the magnitudes be in either order. [f therefore there be anumber of magnitudes how many foener in.one order and as many other magnitudes in an other order, hich being taken two and two in eche order are in one and the fame proportion, they {hall al{o of equalitie be in one and the fame proportion ; which was requt ved to be demonjtrated. Rien HO Coafirn ion. Demonftra- 50M When there ave more then three magns- tudes tn esther order. ABCD EFGH Proportion of ix eqnaltty in perturbate proprotiona- bitte. Coustrxz ion. Lake Dato. - Remora. 820° er eet The 23. Fheoreme: ° "The 23-Propojicion. seat ff there be three maonitudes ii one order znd as many other magnitudesin an other order which Leyny taken two Co two in-echt orderaretn oneandtheyame proportion , and sfalo tbetr proportion be perturbate:then of equalitie they [hall be in one and the fame proportion, | eg te as V ppofe y there be in one order three magnitudes namely, A 1B Cele = ; be take in an other order as many other magnituds which let be D, ELE = whic h being taken two and two,in eche order Vet bein one and the Jaime proportion : | | and [uppofe. that their pro portio be pers turbate. So that as Aisto | B, folet E be ’ Fandas Bis taC, falet D (2 , : beto Be The | | d fay thatas . | + ¥ i, Ais toC,fois. pe D to. P. | : | “3 ee: Be aN A;BD ¢ S quemultiplie ces, andlet. ) : . ~ a , 4 ; . SH a ee, & 3 ser < Be AK: and ~~ | ‘ikewife vnto C,E,F take any other equemultiplices what/oener ,and letp fame be LL M'N...And fora/much as G and 1 are equemultiplices ynto A ana B,but the partes.of equemultiplices are in the fame proportion that their equemaltiple we ss ces are(byy 15.0f the fift)-wherfore as Ais to 'B fo is Gto H.* And by¥ fame _reafon alfo as E.1s to F fo ts Mto N.But as. Ais toB fois Bto F.Wherfore (by Vil. of) fift ) as.Gis te H, fois Mto N. And forafmuch as,as Bis toC fois D ‘to E and bnto Be Dare taken nie Fee K: and likewifeynto C ‘and E are taken certayneother sty sey ices Land M: therfore( by the 4-of 7 the fifth ) as His to Lo is K to Mand alternately alfo{ by the 6. of the'fift) asBisto D fois C tok. And forafmuch as Fl and Kare the eqsemultiplices of Band D, but the partes of equemultiplices are in the fame proportion that Bead bake their of Euchides Elemente. Fol.14.9. their equem -egineniiltiplices are (by the iscop tbe fife) therfore as Bis to D., fois F100 K But as Bis to D fois C to E therfore (by the tr of the firft ) as F2 is 10K, fou€ to B. Agayne fora yfmuch as.Land M. areithe equemultiplices of Cand E, therfore as Cis to Bfois L to M.But as Cis to Bisfoss. 4. to.K i therfore as, itto K fois L to M ,and alternately by the 16.of the fift as His to L fo is K (But as 4 isto B {ois E to F (by fuppofition )wherfore as Gisto H,fois F,{by the rtofthefift.) Agayne forasmuchas and 2 are equemultiplices ynto£aud F, therefore agayne (by the 153 of the filth)ias Bis cok; fos ALtowN, Butas Eis ta F ,fo hauc weproued is G.to H, wherforeasG ts to Ho is e4/ to N (by the a n.of the fift) Aid Yor that as-B isto €; fois DtoL Cby fuppofition)Andyntos andD aretaken e- ghemultiplices Hand KX: andvnto.Gand £ aretakehcertayne other equemultiplices *» : ret Land A7théréfote asV7 is to L fo is K to AZ, by the 4 Of this booke). ~9\ ‘Bat ihissproued thatas Gss.to FL, Yo. in Mite NA Seyng\ therefore that ther re dn oneal thrte rhage vatoelty CexFLyl:) ad as.many other maghitudes in an other order , namely, KSMGN, -whiche bemg ta: ken two and two in eche order, are in one and the fame’ proportion, and their proportion is perturbate therfore of equalitie( by the 21, of the fifth if G excede L.,then fhall Ksexceede'N and if it.be equall it hall be equall: and if it be le[Se it fhall be leffeBut G dnd K are equemtuitiplices vito Aand D and Land N ave cértayne other equenultiplices bhtoC and F. wherfore as A is toC, fois D to F( by the 6.definition of the fifth). If therefore there be three magnitudes in one order,and as. many other magnitudes int an other order, which Sih taken two and two in eche order arein one aud the fameprapartion, and if Wi their proportion be perturbate: then of equalitie they {hall be in one and the fame pros portion: which was Fequired to be proud. — From this marke * firtt to the famemarke agayne,you may if you will in flede Of Theons arpumentes which feeme fomewhat intricate,read thofe argumentes following printed, with an other letter, which gre very perfpicuous andbriefejand followed-of the mott interpreters. : ttt ae This Propofition is alfo true if there be more themthre magnitudes in either order: ~ As forexample,let there befoure.Sothatas: AistoB, foletE betoF, and as Bis.ta Cy. folet D betoE, &asCistoP,foletQbeto |} D. Then] faythatas A is to Pj foils rok. eee For forafmuch asit is beforé proued; that Ae», 440 isto(C,as DistoF taking away B& Exthete «| .. fhalltbe three magnitudes A,C,P, inone of | ” der, ‘and as many other in‘an_othérérdery = f°) {p } 98 nan¢ly ,Q,D.F which beingstaké\intechiore, fi yh) 4g dertwoand two,are in one and thefame pro. + | ortion,and their proportié is perturbate, 7 wherfore'by the forinet PrOpohtiOAIo Pi Opie as QistoF \whichwas.cequired tobe dee...) Li cathy And euen as by the demonftration in three magni.cdes is taken the proofe in 1 foure magnitudes by leauing outoneof 4 8 C P —— Mm. is. o- . ~* ess 79 Ain addition af Camp ante Noles we oS = - —— Se et ee eC e es $ os 4 ._ ¥. ‘ CORT vant, ‘Phe ffthBooke ta _ the meanes xfo by. the, denton{tration in foure magnitudes is taken theproofein. fiuemagnitudes by Jeauyng’ out two of the meanes: and by the demonftration in fiue,the proofe in fixe, by leauing out three meanes.And fo forward cOtinually, whichis alfo to be vnderftanded in'the former kynde of proportion of equalitie, . ol.3 Heel hauevnto the fecond the Jame proportion that the. third hatb.to. the fourth,andifthe fift bane.vnto the fecond - itd — | pmeoes “9 a me | xe ‘A ESEZOS pofuidof ebis E Fd. che fexthath vate F the fourth » Chen J fay,that. the cnn crching Jed together namely,AG bath ynto.C the Jecond the.’ wnultiplices, fame proportion that the third and (ixt compofed topes a ite ther,namely, D H, hath ynto F the fourth‘. For for * | ~ i tharas BGO rs't0 Cfo ts EH to F:then.alfoby-conuers »'s 9 6 * 2 sudes. fron (BC orollary of the 4, of the fift,.as. Cis.to BG, 2 the fa Teh fécondts greaterthen FLD the fourth ¢ by ther 4.of the fift ): And fordfmuch as AG is eqgalt pate Ej 7 CA of Euclides Elementes. Fol.1s0. ST he 25.T heoreme. T he'25. Propofition. Hf ibeve.be foure magnitudes proportional : the greate/! and ‘the leatbof them, (ball be greater then the other remayning. : Die V ppofe that there be foure magnitudes proportional. A B,C D,Eand ‘ Sy F.So that as A Bis toC D,fo let E be to F And let the greatet of then ULAe 1. £B 307. the left of them be F. Then Lfay, that thefetwo ma ienitudes AB and F, are greater then the two magnitudesC D ¢¢ E. Fora/muchas AB is fuppofed to be the greatest of all foure , therefore itis-greater then E.. .T heres fore from the greater A B cut of ( by the 3.0f the firft) vnto Evan equall mage nitude AG Land lkeiwife-( by the fame) from C Dect of dato F an equall mage nitude C FL. ¢ whith may be done, for that the magnitude C Dis greater then themagnituae Ex: for that as. A BastoC-D, fois ito F F therefore alternately asd Bis toB,foisC D toP (by the 16.of the fift). Bit’ 3“ 4 Bis creater then B-Wherfore allo D is. greaterthen. — jo) o.0- EW hich thingmay-alfo be praned.by 7 14,.0fthe fame.) <4 % otis risi a]: Now for that as A Bis toCD, fois EtoF:, but Eis e = | quall puto AG, and F 1s. equall nto CH: therefore As A Bis to CD; fo 1s: AG-to 6 Hand: forafmuch yas. t thé whole AB sto the-wholé CD; fois the part taken ae | way AG, to the part taken away CH: therefore the | refidue G'B( by the 19..0f the fft.).is Wnto the refidue | | ELD, as the-whale. A Bis tothe whole-C.D-+-ButAB 4 | the first is greater then C D the third : wherfore@ Bithe | gequall bnto.F-xtherefore AG a ud,F. are equall,pnto Z C land E . And fora/much as if onto thinges vnequall °° © ) siti be added thinges equal , all fhalt- be rnequall(-by the fourth common Jentence ); therefore feing that G Band D H are ynequall, and G Bis the greater sf ynto GB be added" AG and F : and likewi[e if ynto H D be added C Hee E; there fhall be produced AB and F greater thenC D ox E. If therefore there be foure magnitudes proportional the greate/t and the least of them, {hall be greater the the other remayning : which was required to be demon|trated. C xoddere follow certayne propofitions.added by, Campane,which are not to be.con- temned,and are.cited,eucn of thé beftlearnedjnamelyyof Johannes Regia montanus, m-the Epitome which he writed vpom Ptolome. DAR on = q. 1 be first Propofition. = ; ~ Uf there be foure guantitics,and if the proportion.of the first to the fecond, be greater M.t14. then Demon firetio beading to an o “ge :* . Bui DCfSsb8istie. tin other dee then the iy sy ofthe third-tothe fourth : thencontrariwife by conuerfions the propor- ston of the fecond to the first, fall be lefse then the proportion of the fourth to the third. Suppole that the proportion of Ato B be greater then the proportion of Cro D. Then Ifay-contrariwife by conuerfion, the proportion of B to Ais leffe-then the pro- portion of D to C.Forif theproportion of Bto Abe one andthe fame with the pro- portion of D toC,, chen contetfedly the proportion of 4 to Z is one'and the fame with the proportion of Cro 'D, which is coutraryto fuppo- eo. PON are tara OR G7; fitron. Barifthe . iat | 7 proportionpbmite iit 1edB webs bo hsd heed ts Tp edited kl efbegreaterthem 3 ay CD ash, the proportion of é DtoC;Theh Tet E bewntodtas Duis A TY (O.§ BN ES) T Lh NSIS FEO 3 TY TS to C. And by the 13..0f this booke,the proportion ef Eto 4 fhall. Be leferhen théprox, portion of 8 to 4.Wherfore (by thesirit part of the renth of the fame) £ is lefle thez, And clierforeby the fecod' partofthe 8 of thé fame, Thath vnto Ea greater proportio. thened hath to. Aud forafmuch es bycoritictfe proportionalitic’e4 is vito Eas Cis to'D sther fore by the 13 ofthe? fame,Chathto D.a gtedter proportion themhath 4 to, 4 which is contrary to re PRO Buon tok ef-was fuppofed tohaue So 2 z prenar proportion then hath C t6°D: Wherefore the proportion of Bro-d is neither one and the famewith the propottion of CtoW norgreater then it .Wherfoteit is leffe which was required to be proued.: 3 1.3 %) o, vet ; It may alfo be demonftrated dire@ly. For lerE be vnto Bas Cis to. D. Then co- monfiration of uerfedlyB isto£ as D is te C\And forafmuch'as 4 is etearer then E by the Arh partor the fame affiriatinely. the tench ofthis booke, therfore by the {coond parr of the ofthe fame #hachivate 4 a léffe proportion then hath Bto.£. Wherfore by thet 3\.of the fame B hath vntoed is : . leffe proportion then hath D'to.C : which was required to beproned, si: 2 Refecond Propofition. os. sv oh) 0 a Uf chere be foure quantsies,anaif the proportion of the first to the fecond ke greater then the ptee portion of the third tothe fourthythen alternately the proportion’ of the firftsothe rhea, hall be préna ter then the proportion of the second tothe fourth. r= erly ada lt} eat eoinaes ear 1% Let e-fhanevnto 2 agreater proportion then hath Cto. Di Thend fayalternacely 4 hath to C a greater proportion then hath B ta D. Far.one and the {ame'proportion | Dewonfiratis. it can not haue : for then altérnately e-4 fhould be to B as Cisto D, whichtis contraty- leading toa sbilitie. sm pofst E An other deo moni ration ef the fame atfirinatinely. to the fiippofitio. But ifithauea léfle proportié,let£ be ynto Cas BistoD. Now the by the 13. of this DOOKE wn gee 4 | OF A : ihath.ynto Ca greater ~ "5. = | © pronor SL ON ch a te EAT TT uf eyesore iniat od tOG. Wherefore (by the WRWR Hb Poer PA CY rice J ABST Qh 8498} Giligary of the tenthiaf.§ @ LB me rsliaidd hus “hm Otis hb O etme if igurcatet th. ee a es | Sa Witeriobe bythe Hirt Sw CD sds wstnyyy, “bus GQ hsowbos patrotuhes sdfthelamey:\) (6 Wings 302 bun Tsiorg 901, Anon odor 3 Hembra Be eredtet BID cence, oh ny baximas ancy askin portion thenhathe/#to * 4 £.And forafmuch as by fuppofitio E is ynto C,as B is to D, therfore alternately E is to B29CK 6 Di Whetfore by thet. of the fanie, Chathto D ayteater proportion then hats Byhichiscontrary tothefo ppofitio. W herfore'the proportion of 4 toCis neither one and the fame with the propertionof Bt D, norleffe then it; whereforedé is greater. Which was required to 4 hee! a This may alfo be demonftrated-affirmatinely, let be vnto B as Cis to D.Now thé by the firft part of the tenth of this booke,£ isleflé then 4: wherfore by the firft parte of the 8,af the fame,the proportion of 4 to Cis reater then rhe proportion of B, toc. But alternately Z isto Cas Bas to D. Wherfore(bythe 13 of the fume) ef hath toCa a 2 OO Seo greater of Euclides Felensentes. Fol.15t6 greater propottion lies hach B'to D ¥ which was required tobeproued. gg The third Propofition: Af there be fourequantitiesp and if the qecher en of the first to the fecond. be greater then the proportion of the third to the fourths. then by compofition alfo the prapartio of the (first ana fecond to the feconds/ball be greater then the proportion of the third-and fourth soithe fourth. Suppofe that the proportion of AtoB be greater then the proportion of Cto D, Then I fay that the proportionotthemwleed Sto £ ig greater then the proportion ofthe whole C D to D. For the proportion ofe4 & to B can not be one and the fame with theproportionief£C DtoD.: for.then by dimifion allo 4 thould be yuto 2 as.C is D:whichas contrary tothe fuppofition, Neither , ©, } alfocan it be leffe-For if it be poffible,lerir be: & : tek EB bevnto Bus C Dis toD. Now then(bythe ~ x 993) ayy Tapia yn 12.0f this boke EB hath vnto 2 a greater propor- ) tion then hathc-4 BtoB, Wherefore by the firit o att ofthe 10,0f the fine EB is prearerithen the. Who AB, And bythe common fentence Ets greaterthen 4. Wherefore by the firftpartiof the’8. of this booke E hath to Ba grea- ter proportié then hath 4 to B,But Eas to B as C.isto. Diby dinifidof proportion: for EBisto BasCDis to'D:Wherfore(bythe ¢ 2:0f the fume) C hath to Daigreater pro- ortio the hath 4 to B,which is cotraryato the fuppofition. Wherfore the proportion of 4B to Bisnotone.and the fame with the proportion of CD to D, neither alfo is it leffe then it. Wherfore it is greater : whichwasiequiredto be proued, This may alfo be demonftrated-affarmatiuely.Forafmuch as the proportion of of to Bis greater then the proportion of Cto'D : let £ be vnto BasCisto‘D. And fo by the firft part of the 10. ofthis booke, E thall belefle then A. And therfore by the com- mon fentence EB thall be lefle them.4.2.Wherforeiby'the firft part of the 8.of the famie AB hath vntoB a greater proportion then hath E BtoB. But by compofition E B is toByas CD isto D.For by fiippofition Eis vnto 4 as isto D. Wherfore (by the 12.0 this booke) 4 B hath to Bagreater proportion then hath C Dts D : which wasrequi- ‘reditobe'proucd: * : TE BEARS aL Pet GGL 7] T be foureh Propofitiont: If there Le foure-quantities and if the proportion Uf the first and the fecond to the ‘fecond be greater then-the proportion af the third and, ‘fourth to.the fourth: then by diuifion allo she-proportion ofthe first:tothe:{econd,jhallbe greater shen the proportion of the thirde te She fourth. ~*~ ae ae a eS Suppofe that the proportion of AB to B be greater then the proportion of C D to D.Then I fay that by diuifion alfo the proportion of <4 to Bisereater thenthe PYO- ne non Aratié « portion of C to D. For it cannotbe the fame. For then by compofition 48 fhonid be to BasC Dis to D, Neitheralfo can it ‘be leffe: , for ifthe proportion.ofC to D be greater then 2 the proportion ofe-4 to B,then by the Zo ott ct Samy Ge ee iam me proportion of C Dto D fhould beatba é-eteaterthenthe proportion.of 48 to 3: ¢ D Proncerm “tna oneal? whichis contrary alfo to the fuppofitid.. Wher- fore the proportion ofe-4 to B is neither one | | | gnd the fame with ‘the'!proportion of Cro. D or is tefiedhen it. Wherefokeitss: greater then it : which wasrequired tobe proucd, | COTE _ Demonfiratid heading to an snipofsibilsteen An other dé- monftration of the famé. leading to a am pofssbilitie. ~~ ee =. ae a ——_———-— RIES ag SEP i eg we . = - bm 5 a . =. S 5 ro = — - aN . . —— — - > SS —_——— = ~ a : HR ~" —— ee oe - C ~ — - —.. ss ae, ——— ~ - ——-— = saa - oa ae - a - = -_ “sis = - = * = ———~ ~ a =o a = —_ ‘ — ¥ na ~ >>. -- — anh : = S - — =—~._7t: ——s —. 2S > = -—— — = ~- _ —~ = - == m — = = > ; on = SS — -- - ~ : — 3. = = = 3 7. — ® bh, = Ree Ay Pte - . yo , — 2 : —— = —s = =—= ~ = === = ———— —-- Se SS es ee — ~ — 4 = —_——— i _ ~ - ete - = — ae =. a = - ee an —- — = = = — - o - = = -_—- = — — = S — SSS Ss Ss == ~ - — = —— . > —— — oe ‘ - ’ = SSS Se = = a -- — = = —— a - SSS ——— —— — <=> See a ee a? ~- —=— res . —S - = SS = — — = — — > —— ——= —- —= SS = — - : - = — . z rs a —_. = — = — = = ee ~~ < —- ——— —- —— ————= —- SS + - = = — ——_ — = = sa “ - be = = ~~ — a = s - —_ ae = 2 — = — > < naa i — — - : - -: =. === 22 . =,» — . —— 7 > . = = - . _ =. ~ —= - -- " 2 = — : - ed i — = > _ — — = — <= 7 ——— " = _ ™ = ——- -: = — - = ic _ s —- — — = = _ 7 —_— — — = => > ~— ———_ a a= — = - ae _— - - a = —~ - _ = + e ps in r ~ eS =_ on . . = . => - om * —— — = — . = — — = —— a 7 ~~ _ . = = = - db . a a 8 eens - —— —————— — = = — = a9 . - =——w - em a - — o . : = a a = ~ ~ . a tet — = = - = —- > — Gnd = —_—- ~ - ‘ . = >, e = - = os ——_— ee - ” - ae ae 2 - —s << = — — = - 5% _ —— - —_~ = 7 = ee : = — — — _ ~ = ET ~~. = —. - = = = = C ana = . / we ; P = m —-) = A NTIS, LITT TE ——. fs ee ge = . + - : - > . 3 " > z ~~ a >. SS ss > ¥ se - ~~ a etm ee . eS —- a — Le * + ; % Pa Sears ae =o : Ses , . Se ee : : ———.= = —— ; == = " a 2 ss = te - —<-% : ~= Sn ee area ae ; —— eae =e Ace SS ee = ates: — wees: — - = ~ aoe A = ~ - oe = — = = — — 7 _ —— rr . > x is _—_— —— —— - a ~ 3 -- — * — —— ma - _— — — == — = a ————— woke ‘ : BS —_—- —_ —- Sa — aie — _ e ——— - ————__- = . e Sled ——~e —- — = =—= = = = = —————S— = = — — : = ——_ SS —— = a —— > = == —— = —— ‘ = m- a a =e as = = SS eS eee a es Se — ~< ~ . - = es —= — = - ~ —— —= = pi eee aS ane a ceeenae _ ~- — = = === — ee . a en | | | 4 : ] : q dn osher dew monttrasson of che fame effir wasinely « Demouffra- $102. Demonftra- oo “ip Af there be taken threeg ‘of Ato 8. . oCin the The fame may-aifo be!proned afhrmatiuely. Suppofé thatE2 be ynto B.asCD.is to. D, Now then (by the firft part of the 10, of . a Ee ‘the fifth) £2 thall be leffe thene4’B: and there- fore by the common fentence, E is lefethen et, af. Rade. -» ‘Whertfore by the firft part ofthe 8. of this booke, | oe i ‘the proportion of £ to Bis lefle then the propor: SES nay ¥ tion of Ato By but as E isto Bifois C to'D:wher- PARAS At 4 at Bat fore the proportion. of CtaD,is leffe then the proportion of to By Wherfore the ro persion of ef to Bis greater then the proportion of C to D: which was Required to e proued. le 22 a soima gt be fifth Propofitions. coco. ck nate _ Uf there be foure quansitiesand if the proportion of the firfand the fecond 6 the ferona be greater then the oportion of the third and the fourth tothe fourth : then by. ewer fion the proportion of she firit and [econd to the firit, [hall be lefethen the proportion of thethird and fourth to the third. | ae nas Jiihaiee Suppofe that the proportion of 4B to B:be greater then the proportion of C Dto D. Then I fay that by euerfion the proportion-of 4 B to. is lefle chen the proporti- onofe DtoC:. Forbydiuifion by the former. rd | propofition)the proportion of Ato Bis: greater A oF K hed a3: Becie 191 then the proportion of CtoD. Wherefore by the. Sr firft of thefe ‘propofition's conuerfedly, B hark: airlw, Bo: , : vnto 44 lefle proportiéithé hath D toCz Whers: fore by the 3. ofthe fame by compofition; the. . proportion ofe-f B to 4is leffe thé the propor- tion of C DroC: which was réquiréd to be proved. fom m2idt i og The fixt Propofition. °° etaant rio | ee. quantities in one order,and as many in an other order, and if the proportion of the firft to the fecond in the firft order, be greater then the proportion of the firft to the fecond in the latter order: then alfothe proportion of the firft so the third in she Jirft order, fhall be greater then the proportion of the firfttothe thirdin the latter order. — Suppofe that there be three quatities in one order_4,B,C, & as many other quatities in an other order D,E,F.Andlet the proportion of Ato B inthe frft order be greater then the proportion of D to Ein the fecond order, and Jet alfo'the proportion’ of B ‘to Cin the firftorder, be greater then the proportion of E to F in the fecond order. Then] fay that | dices h sks the pro- portion . . ce» ee ey Ed Yo nolysa firfttor- © — Dsl sits: der, is reater ) Op ae eam | propor- g@ ._, Se tion of See ist) 7 ? L- DtoF vegty pee Ee. | Y Sei asest in the fecond order.ForletG be vrito Cas E isto F, Now then by the firtt pee of the £0 of this booke G fhall. be leflethen B, And therefore by the {ecoad parte of the net. ne ’ bd — ae eo a of Eucliles Elementes. Fol.152. fame, the proportion of 4t6 Gis greater then the proportiot Of Ato BoWhétfore the proportion of 4 to Gis muche greater then the proportion of Dto E. Now then let 47d€ vito Gas Dis to’ Wherfore by the firft part of the 10.0f the fame,e4 is greater the #7. And therfore by the firlt part of the 8.of the fame, the proportion of #4 to Cis greater then the proportion of to C. But by propoftion ofequality His to C as: Dis to F (tor His to Gas Disto E, andG isto C as E is to F. Wherfore by the f2. of the {aime of hath to Ca greater proportion then hath D to F: which was required to be proued. a? | reanihts g. I be feuenth Propofition. ’ ‘If there be taken three quantities in one order,and as many other in.an other order, an if tise proportion of the [econd to the third in the fir(t order be greater then the proportion of the firft to the fecond in the latter order if alfo the proportion of the fir[t to the fecond inthe firft order be greater then the proportion of the fecond to the third inthe latter order : thers fhallthe proportion of the firft to the third in the firft order be greater, thes the proportion of the fir{t-to the third in the latter order. Suppofe that there be three quatities in. one order 4,B,C, and as many othetinan other order D,£,F, And let the proportion of B to Cin the firft order; be greater then the proportion of ‘D to E inthefecond order,and let alfo the proportion of Ato B in the firft order, be greater then the proportion of E to F in the fecond order. ThenI fay that dhathto C agreater propor 4 ———+——+—____ a, Operas in ment nlp cues tion then hath D | to F. Thisper- * tainethtopro- o portion of equa- litie. For let G be vntoC,asD isto MB Tina ern E. And by the ¢ pensmenntnbicieehhe sch teiein dks firft part of the C 10.o0f this boke, we ee ee G {hal be leffe thé | 8.And therfore by the fecond part of the 8.of the fame, the proportid of A to Gis grea- == F ter then.the proportion of 4 to B, Wherfore 4 hath vato Gamuch greater a tend ft then hath £to£,Now then let & be vnto G as E is to F.,And by the firft partofthe 16. of the fame, 4 fhalbe greater then H.And by the firft partof the 8, of the fame,the pro- portion of 4to Cis greater thea the proportion of H toC, Butby the 23..0f the fame theproportion of H to Cis as the proportionof DtoF (for GistoCas DistoE, and His toGas£ isto F,) Wherfore(bythe 12. of the fame) the proportion of 4 toC is ‘greater then the proportion of D to:F, which was required to be proued, “ q Theeight Propofition. If the proportion of the whole tothe whole,be creater then the proportion of a part taker away,toa part taken away:the {hall the proportion of the refidue unto the refidue be greater then the proportion of the whole to the whole. Suppofe that there be two quantities 4B &C | D : from which let there be cutte of thefe magni- d E B. tudes 4 EandCF-:and let the refidue be E Band = F ‘D.Andlet the proportié of 4B to CD be grea. ter then the proportion of 4 Eto C F. Then I Gece _F ng ES TT 2 _ >. == ~ a oe = ~ r ~ ~ ,. Sw 4 er : A - =. Fr: 9 > =a = 3 . ~ a. = wit te et ~ En = = —- ~ —~ =s ~ de hens ~ SS, Sit as > el ——— >< Ptr ot ee *, e x + te ate ee . J = So es z = eens — a Siete ate — ie -: Fad = r ——— —— in SRLS : = — oe os = ee: ro 2 — = —— - PS) CES SOS SF eH. ———S [ym ye ee eee -—-—>- —- - vote. So —_—- —a = ~ =e = — See = > 7 —— ag Se eee ni a Si etn ne et ROR a he tls or ae = ——— . “ ” - - — ete eo - - a = ee I i — a a a - - _* “ === eS ae ental — 2 Se on > oe ee ea ae — ae as . - ~~ ~— - ase eee TL —_ —— _ = _ _— = - - - ~ = os: 72 ~ = Ps = ms - ra e\ -SS—- ars i eg Pe ers Ses —— — 3 - =—S- -x cae tx fe Ph. ee PD. <5 agin. — = = a — a P St v2 ee Pees See , ee Sey a re ee = _ nia = — ~ —_—— ——_ _ ~— —- hone Demonfira- $30 Hs calor’. Lhe fifth Bookes\: of thefe propofitions now-added) alternately, the proportion of 4 Btoaf Lis: greatet hen, the proportionof C Dto C F,And:therfore by evuerfion of propdction éby the-s; pfthefame)the proportion of AB to & Bas lefle then the proportion of:C'D:to F D, Wherforeagayne alternately the proportion of 4 Bto€ Dis lefle then thé proportié OE BtoF Diwhich wasrequiredtobe proved. «> ¥ ad 02 be: | (og The ninth Propofition. If quantities how many foex er in one order be compared to as MARY other ix an other ww- der,and if there bea greater proportion of euery one that goeth before to that wherunto.it is referred, then of any that followeth to thatwherunto it is referred : the proportion of thems “alltaken together vnto all the other taken together, phall be greater, then the proportion of ‘Any that ‘followeth to that wheruntottis compared,and alfa then the proportion of all thei taken together to allthe other taken together, but fhatl be lefethen the proportion of the prt rothe first. AS | A AAG | Suppofe that there be three quantities in one order, 4,B,C,& as many other in an o- cher order DE;F.Andiletthe'proportid of 4 to D be greater thé the proportié of 8 to E,ler aifotheproportio of B to E,be greater then the proportié of C to F,Thé I fay that theproportid of ABC raké al together to DEF také altogether, is greater thé the pro- partion of.B +> | : to E,and alfo then-the pro- Which was requi- InSTg red to be proued. ia nent 15% 0c ——— *% ’ g e 4 \ 2 asia tae Op ceo The end of the fifth bookes sins vos ~ « m wee 7 ~* ea % | : ie ee kh oe >i > " Fol.152. a Thefixth booke of Eu. clides Elementes, | H.tS $1} XTH BOOK E, is for vie and pracafe, a The argument mott{peciall booke. In it are taught the proporti- of this fixth XS}. ons of one figure to an other figure ,& of their fides booke. Yew 4. the one tothe other and of the fides of one tothe ¢ fides ofan other, likewife of theangles of the one to m theangles ofthe other . Moreouer it teacheth the ey defcription of figures like to figures geuen, and mar- <\\ ueilous applications of figures to lines, euenly, or “t) With decreafe or exceffe , with many other THeo- 7 ENES _ remes, not onely-of the. proportions of rightlined fi- Cm) se Sa -gures, butalfo of feGors of circles, with their angles, < : j 1 1 On the Theoremes and Problemes of this Booke depend for the moft part, the compotitions ofall inftrumentes of meafuring length, breadth, or deepenes, and alfo the reaton of the vfe of the faine inftrumentes, as ofthe Geometrical] {quare, This booke the Scale of the Aftrolabe, the quadrant, the ftaffe, and fuch other . The vie of necefsary for which inftrnmentes, befides all other mechanicall infrumentes of rayfing vp, of *¢ v/eofi- mouing, and drawing huge thinges incredible to the ignorant, and infinite other 47"! of 5 aE Ra Pa 4 Geometry ginnes ( which likewife haue their groundes out of this Booke) are of wonderfull and vulpeakeable profitc,, befides the ineftimable pleafirre which is in them. §g‘Definitions. 1. Like rettiline figures are fuch, whofe angles are equal the resp dé one tothe otber,and whofe des about the equal angles are fist, proportional. As if ye take any two rettiline figures, As for example, two triangles ABG y and DEF: iftheangles of the one triangle be e- ig to the angles of the other, namely , if the angle A be equall to theangle D;and the angle B equall to the angle E,, & alfo the an- gleC equall to the an- | zleF. Andmoreoner, 5 C eC oe foe Wehe fides whiclrcon- © 9! Fo ae ae | taine the cquall angles be'proportionall. As ifthe fide A B haue that proportion = 3 Nn,j. € T be fixth Booke the fide B C , which'the fide D Ehath to the fide E Fyand alfo ifthe fide BC be yuto the fide C A, as thefide E Fis to the fide F D, and moreouer,if the fide C A beto the fide A B, as the fide F D is tothe fide D E, then are thefe two triangles fayd to be like : and fo iudge ye of any other kinde of figures . As ifin the paralle- logrammes ABC D and EF GH, theangle A beequall tothe angle E, and the angle B equall to theangle F, and the angle Ce uall to theangle G , and the an- gle D equall to the angle H . And farthermore, ifthe fide A C haue that propor- tion to the fide € D which the fide E G hath to the fide GH, and if alfo the fide CD betothefide DB asthe fide GH is tothe fide H F,and moreouer, if the fideD B be to thefide B A as the fide H Fis to the fide F E, and finally, ifthe fide BA betothe fide AC as the fide F Eis to the fide E G,then are thefe parallelo- grammes like . 2. Reciprocal figures are thofe, when the termes of proports- on are both antecedentes and confequentes in either figure. As if ye haue two parallelogrammes ABCD and EFGH. Ifthefide ABto the fide E F,an antecedent of the firft figure toa confequent of the fecond figure, haue mutually the fame proportion, which the fide EG hathto the fide AC an antece- dent of the fecond figure to a confequent of the firft figure: then are thefe two figures Reciprocall . They are called offome, fi- cures of mutuall fides and that vndoubted- ly notamiffe nor vnaptly. And to make thys definition more plaine, Campane and Pellitarius and others,thus put it: Reciprocall ficures tre,when the fides of either be mutually proportional, asin the example and declaration before geuen. Among the barbarous they are called _Mutekefiareferuing full the Ara- bike worde. & | | 4 | : } | 2 by. Acre Cee eee Manan wh { | | | ; | © SS aca he i Roges I ee | 3. Aright line is fayd to be deuided by an extreme and meane proportion, when the whole 1s to the greater part, as the grea- ter partis tothe lefe. As if the line A B, be fo deuided in the point | C,that the whole line AB haue the famepro- 4 < er portion, to the greater part thereof,namely,to * AC, which the fame greater part AC hath to 5 cise the leffe part therof,namely,to C B, then is the line A B deuided by an extreme and meane proportion , Commonly itis called-a line denided by proportion hauing a meane and two extremes. How to deuide aline in fuch fort was taught inthe 11. Propofition of the fecond Booke,but not vnder this forme of proportion. 4. The Ra Te of Enchdes Flemenites. Fol.ts4. he Lheahtuileofa figure is a perpendicular line drawen from’ ree fours the t oppe tot be bafe. | definitions As the altitude or hight of the triangle ABC, is the line AD being drawen perpendicularly from the poynt A, being the toppe or higheft part of the ttiangle to the bafe therof B C < So likewife in other figures as ye fee in the examples: here fet: That : | which A here hee Vi calleth Bay : the alti- / | \ oh tude or) 2: ¥to ohio yx height 5 D G Ci. of a fi- gs ~ gure, in Saeed > / fe >. 4 the firft | bOOKEMT s\n the 35. \ et Propofi- tion and certaine other following, he taught to be contayned within two equidi- ftantlines : fothat ficures to haueé one altitude and to be contayned within two e- quidiftant lines,is all one. Soin all'thefe examples, if from the higheft point ofthe . figure ye draw an equidiftantliné to the bafe therof, and then f6 that poynt draw aperpendicular to the fame bafe that perpendicular is the altitude of the figure. ‘ y 5+. Proportion is faid to be made of two proportions or more, Adie wi when the quantities of the proportions multiplied the one into ibe other sproduce an other quantitie. i¢Ofaddition of proportions, hath benefomewhat fayd inthe declaration of the 10.definition of the fitt booke: whichin fubftance is all one with that whichis here taught by Euclide. By thename of quantities of proportions, he vnderftan- deth the denominations 6f proportions. So that to adde two proportions toge- ther,or more, and to. make one of them all, is nofliyng els, but tomultiply theit quantities together, thatis to multiply euer the denominator of the one by the denominator of the other. Thysis true in all kindes of proportion, whetherit be 2y the name of equalitic,or of the greater inequalitie, when the greater quantitie is referred to “f SE se the lefle : or of theleffe inequalitie,when the leffe'qnantitié is referred to the STA od sh yes ter : or of them mixed together - Ifthe proportions be like, toadde two togethet minations of isto double the one, to adde 3. likeis to triple theone, and fo forthin like propor- proportionse tiofis, as Was fiifficiently declared’ in the declaration of thé ro. and a1, defnitrons ofthe fift Booke. Where it was fhewed, that ifthere be 3. quantities in like pro- portion, the proportion of the firft'to the thyrd,is the proportion of the firft to the fecond donbledand if there be foure quantities in like proportion, the proportion of the firtt to the fourth fhall be the proportion of the firlt’ tothe fecond atipled: which thing howo'do was theretaught. Likewy{e in proportions: valike ; the _ proportion ofthesirlbextremetto the lait is made of all the meane proportions ict “BSR! JP“ | Nn.ij. betwene ,) eT ae mY - (ar Ai.) , aE na ii NM Me |i | mT fi } an 4 HN a mei Te | | } A tB) ‘a Th ‘Oi he th } H nyt | , f D1 ta at : | is { ' 1h ie: Wy f f i Ht ii t un ' q : Wee ee! Y “lH j ah i} | ' Ph H | } ty ny ' my : ie in : q } | : hit " ite Tih lth, i “ Ta a . i} 1 H { : ate : f j iH} he | i! ’ RY Ta : hae hy ) i hed | | (OLS), | : i] en Le 7] { a i ii i "a 1 ail be Hedh | y | i 1? ii WW ; aye N ee | foe # he ieee’ | : i il i} ; yeaa! Ve : . teed . Hoa +e Hy ; iT | ; } i! \ Pah 1 s b aa i 5 } Ht i 1% ij ye | \ f : ue \ ; Pune i ee ha) || : t bn i Daa : ay RMD Lie | " : { a i} Hea ay ail ii j | ? ih i} Sea De Wh Wi : ine 1 { iia : ee i) | a | ‘ 1] rt : A 7 aru | 4 \ \ | : ; i} \ : ! ele ‘ ; 4.) OD q | : ide - t i | : aut : } "y ‘ - ’ ' : \ a ag lhe be eA? - iV ? ae || aid ; i r Pt ih } ; 4 } y iat Ly waa et 5 | ; : 3. 4 ae | : + H tt - ee oe ; 1 : ¥ } : : i ) aT Hh i 1} aad He i i Moe Daan , ae hh ay vant q ass aa a , i ti is { me iat | : ai } rae a eat | } a ie ’ ‘pte pf! bee te at H : ’ - ! yal : ) if | 1 t > ti Bit iay i AME ; J i - | d . ai Wey > es j Pra, | Hine By a | on i em it eu Tt ely "4 i" 4 a ; “ag ii i a : OM 7 ae eh Di MD. nia ” d 4 A : : j yA : ¥ * Y - aaa | Why An : | tay! J Ne a | | ir 1p ahh it Phe) tak > hika! 1h hy i | ’ i ry ( re ; it il cu Mii tyr De th Sy a A 4) OG ‘ 4 ‘] ‘ ; i ‘ : “Yad Foi ri ai { Wa VERE Litt ‘ « ’ iia? et A t i ! , , bi Ja rH taal be Py ; ee J : i : | ; ' ; it ! } i } ‘iy oie! > ao , nity : : HE! tp? ! ay ey: > Ru | % : : 5 ’ is : 1 eee 1} se Mae : ’ ; bY € . ! - Do \ a) : HP hb: E H ¢ ia - ! hy AE i aE wi, . ABa wh ; ari i! : Aust AnH eh g ) ' ;4 th H } PAPAS AL HH Ett ' ” “ jig epi ht ti : - . ' 4 i} ’ vib : i) ah : iy 4) ae t sf iy ny! f vo 1 ‘ ; iF } 1h +? 4 4h) ia J y f if . AMG : j i e bt AN ‘ ai Pia we ; ae Dear 5 J Th reel Mt gm ih .) 0 Foe ae aah, i aha (i : r 4 ie 8) t er 4 Sh as f mt Bt } | ‘ ; } we Af mitt 4 Sh 9 ae i bake th i r +4 Ae > we re a f 4 ee ’ } q } 1 Ve : - Pal yn | ; SU Baa, y t q San 4 280 Ube , wu iit ” \ : our %y , iB! ’ Ng 4 } ; aT hb oe ‘ hh | , Wa ‘ ‘ nl) alee ay Tea a ee A ae a } i Per) ) ' bi iat >| : y a Oe ed i, i " BE ee) | . 4 ’ » 0 al . ; we 3 pier t! i) BI i rr <€ " ry : at oo le i] e hy - Mie ' f >h # 1% | i, ; ye Le Al bah ny Dee al inet } wid + * hy ary? iRUcH cn iB a0 t ao) site as i | OR eR hi 4 Paiey 4 7 ! iP vey} \ ND aH es “ Mf , Be , i 4 i t ' : SS +: ax> Te - Pa = = RM ES LS. . ee > Se SS. ee ee = 7 a aan =a Example of this definttids Example in wHmoeErs Ze 3-156 13, Another example. T he fixth Booke > \ betwehethem .. Suppofe three quantities A,B,C, fotharlet A hauetoB fefqui- altera proportion,namcely, 6.to 4. And lét B to C hauefefquitertia proportion, namely, 4.to 3. Now the proportion of A to . C,the firft to the thyrd,is made of the propor- 4 tion of Ato B, andofthe proportion of B to Bij to C added together. Ifye willaddethemto- © gether, ye mutt by this definition multiply the quantitie or denominator of the one,by the quatitie or denominator of the other. Ye mutt firft therefore feeke the denominators of thefe proportions, by the rule before geuen in the declaration of the definitions of the fift Booke.As if ye deuide A by B,namely, 6. by 4, fo fhall ye haue in the quotient 1 + forthe denominator ofthe proportion of A to B: likewile ifye denide B by C,namely, 4. by 3. yethall haue in the quotient 1 + forthe denominator of the proportion of B to C,now multiply thefe two denominators r+ and 1 the one into the other, by the . . ° ; - rule before taught, namely, by multiplying the numerator of the one into the nu- merator of the other, and alfo the denominator of the one into the denominator of the other : the numerator of 1 +. or of +_ which is allone, is 3, the denomt- natoris 2: thenumerator of 1~which reduced are +- is 4, i | the denominatoris 3 : then multiply 3.by 4, numerator by nu- 12 merator, fo haue ye 12. fora new numerator : likewifemultiply 3 4 2. by 3. denominator by denominator, ye fhall produce 6.fora . nominators : 3. into §. produce 15 :. and 15. into 6. 10 produce go: which fhall beanew denominator. 2 I $4. * So haue you broughtforth = or — which is pro- $55 5a Pe er portion fubnoncupla:which is alfo the proportion 99 of 2.t018. Wherefore the proportion of 2. to 18. | that is, of the extremes, namely, fubnoncupla,is made ofthe proportions of 2,to 3: Of3.to 15 :: and of 15.t0 18 : namely,ot fubfefquialtera,fubquintupla,and fubfef- . quiquinta. ty es Hy Another example, where the greater inequalitic and the leffe inequalitie are mixed together 6. 4.2. 3. the denomination. ofthe proportion of 6.to 4,is I —, of 4.to 2,is -, and of 2.to 3,18 —+ now ifye multiply as you ought,all hele denominations together, ye fhall produce 12.to 6, namely,dupla proportion. Forafinuchas fo much hath hetherto bene fpoken of addition of. et it fh a of Eucldes Elementes. Fol.1$3 itfhall not be ynneceflary fomewhatal{o to fay of fubftraction. of them , Where it is to be noted, that as addition of them , is made by multiplicatié of their denomi- nations the one into the other:fo is the fubftra@ion of the one from the other done, by ditifion of the denomination ofthe one by. the denomination ofthe o- ther. As if ye will from fextupla proportion fubtrahe dupla proportion, take the denominations of them both."The denomination of fextupla proportion, is 6, the denomination of dupla proportion, is 2. Now deuide 6, the denomination of the one by 2.the denomination of the other : the quotient fhall be 3 :which is the denemimnation ofa New proportion, namely ytripla . {9 that when dupla-pro-. portion is fubtrahed fromdextupla, theré fhall temayne' ttipla proportion ¢ And thus may yedoiall others: MAME POOTSN HHS OSE H Lie Of fubfrattis on of propor tion. 6..A Parallelogramme applied to.aright line, is fayd to want in forme by a parallelogramme like to one geuen : whe the pa- rallelograme applied wantetb to the filling ofthe whole line, Tre ixsh dee. by a parallelogramme like to one geuen : and then is it ayd to PHM exceedeswhen 1t exceedeth the line by a parallelogramme lske to that which was geuen. ute As let Ebe.a Parallelograme geuen, atid let AB bearight © 2 Ska ae eee line,to whom is.applied the pa- — | a rallelopramme A C D F. Now if ic want of the filling of the fine A B,by:the parallelograme DF GB being like to the pa- ‘yallelogramme geuen E, thenis. Snecten enncewer the’ parallelogramme. fayd to. 4 F B want in forme by a parallelo- a grammic like ynto a parallelogramme geuen. _ ~~ Likewilé if it exceede, as the parallelogramme A C GD applyed to ‘the line rallelogrammelike to-a pa- AB, ifitexceede it by the Tallelogramme geucn. ocr Sov EN MOAT LB sine 9D , mi gramme FGBD ~ -. “this definition is added by F/ufares asit feemeth, it isnot in any cOmon Greke booke abroadsnor in any Cominentary. It is for many Theoremes following very being like:to, the parallelo- Heceflary. “= sa ‘ . | a Nin.iij. Sup The | Seale “ec Gos : grainme\ E\which was ge- men ; then is the parallclo-, ; gramme ABGD, fayd to xceede\in forme by apa- — t | . 3 The fixth Booke*: | SeT het. Lheoreme. °° The 1. Propofition: Triangles (9 parallelogrammes which are ynder one (> the Jelffeme altitude:are in proportion as the bafe of the one i to the bafe of the other: | | V ppofe that there be two triangles ABC and ACD 7 KZ || two paralleiogrammes E Cand.C F.Whichilet be fet Due s der one and the felfe fame altitude, or perpendicular line _ He drawen from the toppe A to the bafe BD. I’hen I fay iF \ ‘that as thebafe BC is tothe bale C D fots the triangle Be) be BC tothe triangle ACD: and the parallelogramme HE Cto the parallelogramme.C F... For forafmuach as the VBS rave, Ng | Con stration. lines EAmd AF make both me right line and foalfo do the lines BCadCD: and therefure the lyne | E Fis apacallel vnto: = - a ee the line‘BD.Produce | eae ) | theright lise D-Bon-tche fide direttly-to the pointes F1,Li( by the 2.peticion of the firft) And bnto the bafe BC (by the 2. of thefir[t) put as many equall lines as you will, as for example ,two namely,B G,andG Hand Ynto the bafe CD onthe othe fide putasmany equall as you did to the other bafe which let be D K and K L.T hen draw thefe right tines AG,A HA K and AL. And porafmuch asthe tines CB,BG and. G Flare equall.the one to the de Demonstra. ther ,therfere the triangles allo ATG, AG B and ABC, are ( bythe 38.0f rion ofthe __the fir) euall the one to the other,Wherfore how multiplex the bale F2 Cs to fait parte — she bale BC,fo multiplex alfois the triangle AH Cto the triangle A BC.And by the'fam reafon alfo how multiplex the bafe LC is tothe bafe DC, fo multi» plex alfoisthe triangle A L Cto the triangle A D Cw herforesf the bale Fae be equall nto the bafeC L.,then (by the 38. of the firft ) the tran ele.A FAC 4s equal pute the triaugle. ACL, Andif the bale H ©. exceede the bale L,then aio the tingle AJA C excedeth the triangle ACL,and if the bafe be leffe,the triagle alfifhall be leffe. Now then there are foure magnitudes namely,the twe $ufes Bl ad Dyandthe two triangles A BC and ACD, and to the bafe B Cand to tle triangle A BC namely to the firft and the third, are taken equee Mule of Euclides Elementes. Fil.ts6e maltiplices amely,the bale Fi C and the triangléA HC, and likewle to § bafe CD and to the triangle A DiC namely, to the fecond.and.the forth, are taken certaine other equemultiplices that 1s the bale C Land the triangle £LC..And it hath bene proucd that if the multiplex of the firft magnitude, that's the bafe ELC do exceeae the multiplex of the fecond thatis the bafeC L} themultiplex aloof the third ,thatis,the.trangle A 0 Cexeccdeth the multiplex oy fourth; that is the triangle A LC,and if the faid bafe'FLC beequall.to the jaid bafet Eythe. triangle allo. AH Ciseguall to the triangle AL C,and if tt we leffeitis leffe.Wherf one by the fixt.definition of 9 fifth,as the firft ofthe fovefid magni. aides 1s,to the fecond fois the third to the fourth. Wherforens the bap B.C isto thebafe C.D.fois the triangle ABC to the triangle ACD; And becaufe (by the 4.1. of the firft) the parallelogramme EC 1: double .to the triangle ABC, and ( by the fame.) the parallelogramme F C15 dwuble to the, triangle AC D,therfore the parallelogrammes B.C and- FC are eqremultipli- ces vnto the triangles ABC and ACD. But the partes of equemult plices (by the 15. of the fijeh )haue one and the fame proportion with their equenultiplices. WW herfore as the triangle A BC 1s to the triangle AC D,fous the parellelograme E Cto the parallelogramme F C. And forafmuch as tt hath bene denonfirated, that as the bafe BC 1s to the bafe CD,fois the triangle ABC, to tle triangle ACD; andas the triangle AB Cus tothetriangle AC D\foits the paralleloe gramme EC to the parallelagramme FC. Wherefore ( by the 11. of the fifth) as the bafe BC 1s to the bafeC D ,fots-the parallelogramme E C tothe parallelo» gramme FC, Fhe parallelogrammes may al{obe- demonstrated a pari by theme elucsas the triangles are sif we defcribe yponthe bafes BG,G Han DK @ K L parallelogrammes vnder the felf fame altitude that the, paralldogrammes geuen are. Wh berfore triangles and parallelogrammes ‘which. are bneer one and the jelfe fanie altitude,arein proportion, as the bafe of the one 4s to ybafe of the other: which was required to be demonstrated. Flere ¥lufsates addeth this Corollary. | If two right lines. being geuen, the one of them be deuided how fe ener: therecianele fi- sures contayned Under the whole line undenided,and eche of thefecmentes of he line aeui- ded ;are in proportion the one to the other,as the {eementes are the one to the otver .. For i moavinyng the figures BA and A D in the former deferiptionjto be reéangled,the reGiancle hgures contayned ynder the whole right lyne A C,and the £gments of the right line BD whichis cutinthe poynt Cjnamelysthe parallelogr:mmes B.A and A Dare in proportion the one to the other,as the fegmeétes B.C ard C D.are, “SeThe2Theoreme. Ff he2. ‘Propojition Ff to.any.one of the fides of a triangle be diawen a parallel Nn, ity. right Demo iflras tion of the Second parts A Coro lary added by FlafSates. The firft pare of thts Theo- POEs Derrouftra- tion of the Jecoud parte dew _ Lhe pxth Booke right line,st fhall cut the fides of the fame triangle proportio~ nally.«And sf the fides ofa triangle be cut proportionally, a right lyne drawn from fettion to fection ts a parallel to the o- ther fide of the triangle. K ji} Vppofe that-there be a triangle A BC, “ynto one of the fides whereof, «sh 2 t namely, "ynto BC, let there be drawen a'parallel line D E cuttyng the eA /ides AC and AB in the pointes E and D: Then I fay firft that as BD istoD AjoisC EtoE A.Draw a line from B to Ej alfo fromC to D.Wher fore ( by the 37: of the firft) the triangle B.D Eis equall pnto'the triangleC D E: for they are fet bpon oneand the fame bafeD E) and are’ contained “ithin the felfe fame parallels DE and BC.Confider A a certainé-other triangle ADE. Now thinges equall ( by the-7.of the fifth) hane to one felfe thing one and the fame proportion. Wherfore as triangle BD E is toy triangle ADE jfoisy triangle C D E to the triangle ADEBut as9 triangle BD E is to} trian: gk AD E,foisy bafeBDtoy bale D A (by the firft of this booke. )For they are bnder one and the felfe fame toppe namely ,E, and there z | fore ave vnder one and the fame altitude. And « by the fame reafon as the triangle CD E is to the triangle ADE, fo is the lyne. CEtothehneE A. Wherfore (by the 11. of the fifth) asthe line BD isto the line DA, foistheline CEtothelineE A. | ) “Butnow [uppofe that in 9 triangle ABC the fides A Bre AC be cut propore ridnally fo as BDistoD Ayfo let C E be to E A ex draw aline from D to B. Then fecondly I fay y the line D-Eisa parallel to § lyneB C. For the fame order of conftru€tion being kept for 9 as B Dis to D A, foisC EtoE A, but asBD isto'D A fois y triangle BODE toy triangle AD Eby the 1.0f the fixt eo as C Eisto E A,fo(byy fame )is the triangle CD E toy triangle AD E:therfore (by the 11. of the fifth.) as the triangle BD Eis to the triangle AD E, fois the triangle CD E to the triangle AD EWherfore either of thefe triangles BDE and CD Ehaue tothe triangle AD E one and the fame proportion.Wherefore (by ther, of the fifth) the triangleB D E is equall nto the triangle C D E; and they are vpon one and the felfe bafe namely,D E. But triangles equall and Jet vpon one bafe,are alfo contained within the fame parallel lines ( by the 39- of the first. )Wherfore the line D E is vnto the line Ba parallel..If therfore to ae ny one of the (ides of a triangle be drawn a parallel line, it cutteth the other fides of the fame triangle proportionally. And if the fides af atri ang le be cut propore tionally, aright lyne drawen from fe€tion to fettion,1s parallel to the other fide of the triangle: which thing was required to be demonftrated. Here of Euchdes Elementes. Fol.154. qf Flere alfo Fluffates addeth a Corollary. Af a line parallel to.one of the fides of a triangle do cut the triangle,tt fhall cut of from the whole triangle a triangle Iske to thé whole triangle. For as it hath bene proued it deui- deth the fides proportionally,So that as E Cis to EA,fo is B D to D A, wherfore by the 18 ofthe fifth,as A C is to A E, fois AB to A D.Wherforeé alternately by the 16, of the fifth as A C is to A Blois A Eto'A D: wherefore in the two trian- glesEA Dand CA B the fides about the common angle A are proportional.The fayd triangles alfo are equiangle. For forafinuch as the right lynes ARC and A D Bdo fall ypon the parallel lynes E D and C B, therefore by the 29. of the firft they make the angles AE D and A D Ein the tangle A DE equallto the angles A CBand AB Cin the trianefe A C B. Wherefore by the firft definition of this booke the whole triangle A B C is like vnto the triangle cutofA DE. ST be 3. Theoreme. T he 3. Propofition. If an angle of a triangle be denided into two equall partes,and if the right line which deuideth the angle deuide alfo the bafe : the fegmentes of the ba/e fhall bein the fame proportion the one to the other , that the other fides of thé triangle are, And if the fegmetes of the bafe be 1n the Jame frpenetes that theother fides of the fayd triangle are: a right drawen from the toppe of the triangle unto the fettion, /hall deuide the an» gle of the triangle into two equall partes. Rei hx Sy si ppofe that there be a triangle A BC,and (by the.9. of the frp) let s %: 1 Oe the angle'B AC be denided into two equall partes by the right lyne A D which let cut alfothe bale B.C in the pont D, I hen I fay that 4s a the feoment BD 15.toy fegment DC, fors the fide BA to the fide A C. For bythe point C(by the 3.2. of the fui) draw E ~nto the lineD.Aa parallel line C E.and cxtende the line B A till it concurre with the line C Ein the voint E,and do make the po a BEC. But the bne.B A [hallconcurre with the line C Ef bythe 5. : peticion,) for that the angles EBC and BC E are ef le(Se then two right angles. For the angle EC Bis equall to theoutwarde and oppo fite angle ADB (by the 29.0f the fir{t..)-And the two angles AD Band D B A of the triangle B AD are a then J two right angles y the 17: of the firft) Now fors ” & | afm uch as bpon the parallels AD and EC falleth the right line AC, one y A Corollary added by Fluffatess ConStruion. Demonftra- tion of the firkk parte Demon'tra- tion of the fecoud part, which ts the conuerfe of the first. ~4, “all: Geometrical demonftrations,” ) Thejixth Booke by the 29. of the first) the angle AC Eis equal Ditto theanole CAD. But bn to the angle A Dis the angle BA D fuppofed to be eguall. Wherfore the ans gle BA Dis alfo equall Ynto the angle AC E. Againe becaufe dpon: the paral= lels A Dand ECfalleth the right lneB AE, the outward angle BAD (hy the 28.0f the farft jisequall ntotheinward angle AE C.Butbeforest was pro: wellthat y angle. AC Bas equall yntoy angle BAD, wherforey angle ACE 4s equall yato y-angle AEC Wherefore ( by: 6.0f 5 firit) j fide AE is equall vata the fide AC. And becaufe to one of 9 fides of 9 triangle BC E,namely,to E Cis drawen a parallel line AD, therfore (by j 2.0f) fixt) as B Dis to'D C, fois B d to AE. But AE ssequall vnto AC, therfore as BD is toD C fois BA to AC, But now fuppofe that.as the fegment BD is to the vapsiceyy Wie Sie Jegment D C, fois the fide B Ato the fide AC, draw a line from Ato D>» ThenT fay that the ane ele B AC is by y right line AD deuided into two equall partes.Fey the fame order-of con/tru€tion re- mayning , for that as BD 1s to DC, fois BA | te AC, but asBDis toD C fois BAto AE(by. ) | the 2 «of the jixt) for nto one of the fides of the »- | 1o triangle BC E,namely, vnto the fide E Cis drawn Joie ¢ GEREN a parallel line AD.Whereforealfoas B Aisto A Bas tate C fois BAtoAE (by the 11. of the fifth ) Where 9 BS SON fore (by the 9: of the fifth )ACusequall vnto A E. Wherfore.alfo(by.thes. of the furft,) the angle 4. EC is equall. ynto the angle ACE, but the angle AEC (2) the 29.0f the first) 1s equal ynto the outward angle B AD: and the angle AC Eisequall ynto the angle C A D-which is alternate vnto him: ‘wherefore the angle BA Dis equall ynto.the angle C AD. Wherfore the angle BAC is by thevight line AD deuided into two equall partes. Wherefore Aes of a triangle be denided into two equall partes, and if the right line which denideth the angle cut alfo the bafe, the fegmentes of the bale (all be in the fame propor tion the one to the other that the other. fides of the faid triangle are. And if the Segmentes of the bafe be in the fame proportion that the other fides’ of the ‘fayd triangle are,avight line drawen from the'toppe of the triangle ‘puto the fection denideth the angle of the triangle into two equall partes. aes This conftruction is the halfe part of that Gnomical figure defcribed inthe 43, propofition of the firlt booke,which.Gnomical figure is of great vie ina maner fa -SeThe 4. Theoremes» Thea. Propofttion, 0 In equiangle triangles,the fides which cotaine theequall and gles are proportional, and the fides which are fubtended une der the.equall angles. are of like propoition, — ae ~ = s™“ = . ’ \ ec Suppofe % of Euclides Elementes. Fol.i58. V ppofethat there be two equiangle triangles. ABCand DC E : and let the angle A BC of the one triangle, be equall yntoy angle DCE Dtig,| of the other triangle and the angleB AC equall yntoy angle€ DE, ~~. ana noreouer, theangle ACB equall ynto theangle DE C. T ben T fay ,that thofe fides of y triangles A BC, DC E, which include the'equall angles are proportionall,and thefides which ave fubtended bnder the eqrall an- gles are of like proportion. For let two fides of the fayd triangles, namely, two of Con/truttion, thofe fides which are fubtended “onder equal: angles : as for example thé fides BC and CE, be fo fet that.they both make one right line ..And becaufe the ane gles ALCGAC Bare leffe then tworight angles( by the.a7. of the first): but the angle AC B 1s es yall ynto the angle F q us D EC; therfore y ane ; | gls ABC DEC = are leffe the two right : angles . Wherefore the eT _ linesB Ark D bez f mh ; co oe j : f : ing produced ,7will at / ~ c> 5 ~N ay the length meetetoges \. / ther. Let them meete. 6 oo eee and ioyne together in | | the poynt F. And becaufe by fuppofition the angle DC E 1s equall’yntothe ans Demonfira- gle ABC , therfore the line BF is (bythe 28: of the first.) aparallell yutothe line C 1). And fora/much as by fuppofition the'angle AC Bs equall yntothe angle DEC ; therefore againe ( by the 28. of the first the line.A C is axparallell onto the line F E. Wherefore FAD Ces aparallelogramme . Wherfore the fide F A is equall ynto the fide DC:.and the fide A Crnto thejide FD (by the 34. of the firft). And becaufe vuto one of the fides-of the triangle BF E, namelysto FE is drawen a parallell line.AC , thereforeasB A isto Fy fois BCtoGH (by the 2. of the fixt). But, A Fis equallynto CD. Whenfore¢ by therswof the fift)as B Ais toC D, fois BCtoC E; which are fides [ubiended bnder equall angles. Wherefore alternately { by the16.0f the fift as. A. Bis to BC, fois MC toC E. Againe forafinuch as © Das a parallel pnto BF, therefore-againe ( by . the 2.0f the fixt as BC isto. CE, fois PD to DEWBut FD 15 equall-ynto AC. Wherefore as B( ist0X, E, fois AC to DE, wiichare alfo fides fub> tended vnder equall angles Jxberfore alternately (bytherOs of the fift as B isto (A, fois ( Eto E D..Wherforeforafmiuch as ut hathbene demonstrated, — thatas\ A Bis ynio BG, fois D.C vate. (GE: bat as BC is vnto CA foi aad 42s ag. Je“ | gy — ae ae ’ : 1 / f ! 4 | i) ! i} f : ; } | f . ait i : ahi 7 ; Phy : i i , % ‘ 1‘ fr 4 ; ‘ } | ¥ 1 ¥ * | ye. : } > "i ' j ; ‘ ] ah | : ' f ian \ | 2a Ny > i | 1 } } } i : ia is f H % q 5 Ni mA Hh i { ’ \ iu 7 i) a, Pah } ! 4 it |} i Hh H 1 y! t Pi - ! i ld : At wt | f i ’ a f \ 4 34 7h Re : i } ah} " ; t Daly : | - Vip - | st ‘ 1 4 4 ’ in] hae i 4 y ' ‘ i Ad | | Hy i : q mii ) ei j ('t ) Re “i | i ' \] ‘ $} 4} ; ' | ’ j 1 f » | i] 1) a mii iy ‘ by : i ' the equall angles areof like proportions which Teas required.tove demonsbated, = Cap be ——— === ——— = 2 ee ares -- ~x = ’ - am : He ao eee = — ne = = = —— m ge ot ——— =e = Ts mers ba) ee - AS - 2 = ——<—_— rae b % —_ ¢ on , ~ ~aye - . : = = — — ————— eo we - ergs Nee <2 PRESS. Hy a: =n ee wa z Sina ss a a ee NS RE ™ 3 3 ee = : = _—= = os : ~ : = ee a == ———— f = - = oS —— ao se ey : Ree a Re ge RE ee = —— =e ake, a F a = - —~—— “ _ . : SS ae —— ee ee - - ~ - = — . _ —_— - = 2 = — a ———— - = . om atten ~~ oo Snowe. os sw —_ 2 ae a = _ = = = = = . — ~ _ ——— - -_— = ——= —— = > ———— ose or re om ———— —s —— se : SS = a — ———_—___—— —- — ’ ee =~ ee RE 2S Sah ope meine ee Sa erm enema nme waren nn ee er SS ——e—E_E———————————— = ————————————— SS = —— — = = = = ——_—_== = ————SSSS= : ~ SS — SSE = = = = : — “ Hoe nn en = r. ~~ 2 = ms pues z ve a == sy ae 1g . '< 4 " ee eee _ This it the conuerfe of the former propofttione Conjtru fron, Demon ftra- $807. ts equal ynto the triangle GEF : andthe reftof the angles of the one triangle oo sn Sap Thess Theorenie. » Lhe-s. Propofition. Tf two triangles baue par jides. proportional, the triangles are equiangle and thofe angles in the are equall,under which vare fubtended fidesof like proportion. “ | ia V ppofethat'there be two triangles ABC ee D E F ,haning their fides ea proportionall,as.A Bisto BC fo let D E beto EF :¢¢ as:BCis to AC, ea) folet E Fbeto D F: and moreouer,asB Ais to AC, folet ED be to DE. Then I fay,that the triangle A'BCis equiangle ynto the triangle D EF: and thofe angles in them are equall ynder which are fubtended fides of like pro= portion that ts, the angle A BC is equall vnto the angle D EF: and the angle BC Avynto the angle E F D: and moreouer the angle B AC to§ angle EDF, V pon the right line E Fand'ynto the pointes init Ecg F sdefcribe( by the 23-0f the first) angles equall ynto the angles ABC ez ACB, which let be FE G and E FG namely, let the angleFEG be equall vnto the angle ABC, and let the angle E FG be equallto the angle ACB. And fora/much as the angles ABC and AC Bare leffe then two right angles (by the 17.0f the first): therefore alfo the-angles FEGand EFG are lefJe then two right angles . Wherefore ( by the s.pe titionof y firft)y right lines EG 7 FG fhallaty length concurre. Let the concarreinithe poynt :G.. Wherefore EF G is.a triangle. Wherefore the angle | ant. " vemayning BAC is equall ynto the angle remaye- ning GE (bythe firft Corollary of the 32. of the fir fi: )statherfore the triangle AB Cis equiangle vne tothe triangle G EB wwherefore in the trian oles ABE and EGF the fides, which intlade the eqnall angless( by the 4. of the fixt ) are proportional and the fides which are Jubtended wider the equall ane gles.areiof like proportion . Wherefore as AB isto BC,fois GEER, But aA Bis toBC, (6 by © puppofttion is DE to EF. wherefore as DE is to Eb fais Ck to BF (by the r1.0f the sift Wheres foreeither of thefeDE-and EG hane to E F one dnd the ame proportion . Wherefore( bythe 9. of the = D Eis equallyntoE G . And by the famé rete 2 oN foralfo DF is equallynto F Gy Now forafinuchas~ °° DE is. eyuallto E Gund E Fis common nto them both , therefore: thefe two files DE GE Fare equall nto theferwofiderG Eand E F, and § bafeDF és.equall'vnto the bafe.E G . Wherefore theangle D E F (by the 3.of thefirft) ss equall nto the angle\GEE :, and the triangle DE F ( by the'4.of the first) are of Enchdes Elementes. Fol.159¢ dreequall dato the restof the angles of the other triangle the tne to other , pre der which are {nbtended.equall fides « Wherefore the angle DFE is equall ne to the angle GT Esand the angle EDF ‘ynte the angle EGF. And becaufe theangle F ED 1s equall onto the angle. G EF ; but the angleG EF is equalt yuto the angle ABC: therefore the angle, ABC is alfo equall onto theangle PED. vind bythe famereafon theangle ACB is equall ynta jangle D.E E; and moreouer the angle B AC ‘vnto the angle ED F . Wherefore the triangle: ABC isequiangle nto the triangle DEF i If twotriangles therefore have, their fides proportionall,the tridngles fhallbeequiangle sez thofe angles intthem fhall be equall, vader which are fubtendedyides of ike proportion : which Bas required to be demonftrated. < Se The 6. T heoreme. : The 6: Propofition. If there be two triangles wherof the one hath one angle equall to one angle of the other,¢> the fides tacluding the equal an- gles be proportional : the triangles fhall be equiangle, and thofe angles in them [hall be equal, ynder which are /ubten= o ded fides of like proportion. aia V ppofe that theyre be two tridnoles ABC, and D E F, which lerhane Sythe angle BA C of the one triangle equall'pnto the angle ED F of the =~ other triangle, and lét the fides including the equall angles be proportios nall thatis,as'B Ais to AC, fo et ED be to D F. Then I fay,thaty triangle ABC ig equiangle ynto thetriangle DE F: and the angle'A B Cis equall ne totheansle DE F, andthe angle AC B equal nto the angle DF E , which angles are fubtéded to fides of like pro 4 | portion .Vntothe right line DF, and AK to the poynt in it D ( by the 23. of the first) defcribe bnto either of 9 angles BAC and. BOF ;an equall angle F DG. And vnto the right line DF, and ynto the point init F (by5 fame) . defcribe ynto jangle AC Ban equall © angle OF G.. And forafmuch as the twoangles BAC and AC B,are( by’ the.17. of the firft ) leffe then two right angles: therefore alfo the two angles ge ED.Cand DEG, are leffe then RUUD SABES ae two right.an asePRcerfere lines D.G ee-F G.being produced fhall cocurre (by the s,petition ). Let thé concurre in the point G. Wherefore D FG is a triangle. | 00,7. Wherefare Confirallion. >= = - hte til — S-—= =a ‘ ms ——. Sa oe i tT taal => 2 ee <8 © eee x : < ot } " | i} if | Ry | i | Ve ’ Nt i} ro (Ang ' i] WO h a Iii ' U il ma | | ‘h f id f i i iiah hit it at! bt id vy ~ Hele | Vel } i q i wy ih i] ih Bot 1 , bi t u ij i iG u } i te) 1p! ty : | aidne } { | ' ie | i : : ei mf he hl ill | ) ii [ ¢ : . Sait ‘i \ { iA \ t } mi i 4 ; q ) ; | ! | | f t | ae i a Nh } j ; | ot ‘a f Wk i) ' y We iN ht! ; , { me ii} | f | yi 4 a a | 1 Pe t I] i! 44 | i } ie tt } 1) Ve jit ff we } } if o 1D a - ) i ih / Hy Hii ha ¢ iy : | } M iy 4 hey : ie i q 7 Wa hl i Af \ ! hd | ‘ ' H ; ° { 1 / j f \% | ie ir J I Al ; if ; } if) vi - ' - Mi d : ‘ : j ‘i fe t f ‘ ‘ ie | ie a aie eu ! a an ie ; : : | . Lhe fixth Booke Wherefore theangle remaming A BC is equall yntotheangle remaining DG F. (bythe: 32, of the firft wherefore the triangle A BCs equtangle vntothetri-. angle DG F. wr hereforeas B Assn proportionte AC, fois GD to DFE by the 4.0f the fixt)» But it 1s uppofed; thatas B Aisto AC, fois ED to DF. Wherefore (hy ther1 of the fift) as EDistoD Pj fois GD to DF. Where fore (hy the 9:0f the fift) E D is equallyntoD @. And D F 1s common. vate. them both. Now then there are twos ’ AE ASMOSIONS bith fides ED and DF equall:ynta two\ \: | fides GD and DF sand:the-angle\ ED F (by fuppofttion’)1s equall ato»: the angle GD F.. Wherefore ( by the 4. of the firft ) the bafe E F is equal ‘puto the bafeGF and the triangle D EF is ( by the fame ) equall vnto the trrangle GDF, and the otheranz glesremayning in them ave equall the: . one to the other, bnder which are fubs tended équall fides. Wherefore the ani ele DEG is equall.pnto the angle triangles alfo be equtangle, and thofe angles in them Jhalbeequalljon are /ubtended fides of like proportion: i, . —r tee of Euchides Elementes. Fol.160. Fat V ppofe that there be two triangles ABC and DE E, which let hane | Xd one angle of the. one equal to one angle of the other namely, the an gle | Kh BAC equall ynto the angle E DF. And let the fides. which include “= the other angles »namely, the angles ABCand DE F be proportio- nall, fothatas ABisto BC, folet DE betoE F. And let the other angles ree mayning, namely, ACD andD F E. be finft either of them lefje thenaright angle. I’ hen I fay that the triangle A BC is equian role buto the triangle DE F. And that the angle A BCisequall’yntothe an: :) gle DEF namely the angles -which-are contaie ned vader the fides: proportionall, and thatthe: angle remayning namely,9 angle Cis equall Dns. to the angle remayning namely,tos angle F.For first the angle AB Cis either equall tothe ane |). gle EF or els ee } fe “ angle ABC be j bd | equall to the angle D E F,then the angle:remaie S an ytamely A CB, fhallbeequall Fo the angle lect ener remayning ‘D F E (by the corollary of the 32.0f 2 ou. € ¥ : the firft) And therfore the triangles A BC and DE F are equiangle. But if the angle ABC be ynequall vnto the angle 1) EF then is the one oftthem grea ter then the other.Let the angle A BCbe the greater ,and ynto the right line A Band ynto the point wit B¢ by the 23. of the firft) defcribe vnto the angle D EF anequall angle ABG.. And forafmuch as the angle Ais equall ynto the angle D, and the angle A BG is equal ynto the an wile D EF; therfore the ane gle remayning AG Bis equall nto the angle remayning DF E (by the corols lary of the 32.0f the firft.Wherfore the triangle A'BG is equianglebnto the trie angle D EF. Wherfore (by the 4. of the fixth) as the Jide A Bis to the fide B ~G foisthe fideD Etothe fide EF. But by upppofition the fide D Eis tothe «fide E Fas the fide A Bis to the fide BC. Wherfore (by the 11. of the fifth) as the fide A Bis to the fide BC forts the fame fide A Btothe fide BG. where fore ABhath to either of thefe BC and BG one and the fame proportion, and therfore ( by the 9. of the fifth) BC is equall vnto BG, Wherefore ( by the s. of the firft)9 angle BG Cis equall vnto§ angle BCG : but by {uppofition y angle BCG ss lefethen avightangle. Wherfore the angle BG Cisalfo lefSe then a right angle. Wherfore ( by the 13. of the firft.) the fide angle bnto it, namely, A GO Bis greater then aright angle and it is already proued that the fame angle is wre dnto the angle F. Wherfore the angle F is alfo greater thena right angle, » Dut it 18 fuppofed to be lef[e whichis abfurde. Wherefore the angle A ‘BC isnot . wneguall Ynto the angle 'D E F ;wherfore itis equall dntoit. And the angle A 1s equall Ynto the angle D by ‘/uppofition.Wherfore theangle remaynin 1 names ly ,C ssequall ynto the an ole remayning snamely,to F (by the corollary of the 32. of the firft) Wherfore the trianyle ABCs equiangte vuto the triangle DE F. But now /uppofe that either of the angles AC Band D FE be not fe then | 0. iy. aright The fir ff part cfebis prepof ition. Demonfira- tton leading te an tmpifisbiq lstie. - a na ~ Abd The fecond i ‘ part of this propa/itt0re . € a Lf hejixth Booke: *: swotiphtangle.T hatis let either of themsbeavightanglesor either of them grea tenthen.a right angle Then 1 fay againe that 1 that cafe alfothe:triangle AB ‘Cusequiangle vnto thetriangle DEF) For if either ofthem be.d-right- angle, -oforafnachas all right anglesare(pythe 4, peticion. eqnall the one tothe “other , Praight waywill followthe inkentof the propofition. But if-either. of themsbe \ereater then.a right augle,thenthe -fiande order-of: €onftruclion that sas before “being Rept, wentayantike fort pronethatthe fide BCisequall pnto the fide “© other angle be proportionall sand of either of the other odd RIA Sunelesremaining be either leffe or-nopleffe then.arightan igle,the triangle fhall s be equidnglesand thofe angles in them hall be equall-which are contained ‘bnder » fides proportionall: whith was required to be proued. sas ety $9 The 8.Fheoreme:: L-he'8:Propofttion. 4 “2 » Tfinaretangle triangle be drawen from the right angle yn- “tothe baje a perpendicular line, the perpendicular kine |hall by deuide the triangle intotwo triangles like ynto the whole,and ‘0 . sladfolike the one ro the other. i BS“ ppofe that there bed rectangle triangle ABC, whofe-right angle ROG let be BAC: and (by theta. of the firft:) from the point A tothe line A Ne BC lettherebe drawena perpendicular line AD) which perpendie shar cular line let deusde the ‘whale triangle. ABC mto thefe two triane nogdes YBDandADC. ( Note thatthis perpendicular line\A D.;drawen wyromtheright angle tothe bale must needes fall swithin the triangle ABC 37 __ sxfodeutdethe triangleanto tio triangles For ifit fpould fall without» then pre saci othe fide B Cvato, the perpendicular line,tberefhould bemade'a triangle, : —_ ee \s4T.., whafebutward angle being an acuteangle,fhould be lefSe then the\ inward and | - soppofite angle Whichisarightan olescwhich iscotrary to the réiof\ es dade ete als fper of Euclides Exlementes. Fol.16te ther canit fall bpon any of the fides: A:Bior AC: for then tivo angles of one and the felfe fame triangle. fhould not be leffethentwo right angles contrary to the felfe fame17. of the firfh Wherefore 1t falleth within the triangle ABC). Lhen I fay, thateither of thefe trians\ 900 0\ “AO 9 gles ABDand-AD Cyaretikeyto © > \ | the whole triangle A BC: and mores over that they are like the one-tothe os \°\.: ther:-Firft that the triangle ABD is. . 9! hike wntothe whole. triangle:ADBC ts thas proueds Forafmuch as( by the 4) fo ssa 3\oJ petition )the angle B Alisequallpne Bosvwr Ps | ; roe to the angle A ‘DB, for either of them 1S a right angle. And in the two triangles A BC and ABD the angle Bis common . Wherefore the angle remayning, namely, ACB is ( by the Corollary of the. 32. of the firft equall Dntotheangleremayning namely to B A DA her- fore thetriangle ABC is equianyle vuto the triangle A BD. Wherefore the fides which contatne the equall anoles ,are( by the 4. of the jixt ) proportional. Wherfore as the fide ( Brwhich fubtendethy right angle ofthe triangle AB (, isvnto the fide ‘BA “which fubtendeth the right angle of the triangle A BD, fois the fame fide. AB. Which fubtendeth the angle C of the triangle ABC, vne to the fide B D which fubtendethy angle BA D of the triangle ABD, which as equall pnto thé angle C:'and moreouer , the fide A Cynto the fide AD which fubtend théangleB common to both the triancles. W herfore the triangle ABC is like ynto the triangle ABD (bythe t:definition of thefrxt ). In tke maner alfo may Wwe prone, that the triangle ADC is like bnto the.triangle ABC. For the right angle AD Cts equall to theright uncle BA Ce the angle at the point Cis common to either of thofe triangles . Wherefore the angle remayning, namely, DA Cis equall co the angle réniaining; naiiiely, to BC( by y Corolla ry of the 32. of the first) . Wherefore the triangles A BC og ADCare equian- gle. And therefore( by the 4iof the fixt )the fides-whsch are about the equall ane gles are proportional! . W. herefore as in the triangle A BC the fide B Cis to the fideC A, foin the triangle ADC is the fide ACto the fide D.Crand againe, as in the triangle ABC the fide€ A is to the fide AB, foin the triangle A DC is the fide CD tothe fide Av And moreoner asin the triangle ABC the fide CBis tothe fide BA, [oinrthetriangle A DC isthe fide (A tothe fide AD. Wherefore the triangle AD € Ws like ynto the-wholetriangle AB C.wh erfore either of thefe triangles ABD, A D-Cis like yntoy whole triangle A BC. I fay alfo, that the trianoles ABD and AD Care like theone to the other. For forafmuch as the right angleB D A isequall yntop right angle A D C(by the 4.petition )and as it hath already bene proned the angle® A D 1s equall yn to the angle C: therefore the angle remayning namely, Bis equall ynto the ans gle remayning namely ta D°A ( {by the Corollary of the.3.2.0f the firft.) Where Oot. fore Lemorfita- iC. Confirutticn. LAF Demonftra- £42 bP The fixth Booke fore the triangle. ABD is equianglepnto the triangle: ADC. W herefore.as the fide BD which inthe triangle ABD ubtendeth the angle BAD is bnto the fide-D.A whichsn the triangle. ADC fubtendeth the angle C -which is ee quall ynto the angle BAD, Joss the fide.d D-which in§ triangl ABD ube tendeth the angle B,ynto the fide DC which in the.triangle A Dé fubtendeth the angle D AC which is equall ynto the angle ‘B: and moreoner, fois the fide B A vato the fide AC which fabtende theright angles. Wherefore the trian gle AB Dis like nto the triangle ADC. If thereforein a reGan igle triangle be dvawen from the right angle ynto the bafe a perpendicular line the perpendicus lar line {hall deuide the triangle into two triangles like bnto the whole 5 and alfa like the one to the other : which twas required to be proved. SeCorollary. Hereb yy it 1s mantfeft,that ifina retangle triangle be draw- en from the right angle vnto the ba/e a perpendicular line,the fame line drawen is ameane proportionall betwene the fetti- ons of tbe bafe: and moreoner,betwene the whole bafe and et. ther of the fettions , the fide annext to the fayd fection is the “neane proportionall .. Foris was proued, that as CD isto D A, fo 24s DAto DB; and moreouer,asC BistoB A, fois BA to BD: and __ finally, as B Cis toC A, JoisC AtoC D. | : Sa Ther. Probleme. The 9. Propofition, — of the triangle A BC namely, vntoj fideBC is draw- ena parallell line F Dit followeth by y 2.0f this booke, of Euchdes Elementes. Fol.162, that as ‘D 1s 1m proportion vate D A, fois BF toF A, But (by conftruttion’) CD is doubleto D A. Wherefore line B F is alfo double tothe line F Aw here fore the line B Ais treble ynto the line A F.Wwherfore from the right line geuen if Bis cut of athird part appoynted, namely, AF: which tpas required to be one . Sa The 2. Probleme. ° The 10. ‘Propofition. To deude a right line geue not deuided, like vnto a right line genen beyng deuided. i aK : V ppofe that the right line genen not denided be AB , and the right lyne RS? gener being deuided,let be AC It is required to deuide5 line AB which ee” 1s not deuided like bnto the line AC which is denided. S uppofe the Lyne ig Coxuftrnson. A C be deuided in the pointes D and E.,¢¢ let y lines AB es AC fo be put that - they make an angle at all aduentures, and draw a line from B to C, and by the pointes D and E draw ‘vnto the line BC (by the 31. of the firft) two parallel lines'D Fand EG: and by the point D vnto the line A'B( by the fame) draw parallel lime > D HK .Wherfore either of thefe eures FH \ ; and F1B are parallelogranimes.Wherfore the line D Fi is equall yntotheline FG,and the Saket line HK 1s equall pnto the line G B. And bee Yad caufe to one of the fides of the triangle DK | C namely,to the fide KC is drawn 4 parale > lel line HE, therefore the line CE (by the 2. of the fixt ) 1s in proportion Unto the line E Das the ine KH 1s tothe lne HD: but the line K 31s equall puto the line BG, and the line FLD is equal ynto the lineG F.wher fore ( by thet: of the fift as C Eis vnto ED fois BGtoGFA4 lvayne becaufe to one of the fides of the triangle A G E namely,to G E is drawn a parallel lyne F D,therforethe line E DY by the 2. of the fixth ) is in proportion ‘bnto the lye D A,as the line G Fis tothe line F A. And it is already proned that as € Ets to ED foisBG toGF. VV herfore asC Eis toE D, fois BG toG Fjandas E DistoD A,fois G Fto F A. VVherfore the right line geuen not denided name ly,A Bis deuided like vnto the right line geuen being denided, whichis AC: which was required to be done. ¢ A Corollary out of Fluffates. By this Propofition we may denide any right line geuen,accordyng to the pro- Oo.iuyj. portion Demonfire- t40M. A Corollary out 4, FlufSates, 7 — = one a en oe, ; » Hay 2 HT A, — yah » ' iH! | fbi! ; : | - : f | | Wee | a) | He} i 1} 1 : (Sae \ ay } ’ = ; . » : \ ae ati ). if i q | | ii 1 i Wig i } : if th ; 4 HT : Wi tipi Py mia i iit } WR] hie & | Bit \ if it i i}) Way Bi) / TE j | { uy 1) ee ; | i} : ' 1 — ith iW , f WH i } ] HAH | VW bie 1 } : | a) ti a eit 1 | if ) } | { i J i fi Waa i | aE | m. i fi | nN ie | ii ni | i | ’ : { f | He de! bh } a |i Sih i) my) aia i Hf ' iis | | 1) be j 1) ii ; be 4 ne | | : J 1} 4 : | , a Hi \ 4 ae | ii Wee |} a 1th! VE Mae) | + 1 if 1 i - +h aan iD) aE eee || vie if Wh) i] ‘lB ee a; | i i ve aE :¥ Oni a ’ |i i f f ) ‘1 Viti | ij hi i; f i j a | Boye J nan if : ‘ak i t i) Ih 1" , i HHH! Mi 1 | tit ' wn 4 any! ate t it y | j| rt fj mt ' ah tl hi ie an ‘ 5 i i | ie 11) a { i fl } | ij a | ‘)) , i | HF j ' i ie | iit : ak: : im, ‘ a u Hy) h q H i ' By ehis and the former propoftion may & right dine be deut- ded into what partes foener yes wth. 5 9 Mae 5 “oe : Conitrulions Demounfira- 210. line AC (by thez.of the firft) put an equall line BD, ‘The fixth Booke portion of any sight lynes scuen, Forlet thoferightlynes hauyng proportion be ioyned together directly,that they may make all:one right lyne, and then ioyne them to thelyne geuen anglewife.. And{o proceede asin the. propofition, where you fee thar the right line geuen A B is deuided into the right lynes AF, FG and: G Bwhich’have'the felfe {ame proportion that the right lines AD, DE, and EC haue. By this and thefornier propofition alfo may aright linegeuen beeafily deui- ded into what partes fo euer you will name.As ifyou will deuide the line AB in- to three equall partes, Jet the lyne D E be made equall to thelyne AD, and the lyne E C made‘equall to the fame by the third of the firft. And then viing the felfe {ame maner of conftruction that was before: the lyne A B fhall be deuided into three equall partes.And fo ofany kynde of partes whatloeuer. s é > “Sg The'3: Probleme. The tt. Propofition. Onto tworight lines genen, to finde a.third in proportion “witb thewt. ; Hee ppofe that there be two right lines geuen B Aand AC, and let them be i y/o put that they comprehénd.an.angle howfoener it be. It is required to =I finde pntoB Aandvnto.AC athird line in proportion.Producey lynes ABand AC vnto the pointes D.and E.Aid mntothe | A and draw a lne from B to C. And by the pointe D ( by the 31.0f the first )draw-pnto the lyne BC a parallel Lyne D E, whith let-concurre with the line AC in the point E.Now forafmuch as vnto one. of the fides of the triangle ADE, namely, toD Eis drawne a parailel line BC: therfore as A B ts in proportion bnto.B.D, ‘fo (by the 2. of che fixt) is AC rntoC BE. But. the hne BD.is equall onto the line AC. VV herfore as the Lyne A Bis.to the line AC,fois the lneA C to the line CE. EV herfore dato the two right lines geuen A Band AC 4s found a.third line CE in proportio with them: which Was required to be done. ee Sh ee * * . = =) ee . = : } - , st en 7 _— oe we 2s % = q Another way after Pelitarius, Let the lines A Band B C be fet dire@ly in fuch fort thar they both make one right An other #4}. Jine, Then fré the point A ere&thelyne A D makyng with the lyne AB an angle at all after Peltta ma .* ~ 7 , “ YeH$e Foe * F AX ais fe Lame aduentures, And put the lyneA D equall.tothelyne B:C, And drawa right line from Dto B which produce beyond the poynt B vnto the point E.;And by the poineC draw vnto the lyne D Aa parallel lyne C E concurring with thelyne D E in the point E.Then Tfay thatthe line C Bis the third line proportional with the lines A B ae for 1071 afmuc of Exclides Elewentes. Fol.163. sdfman alas bythesr syafthe firit the angle Bof thes... 2 ee | siangle AB D.is equall to the angle B Siew so eee i angle CB E,and by the 29, of the itie; the angle) oof Vsrviact hes 5 VAs equallto the angle C, and.theangle D to the, angie E : therefore by. the 4. of this booke A B is - toD A.asBCistoC E, Wherfore(bythear. of * the fifth)A B isto B CasBCistoC E: which was required to be done. t* ~ i > b- &% on - ee ab hear hetupstal 4. AR. other Way alfo after Pelitarins. “2° et the linesA'B atid’ B C-befo ioyned toge- ther,that they may make aright angle, namely, A BC, And drawealine fromA to C, and from the” point C drawe vnto the line AC a perpendicular CD (by the 11. of the firft) And’produce the lyne ~ CDtillit concurre with the line A B'produced vn-* tothe pointe-D.Then I fay that the line_B D epee third lyne proportionall withthélines AB and BP C : which thing ismanifeft by the corollary of the 8,of this bocke, Sp The 4. Probleme. The 12. Propoftion. | a Anoitth» . Way after PL elitarius, |" “Unita three right tines geiuen rofinde a fourth in proportion with them. nde‘ynto A;B5C,afourth line im proportio pith them.Let there be taken two right lines D Ew DF comprehending an an leas it [hall Confiru tion. pe ‘happen namely; D F.And( bythe 2.0f the first Ynto Ms “an equall lneD G.And ye“ bs t5acnb.gs ie that the three right lines genen be A, B,C. It is required to elineA put to the late“ BCD the fame)*> Py | | put an equall line G E. And? IN A ea ee moreoueg, toute flineC pati 1 Bee 2 = an equall kak PEL. Ehenayo> wis 5 me 36 draw a line froG to Hi And*** > po9 | by the poynt E (by the 3 of | Eamets the fir[t) draw ‘dnto the line.- © | ® GH a parallell line“ BE \. Now forafmuch as pnto one | of the fides of the triangle’ ~ DEF, namely, vntoy fide . B Fisdrawena parallell ine ) | , oN "GH: therefore (by thez.of : ight} Roce he -_ the fixt)as the line D°Gss to the lineG Exfois the line. D F1.to the line Ht F. Dewnfira- ticks 3 be] But ‘The fixth Booke But the line D G is equall ynto the line A,and thelne GE is equall nto the dine B, and the line D H nto the line C._Wherfore as the line Ais ynto the line B, fois the line C vnto the line FA F . Wherforebnto the three right lines geuen A,B,C, is found a fourth line FA F in proportion with them : which was requio ved to be done. | q An other Yay after Campane. Suppofe that there be three right lines A B,BC,and BiD. It is required to adde vn- to them a fourth line in proportion with them. Ioyne A B the firft, with BD the third, in fuch fort that they both make one night_linenamely,A D.And ypon the faid lyne Berec&t from the point B the fecond ea: ce , : line BC making an angle at all aduen- ise Pee 1883 tures. And draw aline from Ato.C, | sera AG Then by the point Ddraw the lyne.D E parallel to the line A C, which pro- duce yntill it concurrein the point E, with the line CB being likewife pro- ~ duced to the point E, Then I fay that A , | theline B Eisthe fourth line inpro- > gpizaes ie portion withthe lines AB, BC, and | BD :fothatasABisto BC, fo isB ee DtoBE. For forafmuch as by the 15 ee & and 29. of the firft the two triangles - . ABCandD BE areequiangle,therfore (by the 4.0f this booke) ABistoBC, asBD is to B E : which was required to be done, = gi $mThe 5.Probleme. The 13.Propofition. F nto two right lines geuen,to finde out 4 meane proportional. ee Y ppofe the two right lines geuen tobe A Band BC . It is required bee ti! Oi twene thefe two lines A Band BC to finde out a meane line proportios eet all. Let the lines AB and BC be fo soyned together that they both make one right line, namely, AC. And vpe on the line AC defcribe a femicircle AD Cand from the poynt B raife vp vnto the line AC (by the 11.0f the fir/t )a perpendicular line BD cute ting the circumference in the point D : and draw aline from A to D,and an other from DtoC.. |. vad ~ \ Now forafmuch as ( by the 31. of the third ) the ¢ Mater Gs angle in the femicircle ADC isa right angles... and for that in the reftangle triangle AD C is.drawen from the right angle ynto the bale a perpendicular line D B: therefore( bythe Corollary of the 8 of the fixt.) the line D Bis a meane proportional betwene the fegmetes of the bafe A Ber BCWherefore betwene the two right lines genen, A Ba B C, is found ameane proportional DB: which wasrequiredtobedone. ; gg APre Dp of Euchdes Elementes. Fol.164. GA Propofition addedibyPelitarinsy: . ashore | A meane proportional beyng geuen,to finde ont in-a line ceuen the treoextremes: Nom 4 propofitios i behoueth that the meane geucn be not ereater then the halfe of the lyne gewen, added by Pex Be: are CitaveHS, Suppofe tharthe meanegeuen be 4 Band let the right line genenbe BC. Itis re- quired in the line 2 C to finde out two extremes, betwene which 4B thal be the meahe proportional. Sothat yet the lyne A B be not greater then the halfe-part-of the line & C.Forfocouldit not be ameane-loynethe lines .4 ZB andBC directly in fuch fore, thar they both make one right line, name ~ HA Reaivth ly, 4C. Then vppontheline BC de- {ctibe the femicirele B E C.And from the point Zere&vnto thelyne 4Ca perpendicular line 4 D : which lyne AD put equal vnto the line 42. And bythe point D draw vntothe line A Ca parallelline'D £, which vndoub- tedly- fhail either cut or touch the {e- . micircle,as in the point E,for that the line 4.D.is not greater then the femidiamcterj Then from the point E draw vnto the line B Ca perpendicularlineEF (bythe. 12, of the firft) Then Ifay that the line B Cis fo denided in the point F that thelyne..4#is-a meane proportionall betwene'the lines BF and: FC}: Which things manifett (by the 31. of the third) & corollary of the 8,of this booke, For thelineF Zis equalto theline A'D by the 4. of the firft, and fo is equall totheline 4 B: then ifwe draw the ryght lines 8 EandC E£,there fhall be made areGangle triangle B EC.And fo by the fayd.co- rollary,the line B F hall be to the lyne F £ (and therfore to the line 4B) as the line FE is to the lyne F C’: which was required to be done, . 3 Fluffates'putteth this Propofition added by Pelitarius as a corollary following of this 3:propofition, | Lhe 9. Fheoreme: Ther4..Propofition. Fnequal parallelogrammes which haue one angle of the one equall unto one angle of the other, the fides [all be reciproe kallznamely ,thofe fides which containe the equal angles. And if parallelogrammes which bauing one angle of the one equal unto one angle of the other, have alfo their fides reciprokal, namely,tbofe which contayne the equall angles, they hall al- _—fobeequall. . eths Sad Akad 7 Peale that there be two equal Parallelogrammes A Band BC hauing TSS thaancle B of the one-equall’ynto the angle Bof the other . And let the The frft pare SS in’s (DBand BE be fet direttly in fuc D fort that they both make one laos. right line namely, D EF. Aud then( by ther4.of the fieft) halt the lines FB and Crom BC be fo fet that they fhall make alfo one vightline namely, GF, T henl fay, that the fides‘of the parallelogrammes AB and BC, which containe the equall angles, Demenstra- ston of the of the fame. Thefecond pare which ig The fixth Booke angles, are reciprocally proportional: that is,asBDistoB E fois G B to BF. Make complete the parallelogramme F E by producing the fides AF andCE, till they concurre in the poynt FL. Now forafmuth as the parallelogramme AB 1s (by fuppofition ) equall vuto the parallelogramme'B C , and there is acertaine other parallelogramme F E :.therfore( bythe 2.0f the fift) as the parallelograme AB.is tothe parallelogramme FE Ey fo is the parallelogryamme ‘BC to the-pae > 4 ae H vallelooramme FE. But as the paralles : ; , logramme A Bis toy arallelogramme | i FE, Joisthe fide DD tothefide BE 1Aoe ee Oe |B (4y the first of thisbooke ). And (by the ? | % 50 fame ) as the parallelogramme BC is to T the parallelogramme F E, [ois theyide 4 CB tothe fideBF . Wherefore alfo( by | G re the 11. of the fift asthe fide DBis'to the fide BE, fois the fide GB to the fide BE . Wherefore in the parallelogrammes AB and BC the fides which cons taine the equall angles, are veciprokally proportionall : “which was firft required tobe proued. | But now fuppofe that the fides about the equall angles be reciprokally propors o tionall fo that as the fide D Bis tothe fide BE, folet the fideG Be to the fide theconucrfeof BE. L’hen I fay, the parallelogramme A Bis equall ‘yntoy parallelogramme the first. > BC. For.for that as the fide D Bis to the fide BE fo is the fideG Bto the fide BF: but as the fide D B is to the fide BE, fo( by the 1.0f the fice Ms the paralles logramme A Bto the parallelogramme FE: and asthe fide G'B-is to the fide BF, fois the parallelogramme BC to the parallelogramme FE. wherefore alfo (by the 11. of the fift,) as the parallelogramme AB is to the parallelograme F E, fois the parallelogramme BC to the fame parallelocramme FE .W. herefore the parallelogramme AB is equall bnto' the parallelogramme BC (by the 9.0f the jift,). Wherefore in equall and equiangle parallelozrammes the fides which cone taine the equall angles are reciprokall : and if in equiangle parallelogrammes the fades which containe the equall angles be reciprokall, the parallelogrammes alfo fhall be equall :-which "was required to be proued. . $aThe 10.Theoreme. The 15. Propofition. “Lwequal triangles which baue one angle of the one equall un. _, t0 one angle of the other, thofe fides are reciprokal, which in- tlide the equall angles. And thofe triagles which bauyng one angle of the one equal vuto one angle of the other, bane ae ae their of Euclides Elementes. FOl.165e thew fides which include the equall angles reciprokal, are al- — foeguall, 4 One! ppofe that there be two equall triangles ABC,and AD E hauing one mK of angle of the one equall bnto one angle of the other, namely ,the angle B AC equal 'ynto the angleD AE. T hen I fay that in thofe triangles A 5 Cand AD E, the fides whichinclude$ equal angles are reciprokallie propors of this prope uonall thatis,as the fideC Aistothe fide AD; fo 1s the fide E Ato the fide AB. For let y linesC A and. AD be fo put,y they both make direétly one right ine. And foalfo the lines EA and AB jpal both make one right line( by the 14. of ihe firft) And draw a line from B to D. Now forafmuch as (by fuppofition ) the riangle A BC ts equall nto the triangle AD E. And there is a certaine other tris cngle ‘BAD, bnto-which the two equall irtangles being compared it “will follow by he7. of the fifth, that as the triangle AB Cis bnto the triangle BAD, foisy trian- se E AD to the fame triangleB AD. But as the triangle ABC is tothe trige gle BAD, fo by the 1. of the fixth, is the bale C A to the bafe AD: and asthe triangle E AD isto the triangleB AD fo (by the fame) is the bafe E A tothe lafe AB. Wherfore (by the 11. of the Jifth ) as the fide CA is to the fide AD, jis the fide E A to the fide AB. Wherefore in the triangles ABC and ADE tre fides which include the equall angles are rectprokally proportionall, But now fuppofe that in the triangles ABCand ADE >the fides which ine dude the equall angles be reciprokally proportional, fo that as the fide C A is to the fide AD fo let the fide E A be tothe fide AB. T hen I fay that the triangle 4 BCis equall ynto the triangle A 'D E. For agayne draw a line from Bto D. And for that as the line C A is to the line AD fois the line E A to the line AB, Lut as the line C A is to the line AD fois the triangle ABC tothe triangle B 4D and as the line E A is tothe line AB fois the triangle EAD to the trie aigle' BA DWherfore as the triangle A'B Cis to the triangleB AD fois the tiangle EAD toy [ame triangle BAD. Wherfore either of thefe triangles AB Cand EAD haue vntoy triangle BAD one and y felfe fame proportion. Wher jive (by the 9. of the fifth) the triangle ABC is equal vntothe triangle EAD. If therfore there be taken equall triangles hanyng one an igle of the one equal yne tr one angle of the other ,thofe fides in them [hal be reciprokal, which include the equal angles:and thofe triangles which hauing one angle of the one equall nto ove angle of the other haue alfo their ides which include the equall an gles rects pookal,fhal alfa be equall : which spas required to be proued. | Pp. i. The The firft pars litter DemouSirae tion of the Sarme. The fecond part which ig the conuerfe of she fizfi. a rot ' hee <— re > a rt “t--~ hee “ 4? - P © ptyate stent sae fe! fete ! Demoustra- Bren of t be firft part. The fecond art whick 3 the conuer[e of the firit. the line F ne to the line AG , therefore as the line ABis to the line CD, ois the lne CH tothe line AG. Wherefore in the © | | parallelogrames BG and D Fi the fides which include y equall angles, are reciprokally proportionall . But eqinangle parallelogrammes whofe | | (ides which include the equall “e = b gles, are reciprokall, are alfo equal x ( by the Na ibe fixt).W Feifive : | the parallelogramme BO is equa pnto the parallelograme D H. But the parallelozramme BG is that ‘which is contayned ‘vnder the lines A Band F, for theline AG 1s put ¢ ge equal ynto the line F.And the parallelogramme D His that which is contained pnder the lines CD and E, for the line C His put equall nto the line E.Where — fore the reffangle figure contained onder the lines A Band F , ts equall nto the relFangle figure contayned bnder the lines C D and E. | But now [uppofe that the reétanele figure comprehended bnder the lines AB and F,, be equall vnto the reézangle figure coprebended yndery lines CD P, Then ae > F of Euchides Elementes. Fol.166. U ben [fay that thefonre right lines ABC DE and F, ave proportionall,that is asthe line A B is to the hneC D ,/o ts the line Eto the line F | The fame 07's der of conftruétion that was before being kept, forafnuch as that which is cons tatned vader the lines AB anil F is equall ynto that ~which-is contained bnder the lines CDiand FE , but that which is contayned vnder the lines AB and Fis the parallelogramme BG, for the'line AG 3s equall ynto the line F. And that alfa which ts contained rmder-the lines CD eo Eis tho parallelocramme DH, for the line C His equall tnto the line E . Wherefore the parallelogramme BG as equall buto the parallelogramme D H, «> they are alfo equiangle. But in pa vallelocrammes 8 er equiangle the fides ~which include the equall angles are rectprokall( by the r4. of the fixt).Wherfore as the line AB is to the linet D, fots the line C H to the line AG, but the line C His equall nto the line E,and the line AG is equall vnto$ line F..W herefore as the liné A Bis to the lineC D, fous the tine Eto the line F. If therefore there be fonre right lines in proportion, the rettangle figure comprehended dnder the extremes 15 equall to the reétanvle figure contayned vnder the meanes . And if the rectangle fi eure which is contate ned pnder the extremes be equall ynto the rectangle figure which is contained ‘ynder the meanes , then are thofe foure lines in proportion : which was required to be proued. | Le The 12. heoreme. T he (7. Propofition. Uf there bethree right lines in proportion,the rettan gle figure comprehended vnder the extremes, is equall unto the fquare that ismade of the meane.eAnd ifthe retlaugle figure which >» ts mate of the extremes be equal vnto the Jquare made of the s,.,Meane,then are thofe three right lines proportional, ie ppofe that there be three lines in proportion A,B,C; fo that as Ais to B,fo let Bhe toC. Then I Jay that the reZangle figure comprehended ——"ynder 9 lines A and C is equall dntoy [quare made of the line B. Vato the line B (by the 2. of the ae ager D Arf.) put an equall line D. = | And becaufe by fuppofition).. ; ack as AistoB.fois BtoC but | ~~ Bisequallynto D, wheres |: 3 sualh fore (by the 7. of the fifth jas b iA 15 to Bois B to Cybutif | 34. | there be oure right lines pro- portionall, the reétangle ie | ae gure comprehended nder a Y ws) ra > A Th e fis it aie COPE Rb e T he fixth Booke ehe extremes is equall ynto the rectangle figure comprehended ‘pnder the meanes (by the 16.0f the fixt ).Wherfore that which ts contained ynder the lines A and Cis equall ynto that which is comprehended bnder the lines Band D. But that ‘which is contained vnder the lines Band Djs the fquare of the line B, for the line Bis equall ynto the line D.Wherfore the reéZan ole figure compreheded yn- der the lines A andC is equall buto the {quare made of the ne B. But now fuppofe that that pis ow ort which is ita ‘dns 8 connerfeof der the lines Ac C beequal ee onto the fquare made of the line B. Then alfo I fay, that as the line Ais to the line B, fois the line Bto the bne C. The fame order d, construe ; étion that Was before, beyng | kept, forafmuch as y which is contained ynder the nes 4 B® P © A.and Cis equall ynto the Jquare whichis made of the line B. But the {quare swhich is made of the line B is that which is contained ynder y lines B ey é , for the line Bis put equall vnto the line D. Wherefore that which ts contayned yne der the lines A and Cis equall ynto that whichis contayned ‘Ynder the lines B and D. But if the rectangle figure comprehended ‘ynder the extremes be equall onto the reéfangle figure comprehended ynder the meane lynes, the tag right lines fhall be proportional by the 16.0f the fixth )Wherfore as the line Ais to the line B.fois the line D to the line C.But the line Bis equall pnto the lyne D. wherfore as the line Ais to the lyne B fois B to the line C. If therefore there be three right lynes in proportion ,the rectangle figure comprehended ‘pnder the exe tremes,is equall bnto the [quare that is made of the meane. And if the retfangle figure which is contayned ‘bnder the extremes sbe equall’ynto the fquare made of the meane, then are thofe three right lines proportional : which was required to be demonftrated. | | we B A ~. : , —— — - — — —E - > = Se Ne SCN =~. = oe — “~ _ ome a : = ——— — : = — - -= i a + = — = = — _ > = sad it 2 Ls ~ >. —_ ore : iL — < a ee —— a — — ~ = = Smee S =a = ae ee — r ages tee —ae es — = ae ) — — —_— - — a ea = oe taro SF = - - eg ane - Ps = : — a = ae = = : —= — = —_ a - : = — = a neem . = . . “<= ~- - ee ~ = = — ——= - —- =! FF ee te ee, —— —a agitate a peer = = —— - ——--— ———— —— ——— — 7 : = ' q ~—— omy - nn — - - ——- — ae — = — = — == ° = ——— — F _- _ ___~ — — een - ~ aa + - -- - —————EE = —— - : ae a IE or a = == ——— — = 5 = - — —— + + i oe < a . ls = —-- _ : = - - =_—— — a anes = - —_ _ cman = —— —— — - ——— a = ‘ SSS SSS Sse SSS ae , : = = ; Sa eee Se eee % a z Fs comet = We He ; — See => = - ste. St — = = oe en SS + EEE Gta 0 EEE Hi i i | i Wine a te a! + Hit I 1b) aaa Hey ti) Wh i —e “= la a = - a A Corday, q Corollary added by Fluffates. Hereby we gather that enery right lyne is a meane proportional betwene encry two right lines which make a rectangle figure equall to the {quare of the fame right lyne. $a The 6.Probleme. The 18. Propofition, Upon a right line geuen,to deferibe arettiline figure ltke,and in like fort fituate'ynto a rettiline figure geuen. | Suppofe of Enchides Elementes. Foli6%. uN E ppofe that the rie hig &geue ne Aa Bs an dlet t be xetdilme figure Pee g wen bel: G. It is required vpon the right line sewei, 4 B to defcribe a " reéziline figure like, and in like fort Jituatebnto the pettiline fi gure pes yen GE. Drawe aline from F to.F, and nto the right line AB and.tos point init A, make ynto'the aiigle Ean equall angle DAB ( by the » 3.0f the first) and ynto.the right line AB and bnto the pontin it B ( by the Jaine) make vito the angle E FFL an eqiall aiglk AB Dy’ berefore » angle remayning B Al F is equal dato the angle remaynin ig ADB W, heréfore the trianclk HE FE és equiangle Ynto, the riangk D AB. Wherefore bythe 4.ofthe fixt) asthe fide FTF is in proportion © | | to the fide"1) Bs fais °° * dbase Se 5 the fide LE te thé 8 e$ GS a hs ee Jide D'A andy fide* ~~ EF to the fide AB, AN eee Againe ( bythe 23. of ae a 7 ‘eae i\.. v the first ) ynto the Ld de eas as \ / . oe right line BD and , Pf. A. se * | vntoypoint init D, / y? : pare es pee Oey ede ms make ynto the angle on ware: i FdLG anequallane y | | Te eS ee gle BDC sand ¢by the fame) ynto the right line BD and yntothe point init B, make yntothe anele HEG an equallangle D'BC. Wherefore the angle remayning namely, G, isequall onto the angle rvemaynin staimely to C.W heres forethe triangle" LF G is equianele "pnto the triangle D BC . Wherefore( by the 4. of the fixt) asthe fide Hi F is in i poten tothe fide D'B, fois the fide EPG to the fide DC anit the fide G E to the fideC'B. Andst is alveady proned thatas Fi F is to DB, fos’ HE to DA; dnd EF t0 ABW, herefore( by.the 11, of the fift)as E F7is to AD, fois E Fro A Band L1G toD C; and mores oner;GFtoCB: And forafmuich as the angle R Hf P is equal. "bnto. the angle ADB) and the angle F HG is equall buto the angle BOC ; therefore the whole angle E FLG is equall’pnto the whole angle ADC, and by the fame reac Jon'the angle EF G is equall Mnito the angle ABC. But (by conftru€hion,) the angle E: 1s equall’ynto the angle'A,and the angle G is proned equall ynto the ans gleC. Wherefore the figure AC is equtangle ‘pnto the figure EG, and thofe Jides which in it include the equdlltineles are proportional, as we haue before proued . Wherefore the reétiline figure AC is (by the firft definition of the fixt) _ likeynto the recline fixwre peuen EG: Wherefore bpon the right line veuen AB isdefcribed arectiline figure AC like ein like fort fituate vate they ecti- dine figuregenén EG : which was required to be done. OS hee heoremes’ > Lhe 19.Propofition,’ == P pti. Sp Like = > —_ Defeription of the retttline figure te qui« eae Dewon/ira $ 40 Be T he fixth Booke® ~epsiletrianglesare ont to heorherin double proportion that “the fides of lyke proportion are. ae: . SY ppofe the triangles liketobe ABC and DE F, having the angle Noatwicect AOC eae he tangles lke ta, pe ee See B of the one triangle,equal bnto the angle E of the oti er triangle, er jan 3 Re RS las A Bisto BC Jolet DE betoE F Y that let BC er BE F be fides of Se ey fay-that the-proportion of the triangle AB C ynto the triangleD E F is double to the proportion of the fide BC to the fide EF.V nto the two lines B Cand E F (by the 10.0f the fixth) make «, third tyne in proportion BG, fo that asB Cis toE F folet EF beto BG,anddraw.a lyne from AtoG.Now forafmuch as A Bis to.BC,as D Eis toE F therfore alters agg nately( by the 16. of the fifth) as ABistoD E fois BCtro kk. Lut as BC 1s toE F fois EF to BG,wherfore alfo( by thea +" i1.of the fifth )as/A Bis to DE sfois E Fto } BG. Wherfore the fides of the triangles AB Giz DEF which includey equal angles are reciprokally proportionall, But if in triangles hauing one angle of the oneequall toone angle of yather, the fides which include 7 equail ane glesbe reciprokal the triangles alfo( by p15. » the fixth halbe equall Wherfore thetriangle. - >» } | vt AABG is equall’pntoy triangle D EF. And for y asj.line BC is toy line EF, Joss the line E Ftojtine BG; but if there be three lines in proportion, the fife [ball kane to thé.third double proportion that it bath.to the fecond (hy the 10. dee pinition of the fifth ) therfore the kine BC hath mtothe line BG double propore tion that it hath to the ine EF, But as BC is to BG, fo (by the. 1, of the fixth) is the triangle A BC to the triangle.ABG.wWherfore thetiangle. AB C is bnto the triangle A'BG in double proportion that the fide BC is to the fideEF. But the triangle A BGassequalltothe triangleD E F.wherfore alfa the triangle A B (is nto. the triangle D.E F in double proportion that the fide BC is to. the fide E F.wWherfore lyke triangles are one tothe other in, double proportion.that the fides of like proportion are: which was required to be proued, es, $a Corollary. Acotary. >... Hereby itis manife/t that ifthere.be three right lines in pro- = = Nb K-$/ - _ = — a ee = _ = _ = : ~ : = = - ee ———. = = = == = x = sa -- OIE ay = : ; aa <= ar Sons aA 3 : ETA ; = - 5 fe es == aan = eS ee “= ead i <=or Saeed 4 a = = == a > = —_ a Sg as = = = ——— = Toe itieaeal* ac. ~ ~— - = = ees ° —= a: = —— = 2 — = es => — -¥ — = E - _ — - _ —= — = — =— — —— ___ = : : = 5 = — — : enn — ae SSS SS ; —== = : SS === ——SS . 7 = =—s - oe -—x = ~ _ - ~ - —— - _~< a == = = = =’ Re ae re ee - ——= — a <= = ese = ——S— == a 5 gcse ee ———— = = ees SSS ——— —————S— —————————— : SSS = S2-SSe r = = =— 2 5 =a Ss 2 ae = —— = = — (awe ot eee saetneeennied - == - . a —— ae . 12 — == ———— — = —— = —— == = = ———————— = eae any ——_= — ” se = 7 —— a = - — - i —T sa ee 2: = > SS ee ee ee a = == - " pees — se = —— SS ee = : ~- —_ — - —S = E —— ——- = f nS ute ve 4") ih i f $ me |i! it 4 ea i bi) Yl 44 aha iat : qt a] ya i Wit " H CPE bi nh { a op hae EH Oren : waite ; Hi} Pi) it | bat > = cB TAT Tig neg TOE * ree : —- of Euclides Eleméntes. Fol.168. isthetriangle ABCtothe triancleD E F which was re» quir ed t0 bedem anit rated. i A Sap The 14. Lheoremes.The20. Propofition. Like Poligonon figures, are denided into like triangles and equall innumber, and of like proportion tothe whole . -eAnid the cne Poliganon figure 1s to the other Poligonon figure in double proportion that one of the fides of like. proportion 15 to »» one of the fides of ltke proportion. 1 bauing the angle atthe point F equall tothe angle at the point A and MWD iapthe angle at the point G equall tothe ancle at the point Band the an» mea gle at the point A equall to angle at the point C: and fo of the reft. And moreouer, as the fide AB is tothe fide BC, folet the fide FG be to the fide GH, and as the fide B Cis to the fide CD, fo let the fide G Hf be to the fide F1.K ,and fo forth. And let the fides A Bex F G be fides of like proportion. Lbhen td | fay frit; m that the/e Poligouon » yeures BCD. Eur F Ge HKL, are deute ded into C like triane = gles and equall in number . For draw thefe right lines, AC,A DF Her F K. And orefallll as( by fuppofition that is by reafon the figure ABCD E ts like puto the figure FG HK _L ) the angle B is equall ynto the angle G , and as the fide AB is tothe fide BC, fois the fide FG to the fide,G 1, it followeth that the two triangles ABCand FG H hane one angle of the one equall to one. dngleofthe other, and hane alfo the fides about the equall angles proportionall. Wherefore ( hy the 6.of the fixt ) the triangle A BC1s equiangle ‘nto the triane gle FG FL And thofeangles in thé are equa ll onder ‘which are fubtended fides of like proportion':namely; the angle BAC is equall to the angle GF H, and che angle BE Ato the angle\G H F . Wherefore ( by the'4.of the fixt )the fides which areaboutthe equal angles ave proportionall: and the fides which are fube AE Pp. iy. tended - : D H K 4 AV ppofey thelike Poligonon figuresbe ABC D E37 FGH RL, The ferft part of ehig Theorems a cami - = oe SS eS — se a ——s ss ee =. / P ~< | ~ ee ; = See Se : 7” = — = a oer - --—— — er = a - - a a we ~= - == = a = — - =~ — — = = —_ — — —=— — — — ~ . er - : - ts — - = —— ——_—_- — ~ = == eres — SS es eee CU The fecond part demon- rated. ACDis lke to the.triangle F E.K,, and the. Qu Feith Bones tended ynder t a of like proportion Kher foreas0-€ ds to BC y fois HGH. But by fuppofition as BC ts to CD, fois. O.H.to. A K. Wherefore of equalitie ( by the 22. of the fift)as\ AC is to CD, fois FH to ELK. And fora/much as by [uppofition the whole.angle BCD is equall to the whole angle'G FAK \ and itis prouedthat the ingle BC Ais eMail to the ane gle G HF: therefore the an gle remayning ACD js equall to the angle remay= ning EEL 1K (bythe xcommon fentence ). Wherefore the triah tes ACD, aud FAK haue againeone angle of the one, equall to one angle ‘of the other, and the fides which ave about the equall fides are-proportionall Wberefore ( by the [ame fixt of this booke )the triangles ACD pats HK are equangle. And by the ofthis booke \ohefides-which are about the equall angles are'proportio- nall. And by the fame reafon maycwe. prone thatthe triangle ADB is equians gle dnto the triangle F K L.. And that the fides which are about the equall ane glessare proportional Wherefore thetriangle A BCas like to ftriangle FG H, gud the triangle AC D.to the trian lek I K and alforke triangle ADE to the triangle EF KL. ¢by the firft definition of this jext booke ). Wherfare the Pos hganou figures.geuen ABC DE, and FG EK L, are denided into triangles dikeand equallinnimber. cvivot we Gk, shilods wm canosreit bee! . vd Ll gumareoner, that the trianglesave the one) to the other,and.to the-whole A olsgonon fixures propartionall: thatis;asthetriangle ABC astothe triangle EG FH, ois the triangle A C Do the triangle FH K , andj triangldAD E tothe triangle F K L: andas the triangle ABC is to the triangle F Gi fous the Poligonon figure £BCDE to the B | afmuch as the triangle ABC is like tothe triangle FG H, and A€ and FH are fedes of like proportion , therfore the proportion of the triangle A BC tothe triaagle | Pe : | / s€0.3 Five FG H is fk | , double to © | the pre Z 5 portion of the fide AL to. the 4 C - 4 D H ; oc “Ke =. : 2 . : “3% © - « ofition).. fore(by.the 11.0f the fift,) as the yes AB Cis tothe triangle FG A, fois - vs oligonon figure EG HK Lx. For fors ». = ae of Euclides Elementes. Fol.169. is double to the proportion of the fide AD to the fide F K (by the forefayd 19.0f the fixt ). And by the fame reafon the proportion of the triangle A D E.to§ trie angle. K L,,1s double to the proportion of the fame fide AD tothe fide F K. Wherfore( by the 11.0f the fift) as the triangle AC Dis to the triangle FH K , fois thetriangle ADE tothetriangleF KL. But asthe triangle AC Dis to the triangle F HK, fois it proned that the triangle ABC is tothe triangle FG H.Wherefore alfo( by the 11.0f the fift ) as the triangle A BC is to the trie angle FG HL, fois thetriangle ADE tothe triangle F KL . wherefore the forefayd triangles are proportional: namely,as ABCis to FG H, fois ACD toP HK , and ADE to FKL. Wherefore (by the 12.0f the fift )as one of the antecedentes is to one of the confequentes ,fo are all the antecedentes to all the confequentes . Wherefore as the triangle AB Cis tothe triangle FG H, fois the Poligonon figure ABC DE tothe PoligononfigureF GH K L.Where fore the triangles are proportionall both the one to the other ex alfo to the whole Poligonon figures. | Laftly Lfay,that the Poligonon figure ABCD Ebath tothe Poligonon fre The third gure FG HK L adouble proportion to.that which the fide AB hath tothe part fide FG: which are fides of like proportion . For itis proued,that as the triane gle ABC is to the triangle FG. H/o is the Poligonon figure ABCD Eto the Poligonon figure FO HK L. But the triangle ABC hath to the triangle FG Fi a double proportion to that which the fide A Bhath to the fide FG( by the former 19. Propofition of this booke ): for st is proued,thaty triangle ABC is like to the triangle FG 1. Wherefore the proportion of the Poligonon figure ABCD E tothe Poligonon figure FG H KL is double to the proportion A the fide A B to the fide FG : which are fides of like proportion . Wherefore Poligonon figures are denided. exc. as before ; which was required to be proned, Se» The firft Corollary. Hereby itis manifeft that all like rettiline figures What [oem rhe so oe: uer,are the one to the other in double proportion that the fides "2. of like proportion are. For any like reétiline figures whatfoeuer are by this Propofition deuided into like triangles and equall in number. Sm The Jecond Corollary: Hi reby alfo it is manifeft, that if there be three right lines a, — proportionall, asthe firft is to the third, Joss the figure defcri-. Corley. bed ~ppon the firft to the figure defcribed upon the fecond, Jo that the fayd figures be bike and in like fort defcribed. } Foy 2... ~~ — ee as , ¢ 7 re 7 } i / , ’ 7 aut Ping dt a » Paayyy a bs Demonfira- sion. aot Seon efi ab ke > ys Bont thé-cquall angles are proportional, wherfore the » fiztive A, which wasrequired tobe proned. about the equall angles are proportios T he jixthBooke ‘Bor itis proned, that the proportion of the Poligonon fieure A BC D E to: the Poligonon figure kG Et K Lots double toy proportion ‘of the fide AB tothe jide FG. And if (bythe 1 nof the fixt) vuto the lines.A Band FG “spe take a third line in proportton,namely,M N ,the firft line snamely;AB fhall haue vnto the third line, namely, to. MN , double proportionthat it bath tothe fecond line, namely, tok G ( by the to. definition of the fift ). Wherfore asj line A Bis to the line MN, fois the reétiline figure ABC to the rectiline figure FG HA, the fayd rettiline figures being like o> in like _ fort defcribed. | SepTheis.Theoreme.. The 21. Propofition « Reliline floures which are like ynto one and the fame reth = Tine figure;are alfo like the one to the other. WaT V ppofe there be tio reétiline figures Aand B like nto the rectiline INS Zs figure C. Phen I fay that the gure A is alfo like bnto the figure B. | Meek Vor forafmuych as the figure A ts like puto the figure Cit is allo equi. === angle wntost (by the connerfion of the firft definition of the fixth | the fides including the equall angles phallbeproportionall. Agayne foraf- much as the figure Bis like "ynto the. figuie.C3it 0s al/o (by the fame definte | tion }equiangle puto it, and the fides mak? Vicherfore both thefe ficnres Av and Bare equiangle bnto the figures . <5. C,and the fides about the equall angles.are proportionall. Wherfore( by the firf? common Jentence ) the figure Ais equsangle nto the figure Band the fides ae les ai : ine Bis like Ynto the me Lhe x6, Bheoreme..,.. Therz..Propofitions Ff there be foure right lines proportionall,the rethiline figures alfo defcribed vpon them beyng lyke,and in like forte fituate, ~ wwifhallbe proportional. And if che rettiline figuresyppon them y saya\ defcribed be proportional ; thofe right lynes alfo (ball be pro- 7 - 4 2 “J cad % ; % ¢. _ oe -* +. X~ —— 5 meer Pee HOSS or VPS: AMSAT sh: . o's Se. “ N . > ha > SS. -— * Se eS ee, ee 834i Sow ws is kV SS + +a ® : , . 9 i. 4 b - Be SS 7 — 7 u to .% me ean 8 eS — B54 haus * 9 ‘ mn ik @- ~ > 4 . ° Set ee Sieh, |<: fh 233 =v ee es -=N s x % . . - 25 ° _ % ” 2 3 ~ > 3, >, 3 , : f >: | a} > 7 : ‘4 . : 4 “ wi’ t} % * Seb sive % . . ie Sas ies ,F 4 . 7 > = - Suppo e we “ of Euchdes Elementes. Fol. 70. Wa V ppofe there be foure right lines AB,CD,E FandG H, andas AB ‘hoes Sig toC D,fo let EF betoG H. And vpon the lines AB andC D (by SES rhe 18. 0f the fixth) let there be defcribed two reiline figures K AB, and L.C-Diike the one:to the other and mm like fort fituate: And vpon the lynes E FandG H(by the fame) let there be decribed alfo two rectiline figures M Ff and N H like the one to the other,and in like forte fituate. I’ hen L fay that as fois the line EF to the line GH. As the line A Bis tothe lneCD, fo (by the the connerfe 12. of the fixth )let the lne E F be to the lyne OR, and vpon the lyne O'R (by. fechas the 18. of the fixth )defcribe ynito either of thefe figures M Fand N Ha like fie gure,and in like fort fituateS . Now forafmuchas the lyne AB is to the Lyne CD fois the lyne E F to the line QR, and pon the lines ABandC D are dee feribed two figures lyke and in like | ort fituate K AB and LC D,and vpon the lines E and QR are defcribed alfo two figures like, and in like fort fituateM FandS , therfore as the figure K.A Bis tothe figure LC D,fois the figure M F tothe figureS R.:-wherfore alfo( by the-11. of the fifth) as the figure MF is to the figureSR, fois the figure M F to the figure N H, -wherfore the figure MF the figure K AB is . The - | | tothe figure LCD, se oa Hi fois the figure M F Hi to the figure N H. i Vato the lines AB i) andCD (bythe 11. - Z | of the fixth) make 4 es AS D FE third lynein propors | = i tion,namely,O: and | a : i; tntotheliness EF | | = pore 1 { andG Hinkkefort | | | ia make athird lynein | | | | 1 i a line proportion, | | | Ay i {P. And for : a ee ea - iy thatasthelne AB ~ p 7 is to the lineC D, fo if is the line E F to the ii line G H,but as the line C Dis to the line O fo is the line G Hi tothe lyne P. i ' Wherfore of equality ( by the 22. of the fifth ) as the lyne A Bis vnto the line O, | # fois the lne E F to the tine P.But as the line A Bis to the line O fo ts the figure a ; K ABtothe figure LCD (by the fecond corollary of the 20. of the fixth ).And :| Ht as the line E Fis to the hneP fois the figure M F to the figure N 1. Wheres aa) fore( bythe s rof the Fifth jas the figure K_A Bis to the fieure LUD,,fo is the fie ql Hf gureM F to the figure NOH. | 4 ji . But now 'fuppofe that as the figure K A Bis to the figure L, CD fois the The fond q figure M F tothe figure N H,then I fay that as the line A Bis to the line CD, part which i# ae ai * Note that this 3s proued an the afumpt followsng. dn afsumpte ‘Lhe fixth Booke M F hath to either of thefe figures N' Hand S R one and the fame proportion, Wwherfore by the 9.of the fifth the figure N His equal’ynto the figureS R. And it is Dnto it like and in like fort fituate. * But in like and equall reiline figures beyng in like fort fituate the fides of like proportion on “which they are defcribed are equall. Wherforey line G Eis equall’ynto the line QR. And becaufe as the bne A Bis to the lineC D,fois the line E F to the line QR ,but the line QRis equal ynto the line G Fd, therfore as the line ABis to the line CD.fois the line E F to the lineG H, | | If therefore there 7 ‘ be foure right lines ee oe a proportional , the v4 ‘\ rectiline figures als / fo defenbel Dpon Fall : ro them beyng like and ” in lyke fort fituate fhall be proportionall And if the reGiline figures vpon them defcribed beyng like and tn like fort fitue ate be proportional, : thofe right lines alfo p fall be proportional: which was required to be proned. utp. cnnervvaliippmmnteanitidiynaaiinnsssid Mebane MX ies : = 92 wah my Se e+n Afumpt. And now that in like and equall figures ,being in like fort fituate, the fides of Re proportion are alfo equall (which thin ig “was before in this propofition taker as graunted ) may thus be proued. Suppofe y the reciline figures N Hand§ be equall and like and as HGistoG N_fo let R 0 beto QS and let G Hand QR be fides of like proportion.T hen I fay that the fide R Q is equall ynto the fide GF. For if they be vnequall,the one of them is greater then the other, lee the fide R Q be greater then the fide 1G. And for that as the line R Qistothe dine QS, fois the dine F1G to the line G N ,and alternately alfo (by the 16. of the fifth) as the lineR Q ts to the line HG, fois the line QS,tothe lynne GN ; but the line R Q is greater then the line HG.W, herfore alfo the line Q S is crea ter theny lineG Nw herefore alfo5 figure R S is greater then the fi igure FIN but by fuppofition )it is equall ynto st which 1s impofsible.Wherfore > line QR is not greater then § line GH. In like ‘forte alfo may we prone that it is not lefSe then it , wherfore it is equall ynto it: which was required to be proned. The of Euchdés Elements. Fol.171. Fle w{jates demonftrateth this fecond part more briefly by the firft corellary of the 20, of this.boke, ehu b érafatuch a§ che tectiline figures are-by Lappokition imone and the Kame popbitio# tnd the Sai drove ofthe postions deubleto the)proporton of the fides A B.to C D,and. Eto G H(by the ferefaid. corellaty). fecond part af- the proportion alfo of thefides thall be one and the {elfe fame (by. the 7. common fentence) rae AE mex! el he'liieA Bifhall'be\yntoltle line C D as the line E F isto thelline GH. WIS | At WOM GEST he rae Dbeoreme. Lhe 23: Propaktiony °° °° ‘9 ter Flufates. ’ Equiangle Parallelogra innes bake the one to the okber that -oniproportion which ts:compofed of the feces. Al axel V ppofe the equiangle Parallelogrammes tobe AC and C F Kanineethe angle BCD of the one equall tothe angle EC G, of the other . I'henl . ater May j that the parallelogr aimme, AC 1s yuto.the parallelogramme CE wit that proport ion Mibic h 18 C ompofed of the proportion.of thew fidesthat 18 r of that phich the fide BC hath to the fide CG and of that whith the fide DC hathto the fide CE. Letthe lines BC and. G be fo.put.that they both make. one right line ( by the 14.0f the fir{t).Wherefore Pelrsute> 342 ydepnil 1ioit sno eb Thad (by the fame) the lines DC andCE RENE Stats fhall make alfo oné right line. Make complete the parallelogramme DGby producing the fides AD andF Gti?" 7. they concurre in the point 1, and let Bs ncn Gos te there be put 4 certaine right line K. | ‘Aud as the line BC is to the line CG, fo( by the 12.0f the fixt,) put pnto the line K a‘ line’in the fame proportion”, Sphich let be Li and as D Cis to.C B, | fovnto:L, put aline i the fame prow Kh M... | , portion namely, A. Wherefore the proportions of the lines K toLyand LtoM; are one andy fame with the proportions of the fides BC to C G,and D Cto CR but the proportion of K_to M 1s com pofed of the proportions of KtolL, 7 Lt pemensras M: Wherefore the proportion of Keto M, is.compofed of the proportions of the tion. fides BLwCG,7 ECtCD. And for that as the line B C 1s to the line CG, fois the payallelogvamme A Cto the parallelagrame C Hi (by the 1..of thefixt). But as the line BCis tothelmeCG, fois the lime K tothe line L. Wherefore alfo( by the r1.0f the fijt jas the line K is tothe line L, fais the parallelogramme AC to the parallelogramme CH. Agaie for that as the line D Cis to.the line CE, fois the parallelogramme C F to the parallelogramme CF: but as the line Dis to the lie CE; Joss the ine La to the line M:: Wherefore alfo( byy fame) us the line Lo is to the Ime M , fors the parallelagramme C H to the parallelon coramme C Find. fora/muchastts proued,that as the line K. 48 to the line L, o%s.the parallelogramme A C to the parallelogrammeC FA, aud as the line Ls tothe line Msfo.is the paraliclogramme CH ta theparallelogramme C Fs thers * fore of equalitte (hy the 22.0f the fift ) as the line Kis to the line Myfo as the pae . vallelogrannné AC t0 the parallelogramine CF. Bat as it bath before benepra weil the proportion of the line K to the line M,, is compofed of the i of Q 4g. ]: tH , res ree g — on other demon tration afer fiufvates. Demonftration — of this propofitie wherein is frit prowed that the paralle pr amime EG is lsbe to the whole parallele- grame ABCD. 5 | The fixth Booke $A ° the fides BCtoC Gand DC toCE. Wherefore alfo the proportion of the pas rallelogramme AC to the parallelogramme C F is compofed of the proportions of the fides BCtoC G, and DC CE. w herefore equiangle parallelogrammes. bane the one tothe ather that proportion ‘which is compojed of the proportions of the fides : which was required to be proued. | Xe Flufates demonftrateth this Theoreme without taking of thefe three lines, K5L,M,atterthis maners, .) .. | | Fora{much as(fayth he) ithath bene declared vpon the ro. definition of the fife booke, and fift definition of this booke, thar the proportions of the extrenies confit of the proportions of the meanes,letvs fuppofe two equiangle parallelogrames 4B GD, and G F< /sandiertheanglesat the poynt Gin eyther be equall . Andletthe lines 3G and GJ be fet direétly that they both make one : = right line, namely, BGT, Wherefote EG Dalfo\ -4 P x fhall be one right line; by the conuerfe*of the ts. : ofthe firft. Make completé the parallelogramme GT, Then I fay,thatthe proportion of the paral- lelogranimes 4G & GZ is compofed of the pro- portions of the fides B GtoGJ,and DG toGE. For forafmuch as that there are three magni- tudes, 4G,GT,andG?%,and GT isthe meane of the fayd magnitudes : and the proportion of the é eyo ae extremes 4GtoG% confifteth of the meane pro- portions (by the 5.definition of this booke) namely,of the proportion of AGtoG T, andof the proportion GT to GZ: Butthe proportion of 4G to GT is one and the felfe {ame with the proportion of the fides B GtoGJ ( by the firk of this booke) ,And the proportion alfo of GT to Gv is one and the felfe fame with the proportion of the } | Bo el ae oe Se . -other fides,namely, D Gto G E (by the fame Propofition) . Wherefore the roportion ofthe parallelogrammes 4G to G'? confifteth of the proportions of the fides BG to Gl,and DCtoG E. Wherefore equiangle parallelogrammes are the one to the other in that proportion which is,compofed of theyr fides : which was’ required to be proued. * SapT he 18: Theoreme. | The 24..Propofition. Tn enery parallelogramme, the parallelogrammes about the Aimecient are lyke vuto the whole,and atfo lyke the one to the other. V ppofey there be a parallelogramme A BCD, and let the dimecient itherof be AC : and let the parallelogrammes about the dimecient A KC be E Gand H K.T ben I fay. that either of thefe parallelocrames ~~ £ Gand Ht K is like pnto the wholeparallelogramme A BCD, and = are lyke the one to the other.For forafimuch as to one of the fides of the triane ¢A BC namely,to'B Cis drawena parallel le E F, therfore as BE isto E A, fo( by the 2. of the fixt) is C F to F A. Agayne forafmuch as to.one of Tike = oft of Euclides Elemente’: Fol.172. . 4 f = ; se : “e7- 3» mo ; “" of the triangle AD C,namelytoC Dis drawenia parallel tie FGs therefore ( bythe fame )as CH is to'F A fais DG to GA Bul as C Fist EA ois it pio” ned that BE is toB A. Wherforeas DE isto.b iA; Joby thesanoftheffth),. is DG to A. Whenfore by compofition (bythe ig. of the iph ay B Ai L° B,foisD A to AG. And alternately (by the ip.of the fifth Jas BA isto AD, fosE Ato AG: Wherfore inthe parallelagranmes ABC Prand BE Sifades which are about the common anole BAD até proportiondll And becar eF hne® fut) is-equall vnto y angle ADGyex'5 G Fas a payadlel puto the yneD C,therfare theanugle AGE, (hy, the.29, of the : in 3310 anole G F A equall vntoy angle DC A * 208 ot or & olgns slays dio stl and theangleD ACTS common tothe: stockison vis poicnsinsc es oF two triangles ADC and AFG Whersisosis fore the-tridncle "DA CH equiangle? *° * PSS butothe.triangle AGE..And; by. thes, | fame veafon the triangle AB Cis eqni«' angle puta thetriangle AE P. Wher- “a fore the whole parallelogramme ABC D is equiangle yuto the parallelograme EG. Wherfore as A Disin- proportion to DC, fo( bythe 4. of the fixth)is AG [ : toG F, aidas Dis oC A foisG FtoF A. Mid as ACis 0CB, fois AF to FE. And moreouer as C Bis toB AfoisFEtoH A. And forajmuch as it is proued thatas D.Cistol Ajois GF t0.F As but-as ACisto.€ B,fo is AF to FEL Wherfore of equalitie (by the 22.0f the fifth) ax D Cisto CB fois GF to FE. Wherefore in the parallelo \erammes A B CD and EG, the fides. which include the equall angles are proportional Wherefore the parallelooramme ABC Dis (bythe first definition of the fixth ):like onto the paya Nelogramme | And by the fame reafon alfa the parallelogramme-A BC Diis-hke tothe pas rallelogramme K [1: wherefore either of thefé paralleloorammes EG and Ki F4 is like vntothe paralieloeramme A BCD. But reGiline figares which are That the paral- lelogr ame KA ss lske to the whele parallelos | =. f . gramme A B- like to one and the fame rectiline figure ave.alfo(-by the 2. of theyixth ) likethe cp.” one to the other. Wherefore the parallelogramme EC 1s like’ to the paralleloe gramme It K. Wherfore in enery parallelogramme,: the parallelogrammes. ae bout the dimecient are like Ynto the-phole}rand allolike the one tothe other; Which was required to be proned, T An other more brivfe demonjtration after Flyfs. ates. Suppofe tharthere hea parallelostame 4B G'D,whofedimerient ket bea Glabour which let confift thefe parallelogrammes E Kand T J, hauing the angles at the pointes Of aid GC comnie nlwith thesvhole.parallelocrdmme _4B GD; Fheni fay that thof pa- fallelogramnres 2X and TL arelike tothe whole paraliclogramme'D Boand alfo.ate QQ. ii, like That the parale logrammes EGandKH are like the ane to the ether. An other Demon fra- tion after’; F tsiffatess Mai ih ' j Dt Ma n } | +! An accition of Pelitarius. She , ‘ “<°% grammes are defcribed about one & the felfe fame dimetient. And to theendiemight . > : we * : sj 3 likethe-one tothe, other \ ‘For. forafmuch'as BD. EK; and 7,7 are parallelogrammies, therefore theright line e414 2G falling vpon thefe parallell lines eZEBKZT,& | E B aid -D#G,or vpon thefe parallel! lines. K“D,E T,and\ > BT G,maketh thee angles equall the one to the other, . namely,the angle. E 4% to theangle KZ A, & the an- gle E ZA tothe angle K .4Z, and the angle T ZG to tie angle ZG/,andthe angle T @% totheanglelZG,\\ and the angle. Be SiG to. the angle. «4G ‘Dr and finally, , the angle B Ged totheangle DAG, Wherefore, (by the firft Corollaryof the'z 2-of the'firk; and by the 34.” of the firit) the angles remayning are equall the one'to® the other, namely, the angle B to the angle D, andthe, angle E totheangle K, andtheangle T tothe angle J. " . Wherefore thefe triangles are equiangle and therefore’ DT ae ye like the one to the other,namely,the triangle ABG to > | Yk a4 out the triangle G D A, and the triangle AE Z tothe triangle. 7K 4,& the triangle? TG. to thetriangle G/Z. Wherefore asthe fide ef'B isto, the fide BG, foisthe fide AE tothe fide EZ, andthe fide ZT to the fide J’ G~ Wherefore the paralielogrammes: contayned vnder thofe right lines namely the parallelogrammes 4B GD; E.K3& Td, are like the one to the other (by the firft definition.of this\booke) . Wherefore in.euery parallelogramme the parallelogrammes. &c. as before: which was required to be de- monftrated. Sit a A Probleme added by Pelitarins a Teg wiangle. Parallelogrammes being geuen 0 that they be not like, te cut of from : “one of thema parallelogramme like unto the other. : Set eats ©’ “Suppofethat the two-equianele parallelogrammes be 4 BC D and CE F G, which let notbe like the ome to the other, It is required from the Parallelogramme 4 BCD, to. cutofa paraliclogramme like yntothe parallelogramme CEFG. Letthe angleC of the one be equall tothe angle C ofthe other. Andilet the two parallelogrammes be fo fetithatthe lines BC6:CG may make.both oy D L enexightline,namely,2 GC. Wherefore alfo o WP Br ; the rightlines DCand C E thall both make one rightline,namely, DE . Anddrawe a liné front the poynt'F tothe poynt C, and : produce theline # Ctill it cécurre with the AL . a Hne 4D inthepoyntH,AnddrawthelineBR = K Cc = HK parallellto the line CD (by the 31. of rhe‘frit) . Then I fay, that from the paralle- lograinme AC iscutof theparallelograme C'D H K, like vnto the parallelograme EG. Which thing is aanifeh by thys 24.Prope- — fitiom For:thar; both. the fayd parallelo- — the more plainly be feene, I haue made complete the Parallelogramme 48GL. ogpAAn other. Probleme added by Pebtarius... sno] Betwenetma rechline Superpcieces, t0 finde out a meane [uperficies proportionall, . - Another ad- dition of Pe litarittse Suppofe that the two fuperficieces be A and:B, betwene which it is required to place a meane {uperficies proportionall, Reduce the fayd two reGtiline figures dandB Soli | ynto - of Euclides Elementes. Fol.173. ynto two like parallclogrames (by the 18:of this booke ) otif youthirike good reduce eyther of them to a {quare,(by the laft of the fecond).. And let thefaid:two:parallelo- grammes like the one to the otherand equallto the fuperficiecds Aahd B, be CDEF and FGH_K.And let the angles F in either ofthem be equall,which two angles let be placed in {uch fort,that the two parallclogrammes E Dand HG may be'about one and the felfe fame dimetient C K(whichis done by putting the right\lines Z F and FG in fuchfortthatthey both make one tight line,namely, ES EG ).. And make coplete the parallelogrimeC LK At, Then I fay,that either-of the fupplements F.L-& F AZis a nieane proportionall, betwene the fuperficieces CF & F'k,thatis, betwenethe fuperficieces 4 ahd B name- ly,asthe {npetficies AG is td'the fiperficiés F\L, fois’ rhe fame fuperhcies FL to the fuperfigies BD, For: by: RK GW ee this 24. Propofition the line HF is tothe line F Dyas: : the line G'F is to'the line F E ¢ But(by the firit of this hooke) asthe line AF, is tothe line FD, fo isthe fal” perficies /7.G to the fuperficies Ft ands the line.G-E. is to the line¥£, fo alfo (by the fame) is the fuperficies F Eto the fuperficies E D > Whetfore(by the r1iof the fift) asthe fuperficjes AG is'toithe {uperficies FL fois the {ante fuperficies F Z to the faperficie® ED:which was required to be done. | SeThe 7. Probleme. The 25. Propofition. Untoa rethiline figure geuen to defcribe another figure lyke, which foal alfo be equal unto an otber rethiline figure geuen: ee V ppofey the rettiline figure gent, wheruntois required an other to be made like be A BC, and let the other rettiline figure hereunto the Jame is required to be made equal be D.Now tt is required to de{cribe a reétiline figure liReynto the figure ABC, and equall onto the fis gure D. V ppon the line BC | defcribe (by the 44. of the fit[t a parallelogramme BE equall yntothe triangle AB. C5and by the fame Dpon the: line C E;de{cribe the paralles dogramme C.-M -equall ynto the reétiline figure D,and in the {aid parallelogramme let the angle KC; be equall wnto the angle (BL, And forafmuch as the angle FCE is by conftruéfion. equall to the angle C BL adde the an gle BC E common to them both: Wherefore the angles L BC and Bl Eare e Oe ‘ Conftrattion. Dewon|tra- 540% Demouftra- £80 Me ye Meal » Lhe fixth Boake gual vato the angles BC Eand ECF, but the angles BC and BC E are gnall totworight angles (by the 29. of the fir?) -wherfore alfo the angles BC E gndE CF are equall to two right angles Wherfore-the lines B CandC F( by the am. of the fir fl make both one right-line namely ;B F and in like fort dothe lines LF and EM make both one right line, namely, LM. Then (by the 13. of the fixth )take the meane proportional betwene the linesB.C.and CF, which let. be G H. And (by the 12. of the fixth) vpon the lineG H, let there be defcribed a vectiline figure K-H G like bnto therettiline fieure A BC, and in like forte fr tate. And for that as the line BC ,is.to the bne GH, fo is the line GH toy line CF: butif-there be thre righttines proportional, as the firft isto thethird fo 1s the figure which is defcribed of the firft bnto the figure which is defcribed of the fecond, the faid figures being luke and in like fort fituate ( by thefecond correllae ry of the 20. of the fixth )-wherfore as the line BC is to the line C F Jo ts the trie angle A BC tothe triangle K'G H, But.as the line BC is tothe ne C F,fo ss the parallelogramme B E to theparallelogramme EF (by the 1. of the fixth). Wherfore as the triangle A 23g a age BCs to the triangle KGH fois the parallelogramme B Eto the parallelocvamme E F.Wherfore alternately alfo = __« (by the 16.0f the fifth) as the. triangle ABLE isto the pare vallelogramme BE , fo is. the. tyjaugh KG. A, to... ive: the pacalelegrerute EF ex but.the triangle ABC is ge: KAI the paralelograme. .. ywherfore alfo the trians , | gle KG Hisequall ynto the sig AB 7 parallelogramme E F : but the parallelogramme FE is equallvnto the reéfiline figure D.Wherfore alfo the reétiline ficure K G lis equall onto the reGiline figure D, and the reétiline figure K_G His by fuppofition like nto therelzic line figure A BC. Wherefore there is defcribed a rettiline figure KG H bke ‘ynto the reétiline figure geuen A BC,and equall vnto the other rettiline figure geuen D : which was required to be done. yan a. ae The 1 0. Theoreme. The2z6. Propo ition. Ff from a parallelogramme be taken away a paralielograme like ynto the whole and in like forte fet, hauing alfo an angle common with it,then is the parallelogramme about one and , the felfe fame dimecient with the whole. Pee. a nn 7 3 Suppofe a ca ¢ ~ ie. of Enchides Elementes. Fol.174., Kee V ppofe that there be a parallelogvavime PBCD : and fromm the paral PY Yi /clogramme ABCD stake away aparallelogyamme.AF like'vnto the at—parallelogramme A BCD ,and in like Sort fitnate, hauing alfothe aie gle D AB common with it,’T hen I ‘lay, that the paralleloprammes: A BCD aud A F are both about oneand the felf fame ™dimecient AF C,that1s thatthe dimecient AFC of the whole parallelocramme ABCD paffethby the angle F of the parallelogramme AF, and is common to-esther of the parallelogrammes. For if AC do not paffe by the point F then if #4 be pofSible.let it paffe by fome-or ther point as A HC doth. Now then the dimetient AF1 Chall cat -eyther-the fideG For the fide F of ¥ parallelograiiine AF. Let it cat JfideGF in the point FA. And (by the 3 1. of the firft) by the point FA let there be drawen to ets ther of thefe lines A D and BC a parallel line #1 K wherforeG K 1s 4 paralles logramme, and isabout one ind thé felfe fame™ °° | dimetient withy parallelogramme ABCID,..-A-.. -¢ | 5; And forafmuch as y parallelogrammes AB ae 4 | C DandG K are about one and the Jelffame x!__ Nin | dimecient therfore (by the 24. of the fixth)* 8 }- a at | the parallelogramme ABCD is like vnto te | the parallelograimne GK. Weherfore asthe ; RE line D Ais to the ling AB fows the line G.A Ty | tothe line AK (by the connerfion of the firft | ont ohh eet definition of the fixth) And for that the par B = vallelocrammes ABCD; aid EG are (by Juppofition ) like, therfore as the line D-Aastothe ne A Bfois the lineG Ato the line AE. Wherfore the lineG A hath one and the felfe proportion to either of thefe- lines A K.and AF: Wherfore (bythe 9.of the fifth )ehe line A Kis ee quall vatoyhne A E namely, y leffe tos greater, whichis impoffible.T he felfe jame imconitenienctalfo-will follow if you put the dimetient A C to cut the [pe LE Wherfore: A the dimetient of the whole parallelocramme ABCD paf Jethbytheangle and poynt F.And therfore the parallelogramme A EF Gis ae bout one and the felfe fame dimetient swith the whole parallelogramme ABCD. Wherfore if from aparallelogramme be taken away a parallelograme lyke ‘bnto the whole and in lyke forte fituate ,bauing alfo an angle common “with it,then is that parallelogramme about-one-and the felfe fame dimetient with the whole: which was required to be proued. q An other demonftration after Fluffates which proueth this propofition affirmmatinely. From the parallelogramme A B GD let there be taken away the parallelogramme AEZKlike and in like forte fituare with the whole parallelogramme A BG D, and ha- uing alfo the angle A common with the whole parallelogramme. Then I fay that both rheir diameters,namely,A Z anid.AZ G do make oneand the {elfe fame right line, De- nide the fides A Band B Ginte two equall partes in the pointes Cand F (bythe to. of Q q. iiij. the * By the dts metret 15 vynderstand here the dimes tient which ig araiven frem the angle Which 25 com. mon to them toth tothe op- pofste angle, Demonftra- tion leading ta an absurditses An other ‘Way after Fluffates. te aa ed ot “& A ee » ~S - Pond sw ’ Ps ie re lines A ZandC. F ait pis lynes. Now, . toone and the felfefamne lyne, nainely; to Lbesfixth Booke- the firft.) Anddtawe a.line from C to. Feo ying ys: Wherefore the line C Fisa parallel to thé’ 4 tight line AG (bythe corollary added’by Campane after the.29, ‘of the firft), Wher- » fore theangles BA G and B.C Fare equall (by the 29, of the firft):but the angle E A- Zis equal! vito the angle B A G'(by reafo the parallelogrammes arefuppofed: to. be like) wherefore the fame angle EAZ is ¢- gull to the angleB’C F, namely, the out- ward angle to the inward and oppofite an- gle. Wherfore (by the 28. of the firit): the ~ then thelines AZand A’G being parallels F | 3 CF.do conciittein the point A-Whérefore . they arefet dire@ly the one to the other, fo that they both make one right line( bythae which was added in the ende of the 36, propofition of the firft) wherfore the paralelo- | grammes ABG.D,and AEZ K’are about one andthe felfe fame dimétient : whicl was . required to be proued, ae | “i . dm this propofrsa are two Cafes in the frp the pa~ rallelogramme compared to the parakelograme defcribed of the halfe line ss de- feribed Gpen a dine greater thé the halfe tine: inthe [icond G70 « lene lefSe. The fir cafe swhere the pa- vellelogramme compared name ly AF $s defers~ bed Gpon the line A K which 6s greater then the halfe line. AC, . parallelogramme A Dis greater then the parallelogramme AF. For forafnuch ori | | i és gramme AF wanting in figure by the The 20. Theoreme.... The 27. Propofition. ih Ofall parallelogrammesapplied ta.a right line wanting ia fi~ gure by parallelogrammies like and in lke fort fituate to what parallelograme which is defcribed of the halfe line: the giea- test parallelogramme is that which is defcribed of the balfe \ line béing like unto thewant. Seen ha Warn bes right line AB; and ( by the 10.0f | the fir ft denideit in Z to equal partes in the point C. And pnto the right tine A B epply y Va parallelogramme AD: wanting in figure by the parvallelogranme © B which let be like and in like fort defcribed ‘vnto the pardlleloe oramme defcribed of halfe the line A B,whichis BC Then lfay, that of all the parallelogrammes ‘which may be applied yntothe line AB-and — E which wat in figure by parallelogrames | like and.in like fort:fituate ‘ynto the pas © rallelogramme DB the greateft is the parallelogramme AD. For yutoy right line AB let-there be applied a parallelo- | | eat parallelogramme FB, which let be like | and in like fort fituate bnto the parallee = “a logramme DB . Then I fay, that the wo of Euchdes Elementes. Fol.175. as tle paralleiogranmme DB is like vnto theparallelogrammeEB therfore (by the 25.0f the fixt ) they are about one and the felfe fame dimetient . Let their dic metint be DB.and make complete the Agure. N ow forafmuch ast by the 43. of pyirft ) the [upplement F C ts equall vnto the ‘[upplement F BE :adde the fieure F Beommion tothem both . Wherefore the whole figure CR is equall bnto the whoe fictive K’b. Butthefgure CR is equall nto the figure G € by the 36. of tle firft,) for that the bafe.A C isequall ynto thebafe.€B. Wherefore, the fiz Lure us equal ynto the figure KB . Adde the figure CE tommon vata them both. Dann the whole figtre A Fis equall ynto.the whole Gnomon L MN. Butthe-whole parallelogramme DB 1s greater then the Gnomon LE MN ( by the 3.commion fentence). Wherefore alfo it is oréater then the parallelogramme AF But the parallelogranme AD is equall "bnto the parallelogramme, D B (dy the a6.of the firft.). Wherefore the parallelagramme AD. is greater then the parallelogramme ‘A F . Wherefore of all parallelogrammes applied to avight line vanting in figure by parallelogrammes like and in like fort ‘fituate sto that paralelogramme which is defcribed of y halfe line the greateft parallelo erame 4s that which ts deferibed of the halfe of the line’ bein 1g like nto the want : which was equired to be proned. | | , Jeane tet AB be denided into two equall partes in the point C, and let the pardleloeramme applied pon the halfe line be ‘AL, ‘panting in figure by the pardlelogramme LB, which let be like and in like ‘fort fitnate ynto the paralles logranme AL’ “Againe bute the line AB let there be applied an other paratle- loorimmeA E ‘Waziting ih figure by y pa = pee a: ralleograme EB being like and in like fort Sa in 5 fituate vntothe parallelograme LB which “ INbe hoy is decribed pon halfe of the line AB.Thée \ I fay,that the parallelogramme A L apple ed ‘Into halfe the line is greater thenthe © | | paralelogramme AE. For forafmuch as the parallelogramme EB is like vnto the pardlelogramme LB, they are (by the 26. of the fixt ) about one and the fame dimetiz ent .Let their-dimetient be E Band make complete the whole figure , and for that the figure L F is equallynto the fi gure LH (by the 36.0f the firft) for the bafe FG is equall puto the bale G H. thers forethe figure L F is greater then the fieure KE . But the figure L Fis equal vntithe figure D L ( by the 43. of the firft) . Wherefore the figure D Lis Lreder then the figure KE: put the figure KD common to them both hers forethe whole parallelocramme A L is oregter then the whole parallélogramnte o AE: “which was required to be proued. SepThe Dcmonfira- tion of this cafes The fecond cafe wherethe paralielo- Lramme compared namely 4 & ts defersbed Ypon the tine AD which ts defse then the line AC, Demonfirati~ on of the fe- cond cafe. 12259 “=e , ‘ F ee, \ Pe a ae ' a Sept Te a 2 ; — ee : . a eee el Er E sem i Vee eee = 7 Pe BEre se hte ———— ——— eee : ae ee ee a ee = ’ =~ oF ae a ee —— » — ee “e ee ee : : ee See FS = ——————————————S : ; = = —— = = == ~ oy —— apart es ~ —_ S— — i a Roe a etree bg ae ~ - - = — - ° ‘ : ies ; SS ——— : ar ee : = ao —-- Para — 2 i ~ ~~ — = - - — - -< _ —— — . - --- ——— : : = ————————— ————— se > — es - ie — - = — - ——1 q an a E : —=—— - — — : = = a won = — _— ——-. = a ——~s_ SS | - : 2 2 ~ aes =» PhefixtBéokes* sdoow s)he The 8sProbleme,\s «Lhe 28.Propofttion, Upon is right line peuen; toapply a:parallelogramme’ equal S¢6 afeciline figure geuen, (7 wanting sn figure bya paralle~ ». dogramme like-ynto aparallelograme geuens Nowit behor <> \nech tharthereétiline figure geuen, hereunto the parallelo- <\ oramie applied muft be equal be not greater the that paralle-. < “parame, dnbich (is applied vpon the halfe yne, that the soonddofedtes /hall.be likesnamely, the defekt of the parallelograme °° applied upon the halfe line; and the defetl of the Petar grammé £0 be applied whofe defetlis required to belike vite wi tothe parallelogramme geuen on. si guinea ; 4 . % aK V ppofe.the right line genuento be A B, and let the veftiling figure ges /- A40% nen wherunto is required to apply pon the right line A\B.an.equall , ye reétilinefigurebe C,-whichfigure Cet not be greater then. that pas = rallelograme-which.s [0 applied ypon the halfe ine, that the defectes an Poll he, like, namely, the defec# of the parallelogramme applied ppon.the halle = Gae,and the defect of the parallelogramme.to be applied (whofe defect is, requis, xed to be like vnto the parallelogramme.geut ). And let the figure whereunto the 5, defe£ or-want of the parallelogramme ts ak oe ae eae se yequired to be like be'D.Now it ts requis ~ ved ppoy right line gene A B,to defcribe . ‘vate the rettiline figure geuen€, anee. qual parallelogramme wanting in figure. by a parallelogramme like vnto D. Let the line AB ( by the 10. of the firft).be deuided into two equall partes in.y pommt E. And (by the 18. of the fixth ) bppon. the line EB defcribe a reétiline figure EB | Ae B FG Ahke puto the parallelogramme D A. ) § ea Meee andin like fort fituate, which fhall alfo | es bea parallelograme.And make complete the parallelogramme AG. Now then the parallelogramme A Gis either equal vato the recziline figure C,.or greater then it by fuppofition. If the paralleloe ramme A G be equal ynto the reétiline figure €, then is that done which we KN? Seuen Thefirft cafe. fought for.For then vpo the right hne A Bis defcribed vnto the rectiline figure 7 ' 7 NY “e of Enclides Elementes. Fol.t76. geuen C an equal parallelogramme A G-wanting in fioure by the paralleloorante GC B,whieh-+s-like bnto the parallelogramme D. But if A G be not equal bntoC then is AC greater then C but AGis equall yuto GB (by the firft of the fixt ). Wherfore alfoG Bis greater thenC. Take the exceffe.of the rectiline ponre 'B G aboue the reétiline figure C( the firft ) A na ‘ynto that exceffe (by the s.0f the fixt') defcribe an equall rettt« line figure K L MN like and mike fort fithate onto theeiline figure’ D. But the rettilme figure Dis like vnto the rettiline G B,wherfore alfo the retti= line figure KLM N is like vnto the rectiline fieure G B(by.thezs.of the fixt) Now then let the fides K LandG E be fides of like proportion, let alfo_y fides LM and F be fides of like proportion. And ‘forafmuch asthe parallelogrime GB is equal vnto the figures Cand KM, therfore the\parallelo eranimeG B is greater then the parallelogramme K M. Wherefore alfothe fide G Eis greater then the fide KL, and the Jide G F ts greater then the fide L M, nto the fide KL put an equall line G O (by the z. of the firft.) and ikewifeonto. the fide Le M put anequalllineG'P. And make perfeé¢ the parallelogramme 0GP-X. Wherfore the parallelogramme G X is equal ¢7 like Ynto the parallelogramme KM. But the parallelogramme K Mis hke vnto the parallelogramme.G B. Wherfore alfo the parallelogramme G X 1s like bnto the parallelogramme GB. WV herfore the parallelogrammes GX andG B are ( by the 26. of the fixt about one and the felf fame dimecient. Let their dimecient be G'B,and make complete the figure. Now forafmuch as the parallelggramme BG is equall vnto theretfie line figure C and nto the parallelogramme K.M, and the parallelogramme G X ;whichs part of the parallelogramme G B,1s equal buto K M.Wherfore the Gnomon remayning YQ V is equall ynto the reé#iline figure remayning , name: ly,to C. And forafmuch as the fupplement. PR is equall puto the fupplement O S put the parallelogramme X B common puta them both, Uherfore the whole parallelogramme P Bis equall vnto the whole parallelograme O B.But the pae rallelogramme O Bis equal ynto the parallelogramme I’ Eby the 1.0f the fixt, (for the fide A E is equal ynto the fide EB) wherfore the parallelogramme IT E is equal bnto the parallelogramme P B. Put the parallelogramme O S come mon to them both.Wherfore the whole parallelogramme I S is equall nto the whole gnomonY OV. But it is proned that the gnoman ¥ Q Vis equal’bnto the rectiline figure C. Wherfore alfo the parallelogramme T'S is equal buto the ree Etiline figure C. Wherfore vpon the right line geuen A Bis applied a paralleloe gramme I'S equal ynto the rectiline figure geuen Cand wanting in figure by a parallelogramme X B-which is like bnto the parallelogramme genen D for the parallelogramme X Bis like vnto the parallelogrammeG X : which -was ree quired to be done. . . q A Corollary added by Fluffates, Hereby it is manifest that if upon a right line be applied a parallelogramme mantyng 272 by that Which Pehtartns.addeth after the 45::of Calle : The fecond a 2 _ _ , . =~ a ‘ fa eA ’ e 7 iets sit. _ \. PS “> Pree ~ / yd nat ete > ee a '@ ae a hide tin =~ ee he me « i, ™ ate). — ‘ . , . a SE ee = . = ane =, — — ee ——— _________, _ ——— ——— a _ ae ’ en ae Spi, shat a Fis See ema ee ce | : a Ps - ————= : == ——~ ~~ = ‘gal SS ee Sn me oda ¥ aE ——— : > > ~ = > See — Se — : . es , ~ coepiomenpeed ee ee ++ a - + oe? Sr --+ — = = = ~ = == +e ~ Ss Me = . = — SF =a ato ae aie oe - Speer ioe 5 — - =~ = —-— = reer , “ee - a = r ——= =x — - ————————— =a — — = = - = = ~ ——————— Mie =~ ——— : —— a cannnieennnnne nena = ———E eatin a = = : : = ——— ty Sipe eee athatinnhiinartll EE ~ <= = =< = === —- —_—- —- —————— a as “ — —— rs a " - : — - = = ee " OS ee ae —~ nisi ed, EES - + fie =e Se ee —S = A Corollary ad- ded by Flues Se3, And 65 pet of Theon asiam afSumpt before the 17. propofi- tion of the téth booke: which for that st falle- weth of this propofitsen I thought it mor amifse here to place. Confit raion. BTA Lhe fixth Booke ~€es 2. ae% 4 - SY co % ‘er 4} ? a ag > . ° 4 . . a ~ ia lasers aneee the pes iogr anon a pled [hall be equal l. to, the. rectangle figure. | whichis contayned under the-fegments of the line ener which are made by the application, — Fofthereft of the line is equal! to the o- ther fide of the parallelogramme applied. YY ) yure ae ee For that they are fidesof, one &.thefelfe. 4 \ayynsy De OB fame {quare; as the parallelogramme A G | Be as is contained vnder the lines A D and D B,or D Gwhichis equall toD B. : a ¥ P . _ - > “See The 9. Problenie. The'29.Propofition. ste pon a right line geuento apply a parallelogramme equall 0 8 yntoa rettils “rallelogramme like vnto a parallelogramme genen. - SEA ppofe the'vight line genen to be AB, and let the rectiline figure geuen Sy i Whereunto- 1s required vpon'the line AB to apply an equall parallelos eae teramme be C? let alfo the parallelogramme ‘wherunto y excelfe is requie redto be like be D. Now it is requis ee | red bpon thevight line AB to apply © aparallelogramime equal ‘tnto the retiilne figure Cand exceeding in fic vure bya parallelogramme like nto the parallelogramme 'D . Let the line AB be ( by the v0: of the firft ) denis ded into two equall partes in the point E And vpon the line EB ( by the ~ 18..0f the fixt ) defcribe a paralleloe ‘eramme 'B FP like ynta the figure D, and in tke fort fituate. And vnto both ne figure geuen, and exceeding in flowre by a pa- of Euclides Elementes. Fol.197. thefe figures BE and Cjdefcribe an, equallreilinefigure GF like bute the figure D_and nyhke fort fituate( by the, 2s.of the fixt )i Wherefore the paralles logramme (Cc Fl wee Lae wr - — ea = —— - Seo aoe : = - b Cr 5 - . : Y ; : - WG kp, - - : < rt “~~ Pine ef . S Kppaferhe right line genen tobe AB It is reqnired to denide the line Sie 4 B by'an extreme and meane proportion. V pon the line A ee iy ConStruflion. BO by the 4620f the firft') afquare BC And vpon the line AC by the Rrj. 29.0 » 6M a* fs in P ae Ee ete 4 — - ‘ i ye ‘ye The fixth Booke® DemonSira. 29: Of the fixt )apple'a parallelogrammeC D egualkynto the'{quare BC and tion. exceding in fiourt by the fieure AD like bnto the figure BC, Nie BC isa Jquare: Wherefore alfo AD is afquare? And forafmuchas BCis equall ynto CD, take away the figure C Ewhich is common ~\" ) eothem both. Wherefore the figure veniayning ;* namely, BF iis equall tothe ficure remayning ; naniely,to.d D, and the angle E ‘of the one is ve oe alley quall Ynto the angleE of the other . Wherefore’ € by the 2.definition-of the fixt; and by the 14: of Bas the'fixt!) the fides of the figures BF and DA, F ‘which containe the equall aniglesarereciprokall. 8 TB | age Wherefore as the fide F Eis to the fide ED, fois *. °°” \ a the frde AE tothe fide EB: But the fiderE po | a Ra éseqguall pnto the line AC, that is nto the laie | ta AD, and the fide ED 1s equall wnto the line | | AE (by the 3 406f the firft). wherefore asthe Asean wi. tine BA is to the ine AE, foisthéline A Eto. ~ > thelne EB. Bit-rhe line AB is greater then | the tine ‘AE. wherfore alfo the ine A Eis grea’ si yh ter then thelove' I: Bz Wherefore the right line AB is denided by an extreme Fil a) i and meine proportion in the pomt'‘E:and the oreater fegment thereof i is AE? Raia |e | which was required to be done. | es put ie Amotherway. i | Another _.. Suppofe thevight line geuen tobe AB. Itisrequired to deuidetheline AB it way. by au extreme. and meane proportion-Deuide the line A B inthe point C( bythe wil raf the fecand\).in.fich fort that the recLangle figure comprehended. vnder the 1 ql kines AB and. B C may be equall’pnto. the Jquare dee | hig fortbed of the line CA. And forafmuch as that Which. Bowe Al a Al 1 is comprehended ynaer the lines.A Bar BCs equall : A | data the {quaremade ofthe line.A,C, therefore asthe Atte shut | 4 line BA 1s to the line AC, [o (by the 17.0f | the fixt.). is the line A C.tothe line NE i CB. Wherfore the line AB is denided by an extreme ¢ meane proportion in eid ] the point C. which was required t be done,.s\; A Vee CREME He fn retlangle triangles the figure made of the hide fubten ding ws ee T ihe abigle,is equal Onto the figures made of the fides c0- ny pr chending ther ight angle; Jo that the fayd three figuresbe o y | like of Euchdes Elementes. Fol.178. ‘belike and in like fore defcribed. | OSG V ppofe that there be a triangle ABC, whofe angle BAC let bea WZ. right angle. he I fay that the fieure which is defcribed of the lyne B Manatee (Cis equall vato the tivo figures which are defcribed of the lines B.A o GAC, the faid thre figures being like theone tothe other and in like fort defcribed.From the point A( by the rz. of the firft let there be drawne Yne Ay to the line BC a perpendiculer lne AD.Now forafmuch as in 5 vettangle trian® Confiruttion. gle ABC is drawen from the right angle A vnto the bale BC-a perpendicular line AD therfore the triangles ABD and. A DC Jett pon the perpendiculer line,are like vato the whole triangle ABC and aWolike the one tothe other( by the 8.of the fixt ). And fora/much as the triangle ABC is like yntothe triangle ABD,therfore as the line C Bis to the lineB A, forts the lne AB to the lyne BD. Now for that there are three right lines proportional therfore (by the 2: correllary of the 20. of the fixth) as the firftis to the third, foisthe figure made of the firft, to the figure made y of the fecond, the [aid froures ~~ <——-ganastitiin) acl 378 ost being’ like and in tke forte de [bs x feribed: Whereforevas the lyne BCis totheline BD, fois the jroure madeof the line BE to the figure made: of the ine B A, they beyng like and in lyke fort defcribed. And by 5 fame reafon as the line BC is to the : ofthe line BC, to the figure hey bes - > “22: Bp & Fad Ry. jf fore Demonftra- tion. = = TSS 7 - a ke Sad . Meee eres 5 ——— oo Se Hit | | N ; ; fy ' { } ! i au (} Bi | ot | ; ' Hit } f : | ~ j J ; | ] Hid il | H » ay - ie ie ; : H Hi | 4 q : if iy i) i 0 iy j Nae 1, nae | 1h ib} : } if i ; Wate I a 1 t ei) - ! 4) i : a 4 ae aie ti | i } ‘i rf y a tt i || ; 5 ; } ji! mi | | z ti } h im i 4 |) =. wanE: | ' 7 a ‘| ae H diy fan Me eG ¥ 4 ‘i | | An thi 1) q V4 i. ' iil f| #} Bye!) ; ; wi ‘ t Be is Hah aa i Pe) jt ! 1) en 1) he t #), P na i r ny ! AY | eR i im 31 ‘ th \ i iy Het Ly a if } if i t f } { : ae Asi ; He: + ay ) ee | 1 : ' : { : f ‘} 4 : Be. ; ft. : ‘ i i 1) j Lh {va} 1: laste Bh § a A ~t - ~ ae a = ; > ay ‘ : SS be Say = = A 2 iA =z = = . . B-. Me eae eC > = “2 _ = — mee —- . , ‘ = - el ey 6 a Le sas SET rs Sa “ " eee 5 _ = = se 2 Sr eas Se ery Pe: ny “ z KOPPERT - - ew —— = are o = - —— oo one a -- The conuerfe of the former propofttion. The firth Booke > fore the refiline figure made of the line B Cis pnto the reétiline figure made of the line B Ain Tenble proportion to that that the line CBis to the line BA ,and (ty the fame) the [quare alfo made of the line BC i8"bnto the fquare made of thé. ine B Ain double proporiuon ie to that that the line CB is bn ‘i Le BPR m= tothe lneB A.W. herfore alfo é | : as the rectiline figure made of the line Bis to the reiline nre made of the line BA, fo ise [quare “ of the ro ‘ e i SR CB to the [quaremade of the ~ ~ E one rs bore line BA, And hy the fame rea Magen ek. £4 3 Ass fonalfo,as the reciline figure made of the.line BC is to the reGiline figure made of y line C A, fois the {quare made of the line BC to the [quare made of the line (A.Wherfore alfo as the reétiline figure made of the lne BC is to the. veCHline figures made of the linesB A and AC, fois the {quare made of the line B Co the /quares made of the lines B A and AC. But the [quare made of y line B C is equall yato the {quares made of the lines B A and AC (by the 47. of the firfh) Wherefore alfo the reétiline figure made of the line BC is equall ynto the. vettiline figures made of the lines B A and A.C,the faid three figures beyng tyke. andin like fort defcribed. < iy, , 4 : ‘ ; ; ‘ “ ; The conuerfe of this Propofition after Campane. a) | iF the fenre defcribed of one of the fides of a triangle be equall to she fieures which are. defcribed of the two other berth fayd figures being ke tad in like fort de{cribed. | the tri-. angle hall be a rechangletriangle. i ares ; \. Sappofe that A BG bed triangle, and let the fi- fot. gure defcribed of the fide BC be equall jtothe Cc ake tuo figutes de(cribed of the fides AB and AC, thei(ard figurés being fike,andin like fort deferi- bed: Then Lfay thatthe angle Ais aright angle. Let theangle CAD bea righe angle,and put the fine'A D equatt tothe fine A B, and drawea lyne from Dso'C.Now then by this'3 1; propofition, » the fictive made ofshicline CD. is-equall to. the... / two figures made ofthelines AC and AD, the «7 » faid figures being like and in like fortdefcribed. Wherefore alfo it is equall, gare the figare « De - A B made of the line B C, whichis by fuppofition e- Ae a tothe two figures madeof the lines A Cand A D.(for the line A D is put equall to x eline AB) wherforetheline D'Cis — to theline BC. Wherfore (by the 8.0f the firtt) theangle BA Cisatightangle, which wasrequired to be proued. > ~ bp ease ~~ 6% aM cry e * . \ “ +. ‘}* ae, co of Euclides Elementes. Fol.179, Sap T be 22. Theoreme. © The 32. Propofition. Iftwvo triangles be fet together at one angle,bauing two fides of the one proportional to two fides of the other, fo that their hides of like proportion be alfo parallels: then the other fides remayning of thofe triangles {hall be in one right line. pole the two triangles tobe ABC, and DCE, and let two of their pI /tdes AC es D C make an angle AC D,and let the faid triangles haue =e two fides of the one namely,BA and AC proportionall to two fides of : the other namely, toD Cand DE, foy as. A Bisto AC, folet D Che to DE, Demonftrae And let AB be a parallel ynto D Cand AC a parallel wnto'D ET hen rae that the lines BC and CE are'in one right line. For foralmuch as the line AB is a parallell ynto the line D C,and vpon the lighteth aright line AC: therefore ( by the 29. of the firft )the alternate angles B ACand ACD ad are equall the one to the other... And : es | by the fame reafon the angle C D E is equal pntoy Jame angled CQ. : ty yh | Wherefore the angle BAC is equall \ re \ dito the anole C'DE. And i, SP Se eee See ee muchas there: are iwo triangles’ >? 90° ** etsy ABCand DCE, baning the angleiA of the one ‘equall to the anele: D of the other , and the fides about the equall angles are (by fuppofition ) proportional, that ts as the line B Ais to the line AC, Joss the ine CD to the line D E,there harothe triangle 4 BEC is (bythe 6.0f the {ext )-equiangle Syute. the triangle DCE. Wherefore the angle ABC is equail yntathe angle DCE. And itis proued, thattheangle ACD is equall ynto the angle BAC. Wherefore the wholeangle ACE is equal Yuto.the two.angles.ABCand BAC Purthéanz Ze AC Bcommon tothem both, W, herefore the angles. AC. Band A CB are ee qual ynto the angles € AB AC Bi CBA, But theangles CAB,ACB, and C BA, are ( by the 32.0f the iv /t.) equall vnto tivo right angles.oW herefore alfg the angles ACE and ACB are equallto tino rithtaneles. Nowthen nz tothe right line.AC, and wntathe point nat C, axe drawentworight linesBC and C E., not on one and the ante [ide making the fide angles ACE ¢7 ACB equall to two right angles WW herefore.the. lines. BC and C EE by the 14. of the jth) ane fet direcély.and do make one vight hnes Tf therefore two trian lesbe fet together at meangle hauing two fides of the.ang proportionall tonpofides * A of K other, [oy theer fides H like proportion bealfoparallels : then pyjides remaye mug of thofe t rtd agles tha in onevight Lue: which was required to be proued, = : Rr.iij. Although - it bi That the an- gles at the Ces ter are in pro- portio the one to the other,as the circumfe- yences wheron they are. T he fixth Booke Although Enclide doth not diftnidily fet forth the maner of proportion of like rectiline figures, as he did of lines in the 10.Propofitié of this Booke, and in the 3, following it , yetas F/u/Jates noteth, is that not hard to be done by the 22. of thys Booke . For two like rectiline figures bcing geuen to finde outa third proportio- nall : alfo betwene two rectiline fuperficieces geuen to finde out ameane propor- tionall ( which we before taught to do by, Pelstarius after the 24, Propofition of this booke ) : anid moreouer three like rectiline figures being geuen to finde outa fourth proportional like and in like fort defcribed,and fuchkinde of proportions, are eafie to be found out by the proportions oflines. As thus. Ifvnto two fides oflike proportion we fhould find outathird proportional by the 12. of this bokes, the rectiline figure deferibed vpon that line thall be the third rectiline figure pro- portionall with the two-firtt figures geuen by the ‘22. of thys booke . And if be- twené.twoides of like proportion be:taken a'‘meane. proportional! by the 13, of thys Booke : the rectiline figure defcribed ypon the fayd meane fhalllikewife bea meane proportionall betwene the rwo rectiline figures geué. by thefafie22, of the fixt .. And fo if vnto three fides geuen be found out, the fourth fide proportional (by the r2.0f this booke) the rectiline figure defcribed.vpon the fayd,fourth line {hall be the fourth rectiline figure proportionall’: For ifthe right lines be proporti-_ onall, the rectiline figures defcribed vpon theim fhallalfo be proportionall fo that the faid redtiline figures be like & in like fort deferibediby the faid 22 .of the fixt. SapT he 22. Theareme. TLhe33iPropofition. fn equal circles the angles haueone and the felfe fame pro- portion that the circumferece haue,wherin they confit, whe- - ther the angles be fet at the centres or at the circumferences sondin likefort arethefettors which are defcribed vppon the ManGemerens wors\oas\ eS ) St Aon | | ed “ppofethe vquallcirchesto be A BC and D E F whofe centres let be G- Sy yiand Al anditer the anvlés fet'at their centres G and H,be BG Cand E SSE Fi antblert he ang les Jet at their circumferences be BAC andED FxPben Tay shat av tbe cirenniferonce BC is ‘to the circumfirence E F fo ts sheaigle BsC.to theangleE FP eand the angle BAC to the angle E Drs dud moreoner the fetter GB C0 the feaor’ HEF . Vito the carcumference Bhi the.28s ofthe third) ,put as many equal circumferences i order as you smell namely C Keand KL avid Ynto the circumference E F put alfo.as many e? guallcircamferencesin number as you-will;namely,F M. and MN. And drawe thefericht lnesG K3G-L,H Mand HN) Now forafinuch as the circumfes rences BOSCK. and K:Leare equal the one to the other sthe aneles alfo BG €, and SG:K wid KGL are by the'z7. of the third ) equall the one to the other: T herfore bowmulriplex thevcircumference B Lis to the circumference BL, jo multiplex ts jangle BG Lo't6 theancle BOC. And by j fame reafon af, Fes multiplex the-circamferenceN Eis to the circumference E Fo multiplex eg iden: nee, angle of Euchdes Elementes. Fol.80. angle N 1 E to the angle EH F.wherfore if the circumference BL be equal ‘ynto the circumference EN ,the angle BG L is equall vntatheangleEH N,, and if the circumference B L be greater then the-circumference EN, the angle BG L isigreater then the angle N FLE, and if the circumference be leffe, the angle is lefJe.Now then there are fonre magnitudes, namely, the two circumfe- rences B (_ and E F and the two angles that is ,BG (,and EHF, and to the circumference B (_and to the angle BG C,that is,to the firft and third are také equemultiplices namely ,the circumference B Land the angle BG L, and like- wife to the circumference E F and.to the angle E HF that is ,to the fecond and ourth are taken certayne other equemultiplices namely the circumference EN and the angle EA N.. Andit is proued, that if the circumference B L exceede the circumference EN the angle alfo B G L exceedeth the angleE HN.And if the circumference be equall,the angle is equall and if the circumferece be leffe, the angle alfo is le/se. Wherfore (by the 6.definition of the'fifth )as the circumfes. vence'B Cis to the circumference E F,fo ts the angle BG (to the angle E:'H F. ne Butustbeangl BG (:sstotheanglek ELF, foisthe angle BAC tothe oy 5, oy: ance EDF forthe angle B.G Cis doubletothe angle BACyand the angle E ples at thecir- Ei F isatfe double to theangl\E.D EC by the 20.0f thethird) Wherfore asthe swmferences circumference BCis to the circumference F5 [vis the angle BGC tothe angle pa EH F and theangleBACtothe anoleE DT Wwherfoyreinequall circles the dugles rein onéand thefelfe fame proportion that their circumferences are, abherher the angles befetatthecentresjoriatthe.arcninferencess- which was ree quired to-he proved. \ sayy Wi Wi ait, v —— * o ae ~~ Lfay moreouer that as the crcuniference Blis to the circumference EE’, {0 That the ec- » ai ae ’ | _ { : i] ey ¥ Hi lie | lJ ' iu } - " Hii i eal | i a] ii wii ol ’ wit mo) : : "a 1 i) ! it { A 5 4 : ri M4 ; ’ | *s mig iI i ai ; hy] } } “4 q ! | f q ef } nf et wh i ; 4 ah ih t t Mt "| a : Ot ; Uh i } vat} : | it mah Wil i) : 1) { Lhe . |) \ { | } a ~ } } , ) |e : 1 UR) ; ii A) at lt { | r : ij iit ; Wis raty. t 1h) ie ih | ip } q | i} i ] } j : hh a ay : eiir Rae tl ; ' iy ti) } j ae i} et ii } i 7) ; ah - ‘ asi 4 4 ql y. bid ‘ " t i id 4 | 2 ; | eee - } j ¥ : 1a “ ; i : qf PY a} ia. i ae | a 4iq a tT ‘ yy |. Te : Vie | ~ ir, } 7 Ne » | | ihigiaag ; ne iia u 1 Pte , : his : A) : Ch, f At : | } a ay He BAS | iB ‘ At) Nah! | \ eae : by! 4) i} By. 4 y ‘a, ah 1 F 4; fae Hy) in! Slt ! wy ‘ ‘ i t \) | dat 4 ' ! 8 ae We et ‘i t MH 1 + " & uy I ht ay a 4 ¥ ul We mm) - i" f f n | iF, q 3) ty | ee r it vi tI ie | \ } i, il : \, ay 7 . og ef | Bel : f ae Wi a Se ny}. - x ‘ ae ol i 4 qf - _ ) 4 } q 7 j 7 ity Fs uh, ie | i ' F 1 tae ; aa , % P { { } i ; a a) / ; t } ; so : - ' { : ' y ak i eT! i ° ' ¥ ' ) ; | i; a y ‘ i : | BE ma ts ' f ’ { nn » we rf ‘ Ho» ug He y ‘ 3 |} ty ia j 4 a, | " M - i ° r O And draw lines from B to 'P and from. to.C from C.to0 >and from 0 to K. Se *s * ‘An dfocekapegs (by Ce 5, definition of the firft,) the two lines BC.and Gl PS ee ee aoe) oer sitet << ee te ert > Geer 2”. eee SS SSS ee “Se eee es = ~~ 2 . — —— ~~ = * i= a - 4, ek, > ee 7 by = Ss ae 3 Te Ne ee a eee ae ~~ va = rf ee ee ye r PP 9 me EI p+ ———-- nee SS = = Se SS i : ~= = —e ee —i4 ~— = ae Sati eS aa ee SE = = — ee od x > a a by Ly —---F---- ~~ : “ re . : a _ _ - _— _— — — - — —— eee — y =" > ._— ~. ~ : . = —= - _ the triangle G'B Cis equall vnto the triangle GCK.And feng that the circum. oles therfore (by the 4. of the firft) the bafe BC is equall ynto the bale Ker. Ference B (is equall Ynto the circumferente CK, therefore the circumference remayning of the whole circle ABC namely the circumference BL AK OC, as equall pnto.the circumference remayning of the felfe fame circle.A BC names ly,to thecircumferenceC P BLA K. Wherfore the angle BP Cis equall vnto the angle CO K_( by the 27.0f the third) Wherfore (by the 10. definition of the third ) the fegment BP C is like bnto the fegment (0 K and they are fet bpon equall right lines BC and K (. But like fegmentes of circles which confift pon . equallright lines are alfoequall the one to the other (bythe 24. of the third). Wherfore the Ieag BP ( is equall vnto the segment (0.K. And the trians gleGB (ts equal pnto the triangle GC K.wWherfore the fetfor G BC is equall puto the fettor. GC K. And by the fame reafon alfo;the fector G KL ise quall ‘ynio either of the feétors G BC and GC K..Wherfore 5 three feffors G | Wherefore viow multiplex. thecixcnmference B Ls tothe circumference BC Se 4. “~% ? << . 1 kee Jas 4% Al a , t : bow 4 4 d ¥ # he A ad . . 7” aS * of Euchides Elementes. Fol, 181. tor FL EN. And it is prouéd that if the circumference'B L excede the circumpfes rence: N., the feétor alfo BG L excedeth the Jettor EIN. And if the circumference be equall, the feoment alfois equall, and if the circumference be lefse the fegment alfo is lefJe. Wherfore (by the conuerfion of the fixt definition of the fifth )as the circumference BC is to the circumference EF, fo is 9 feétor GBC vnto the fetfor H EF : which is all that was required to be prowed. Sw Corollary. eAnd hereby itis manifeft,that as the fethr is to the Jector, Jois angle toangle by the 11..0f the fifth. Fluffates here addeth fiue Propofitions wherofone isa Probleme hauing three Corollaryes following of it, and the reftare Theoremes : which for that they are both witty, & alfo ferue to great vie, as we thall afterward fee, I thought not good to omitte , but haue here placed them : but onely that] haue not put them tofol- lowé th order with the Propofitions of Exclide as he hath done. qt he first Propofition added by Fluffetes. T 0 defcribe two rectiline figures equalland like unto a retiiline figure genen and in like fort fituate, which fhall dae alfo a proportion genen. Suppofe that the reiline figure geuen be..4 BH. And let the proportion geuen be the proportion of the lines G Cand C.D. And (by the 10.0f this booke)deuide the line AB like ynto theline G D in the poynt £ ( forthat as the line ¢-C is to the line CD, {fo Jet the line 4 E be totheline EB). And vpontheline 4B defcribe a femicircle AFB. And fromthe poynt £ ere& ‘(by the t1.of the firft) vnto thelineeAB a perpendicular line EF cutting che circumferencein the poynt F ,And drawthefe lines 4 FandF®. Avid ypomeither of thefe lines. defcribe reétiline fignres likevnto the reGiline figure AH B and in like fort fituate (by the 18.of the fixe): whichletbe 4 X F,& F1B.Then Bay, that the rectiline figures 4K F,and F/B, haue the proportion geué(namely,the propor- tion ofthe line GCtotheline C'D ) and aree- quall to the re@tiline figure geuen..4 BH. ynto which they are defcribed like and in like fért fl- tuate . For forafmuchas 4 F Bisa femicircle, thercforethe argh ¢4 FB is atight angele (by the 31.0f the third and Fis a perperidicular Hfé. Wherefore ( by thé 8.of this booke) the tridielés AF Eand FRE are like both tothe whol¢ triangle «74 F Band alfo thé one té the other. Wherefore (bythe 4.of this booke )as theline 4 Fistothe line FB, fois the line AE to the line E F,and the line E F tothe in€ EB; which aretides cOtayning equal! angles. Wher- fore(by the 2 2.of this. booke)asthe reGtiline fi- gure defcribed of the tine AF is to the reQiliné pure deftribed Of the line FB fois thé reQie line hgurédeferibed of the ine 4 Eto thereai- | line figare defcribed of theline E F, the fayd rectiline figures being like and.in oe fort ituate. ConftruGs0t of ebe Pros blemes Demonftha- t80n of the OF1€. t er fi “ , . y ] , . epee ck ae ~~ eee a ~ \. rt #“f-- Pree ~ > # * eee ~~ ee ast The fivft Co- wollary. The fond Corokary. The third The fixth Booke firuate .Butas the reGiline figure defcribed ofthe line eE being the firft, is tothe réGiline figure defcribed of the line E F being the fecond,foisthe line ¢4E the firft,to the line EZ the third (by the 2.Corollary ofthe 20.of thys booke). Wherfore the recti- line figure defcribed. of theline 4 F isto the rectiline figure defcribed of the line F 3, astheline «7 £ istotheline EB, Burthelineed Eis to theline EB (by conftruGion) astheline GC isto the line CD. Wherefore(by thet 1,0f the fife) as the line GCisto theline CD, fois the rectiline figure defcribed of the line e4 F to the rectiline figure defcribed of the line F B, the fayd retiline figures being like and imlike fart defcribed. But the rectiline figires defctibed of the lines 4 F and FZ, are equalt tothe rediline figure defcribed of theline 4B, vnto which they are (by conftrudtion ) defcribed lyke and in like fort fituate , Wherefore there are defcribed two rectiline figures ef K F and F JB equalland like vnto the reGiline figure geuen ef B Handin like fort fituate and they hauealfo the one to the other the proportion geuen, namely, the proportion of the lihesG 6 to tlieline.¢C D: which wes required to.bedone, | qf Uhe fir ft Corollary. To refolse a rethiline figure geut into two like rettiline figures which fhall hane alfo a proportio ge- de orif there be put three rightdinesin the proportid geué,andif the line 4B becut ibthedaneproportiduthatthe firitlineis tothe third, the. rectiline figures defcribed ofthelines e4 Fand#B(which figures have the fame proportion thatthe linesud £ and E Bhaue) {hall bein double proportionro that which the lines 4 Fand F Bare(by; the firft Corollary of the 20,of this Eaore) » Wherefore the right lines e4 F and FS are the one to the other in the fame proportion that the firft of the three lines put 1s to the fecond. For the fr# ne to the third;namely,the line 4E to theline £ B is indou- ble proportion thatit is to the fecond, by the 10,definition of the fife, gq I he fecond Corollary. Freréby may we léathe,bow from avettiline figure geuen to take aay apart appointed, leaning the rele ofthe vettiline figure like unto tle Whole. For iffré the right line 4B be cut ofa part appoynted iameély,£ B (by the 9.0f this booke) as the line 4 £is to the line EB, fois the reGiline figure deferibed of the line AF tothe reailine figure defcribed of the line FBitthe fayd figures beitig fuppofed to be like both the oneto the other andalfoto the rectiline figure defcribed of the line 4 B, and being alfoin like fort fituate ). Wherfore takineaway fromthe reciline figure deferibed of the line 42, the rectiline figure de- Kribedof theline F-B the refidue namely, the reiline figure defcribed ofthe line 4 F- {hall be both like vnto the whole rectiline figure geuen deferibed ofthe line «4B, and in like fort fituate. BORE ERE) ; . q Che third Corollary. T ocompofe two like rectiline figures into oneyettiline figure like and equall tothe fame figures. Let their fides of like proporti obe fet fothatthcy make aright angle,as thelines e4 F. and FZ are . And vpotheline{ubtending:the faid angle,namely,the line AB,defcribe are@tiline figure like vnito the reGtiline figures geuen andinlike fortfituate (by the 18. of this booke)and the fame fhall-be equallto the two re@tiline figures geuen(by the 31. of thisbooke) : | ‘* MS Ee | iy a T he fecodd Propofstion: * > If two right lines tut the one thé other obtufeangled wife, and from the endes of the lines which cut the one the other be drawen perpendicular lines to either linethe lines which are Letwene the endes and the perpendicular lines ave cut reciprokally. © -.:: vg} aif ai bans stil ened goisee sautioot bye z °5 sSuppofe we of Euchides Elementes. Fol.1$2. Suppofe that there be two right lines .4:B.and\G:D cutting tht oné.the other in-the point £, audtmaking an obtufe angle in the (e@ion £. And fromthe endes of the lines, namely 4 and G, let there be drawen to cither line perpendicular lines,namely, from the point 4 to theline G D,whichlet be 4 DY; and fron the-point G.to the night line “4 B: which let be GB.Then Tay,that the : : | ¥y rightlines 4 Band G Ddo, betwene thes ~~ a end 4 and the perpendicular B,and the his on aii iereies OS Sail indoles end G and the perpendicular D, cutthe =\) — ™ one the other reciproka!ly in the point E: re i, ~~ fre fo that as the lire AE is to the line ED, SMe: sh aire fo isthe lineG Eto the line E Bor foraf. es 1d of BSE Deweufira- muchas the angles 4D E and GB E, are err : Zr tion of this right angles,therfore they are equall. But’ | Le /. propo/itione theangles e¥EDandGEBarealfoc. 3 guall (by the 15.0f the firlt) . Wherefére : | the angles remayning namely, A.A Di& BG Bare equal! (‘by the Cérollary ofthe 32. ofthe firit,)... Wherefore the triangles CM EDandGEByate equiarigté. Wherfore the fides about the equall angles fhall be proportionall(by the4.ofthe {ixt) » Whetfore as the line 4 £ sto the line ED, fois theline GEtotheline EB, If therefore.two right lines cut the one the other obtufeangled wife:&c : which was required to be proued. q Lhe third Propofition. If two right lines make am acute anele,and from their ender be drawen to ech line perpendicular lines cutting them : thé tworight lines geuen fhall be reciprokally pra- portionall as the fegmentes which aveabout the angle. | Suppofe that there be two,rightJinese--B.and GB, taking amacute anglé AB G. And from thepoyntes «4 andi Gdet there be drawen ynito the lines ¢7B and GB per- péndicularlines 4C and G Zjenttingtheilines of B and @-Bin'the poyntes: B:andt, Then I fay,that the lines, namelyy4B to.G Bare reciprokally proportionall,asthe fee- meéntes, namely, CB to EB which ate dbent the acute:>: is% CPO? Demon ras angle B. Por forafmuchas theiightanglesof CB and tion of this .. GE B are equall, and the angle.4BG is common tothe propofitiens triangics e4 BC,and GB E ptherefore aheeaveles re 4: xt Sua mayning BAC and EGB areiequall.(by ithé:»Gorotla-: - — ry ofthe.3.2. of she firh:)s, Whenfore the triangles!s4 BC) 3) and GB E.are,equiangie o> - Wherefore thenfide® about the equallanglesare propontionall (by the giafthefikt): {fo that, as.theding.e-fB.isto:the dine BG, fois thie line: GS to the le BE, Wherefore alternately asthelane:: © 2) e7 5 isto the ling GByfoasithe ding C A -eoithelines B.bax ; If therefore two right linésmake: au acatean gle’ Gace which wasrequiredtobeproued, _hssexfinome . 4 oog The fourth Rrapofition. T 42 Lf ina circle be drawen two right lines cutting the one the other the [ections of the one to the fectiens. of the-orher-fhall be reciprokally proportionalls: t wk - . ; a. In the citclelaG Bletthelftwo rightlincs AB and GD cutthe onetheotherin P —— athe the poynt £, Then i fay that reciprokally-as thefine .4 E istothéline ED, fo is the ae : + ie line GE to the line EB’. For foralmuch as (by the 35. of the third ) the re@angle fi- P9/0% ons gure Demouflrati- ne 5 ‘ a = we es = a ee a a = on sion of the fea : Phe fixth Boke) lire contayned wnder the lines ef E and EB isequallto therectangle figure contay=** doe antares | ned vnder the lines G Bund ED, bur iti e: ae: eNO. ‘qual reGangle parallelogrammes thefides~ about the equallangles are reciprokall (by’ the 14. of the fixt) . Thereforethe line 4£ is to the line E D_-reciprokally as the line® G E istotheline EB (by the fecond defi-”':"” | nition of the fixt) . If therefore ina circle , be drawen two right lines,é&c: which was \ required to be proued, *~- gt he fift Propofi tion.. If from a poynt genen be drawen in a platne ee oftperficies tworight lines to the concauecircumference of a circle : they-fhall be recis prokally proportionall with their partestake without the circle. And moreouer vicht o dine drawenfrom thefayd poynt ¢y touchine the circle; betwene the whole tine ana the viter fezinent. Suppofe that there be a circle ef BD, and withoutrit take a certayne poynt,namely, G, And from the point G drawe vnto the concaue circumference two right lines: GB and G D, cutting the circle in the poyntes Cand E, Andiet the line G & touch thecirclein the point 4, Thé T{ay, that the lines, namely, GB. to GD are reciprokally as theit parts taken without the circle namely , as GC to G E .For forafmuch as (by the Corollary ofthe 36,of the third) the reangle figure contayned ynder the lines GB AndGE is wenell to:the redtangle figure contayned yn- der thé lines GD and GC; therefore ( by the 14.0fthe'” fixt)rediprokally: asithe line GB isto theline’G Ds fois. the liriesG.C.2o thelineG £, for they are fides contayning equall angles. I fay moreouer;that betwene thelinesGB | and G E, or betwene the lines G:D and G-Cthetouch line G Ais 2meane proportionall, ‘For forafinuch as'there-° &angle figure comprehended. ynder® the:lines G B-and * G E is equallto the oo made‘of the line 'e4 G (by the 36. of thethi Shall be a meané proportional ) it followeth:that the touch line GA is a meane propor. tionall betwéne the extremes GBand G E\(by the fecond partof the 17.0f | _the fixt ) for that by that Propofition the lines GB)G A, and GE are‘: proportionall, Andby the famereafon may itbe proued that the line Ge isa ere me gy betwene the lines GD °° « and GC,and {oo rallcothers . If therefore from @ poyntgeuens ce 2 which was: required to be demonftrated. > e¢ 6 Se Theeéndof the fixith booke of Euclides Elementes. : ) ae 7 RCSA ES | Gy aleg,’ | + ee ~ ee %,i Tye ; . 7 + Ht rt *4 } € ee 1¢ feuenth'booke of Eu- chides Eleimanrées: . - . = . peed call bookes which foltow asin the ténthelewenth, and fo forth, he ve could by nomeanes fully and clearely make plaine & demonftrate; rc a6Kh. * E, Me Bin | Alaa | 1 ire - ] pore p ney Jos. Without the helpe and ayd of nombers | Inthe'tenth is entreated of Sy} lines irracionall andy heertaine,and that ofmany & fondry kindes: 0 “toy = Ast) yr “Zh 's ow = eit yh, 2 . mrs, * Me nn St ns, en om en BASIE EIT my) . k eV Sx andin the eleuenth 8 the other following he teacheth the natures Sb dee EM of bodyesyand compareth theyr fides and lines together, All which 2 Ee ‘ at forthe moft partare alfo irrationall: ‘And as rationall guantites,and ; , AN es x 4 thecomparifonsaad proportions of them; cannot be knowen, nor FADE OEE 27 exalily ‘tried , but by the meane of homber jin which they are firft me. hd : hn pars . Sek fi) ken fenie and perceined + ewen fo likewife Cannot irr tionall quantities be knowen-and fount out without nomber.As ftraightnes isthe triall of crokednés, and inequalitie is ried by equalitie : foare quantities irrationall perceiued and knowen by quantities ratienall : which are firft and chiefely found among nombers., Wherefore in thefe three bookes folowing’, being as it were in the middelt cf his Elementes’, he'ts compelled of necéffitie to entreate of nombers » although not fo fully, as the nzture of nombers requireth , yet fo much as fhall feme to be fit,and {ufficiently to feruc for his purpofe W herby isfene the neceflitie, that the one Arre;namely,G eometrie, hativof the other,namely,of Arithm eticke.And allo of what excellécy.and worthines Arirhmeticke is aboue Geo- metrie:.in that, Geometric boroweth of it principlées,ayd,and fuccor;ahdisas it were maymed with outit. Whereas. Aritameticke is.ofitfelfe, fuficientand neadeth norat all any ayde of Geometrie;but is abfolute and perfitin it felfe,and may-Well be taughtand attayned vnto: without it. Agayne the mat- ter or {ubiect where cbout Geometrieis.occupied which are lines, figures,and bodyes , are fuchas:o£- fer them felues to the fences,as triangles,{quares,circles,¢ubes,and othet are fene& iudged to be fach as they are, by.the figat: but nomber, whichis the fubiedi and matter of Arichmeticke,falleth ynderne fence,nor is reprefented by any dhape,forme, or fignte = and therefore cannot be wdged by any fences butonly by-confideraion.of the minde,and vnaderitanding, Now thinges fenfibleare fatre ynder in de- gree then are thingesintellectugll-aad are of nature much more grofie then they. Wherefore nom ber, as-being onlyantelleciuall, is more pure,more immatefiall,and.more fubtile; farre then is magnitude: and.extédeth irfelfe farther ForAsthmetcke,notonely aydeth Geometric: butminifireth principles, and groundes.to,many other nayrather to all other {ciences aid.artes.Astomuficke; Aftronom y; nae surall ph uofophy,peripectine, witht others, What other thindds in mufickeen treatedof , thennomber coatracted to found and veyce? In Aftronemic.,.who wichout-the knowledge-of nember can doo any thing. either in {earcuag out of themotions of che hé@auenss.and their courfes yeitherin judging and forefhewing the efedcs ofthem?Innatural philofophie,itis.ef no fmall force;The wifeltand belt lear» ned philofophers'tha: haue bene,as Pithegoras, Timexs, Plato,and their followers;found out & taught mo pithelyand purey, the fecret.and.hidden-knowleédge of thé nattreend.condicion of all thinges, by nombers,and by. the propricties'and.padionsof them; Of what force nomiber isin perfpediive:, let him declareandiudge,who hathrany thing traueled therein. Yéa to befhort;whatcan be worth elyand with prayfe prattifed in common. life ofany man ofany condition,withoutthe knowledge of nomber. Yea it hath, bene taught.of the chiefeit amoneit philofophers,;.thacall natural! thinges areframed , and hauvetheircon {iricton.of nomber,Beersusfayth Hee [ust principale in anime. co#ditoris exemplar: Nomber (Cfayth he ) was the:p‘incipall example and patron in the minde of the creator ofthe werld:. Doth aot thar great philopher Tsmeusinhis booke,& alfo P/aroin-his Timeo,followiiig him sfhew how'thé foule 4s.compoled of harmgnicall. nombers.,and confonantes of mitficke ; Nomber compaferh all chitges, and is (after thefe mea) the beingand very. eflence.ofall thinges, And miviflreth aydeand helpe;as to af other knowledges,{o alfo no {mallto Geometrie +, Which thing caufeth Excide in the midett of his boeke of Geametric,to inferte and place thefe three bookes of Arithmeticke-:!as Without the ayde.of swhich,he could not well pafle any father. | _»o att this fenenth booke., he firlt placeth the generall principles, and firft groundes of Arithmeticke, and férteth the difinimous of nhomber orraulutude andthe kinds therof:asinthe firtt boke, hedid of magnitude and the‘kindes and partes.thereof ... After that he entreatethofnombers generally »and.6f they:partes2 and {earcherhand demonftrateth. in general! the moft common: pafftonsand proprietics ofthe fame, and chielely of nombers,prime-orincommen{urable’, and ofnombers compofedercoms meniurabie and of their pro prieties aad pattely.alfo.ef the: comparifon. or proportion.cl one nember toan other, odii>. 2 Se pik cadell =! 224 : iy bsdissieb 3} » RIDE ots ti 2A 7 Definitiens e-« Why Exchde’ tn the middefe of hrs workes Was compelled to adde thefe three bookes of numbers, Arithmetike of m10F€ excellency the2 G COMESTY» Thin ws stelle en fi ” Soy fsteale OF paere AB >, j Worthiness ther th $72 ge <> $f en fre le. fr ithmnetske weinifireth prin ¢ip les asd £7 OF des 49 Hantr to alé [ctences. Boetias: Cap .ts Lib prim, Arithmetis, Timaus. The argument of the feuenth booke. The firf defi WSEIO7, | : | Wil i | Without Gnity A AD | fhould be confu- AW fion of thinges, Boerwussn bis i it ii ) booke de Gnita- ( ce €F Yno, Pal i t ij in other Aefi» i nition of Guiry, Wil The fecond de~ it frritton, Difference be- |e ig fwene a point and Gnity, Boetsus, ia tl: | f, i up ani if Ny ¥ i) . D | ah) a a | af 4 NA fii iH) Bit | 3s orher deft- 12 i i | nition of num- TBE | ber, ; t an. ' i iy f if A oe 1 Ba NPR) | in|] | ui Hi lordanée, “tn other defi- nition of wmsms bers, Vniry hath in st the Gertwe and power of all numbers, Number confs- dered three ma- wer of wayes. The fenenth Booke q Definitions, 1 Vanitie is that,wherebyenery thing that isis fayd to be on. Asa point in magnitude,is the leaft thing in magnitude,and no magnitude at all, & yet the ground and beginning of all magnitudes: euen fo is vnitie in multitude or nomber, the Jeaft thing innomber and no nomber atall,and yet the ground and beginning of all mombers.And therefore itis here in this place, of Excsde firlt defined:as in the firlt booke,for the like reafon and caufe was a point firft defined, Vaitie laycth Eachdeyss that whereb y ewery thong s3fayd to be one: thatis,vnitic is that,whereby cuery thing is deuided and feperated from an other,and remayneth on in it felfe pure and diftin@ fro all others.O~ therwife,were not this vnitie, whereby all thinges are feioyned the on from the other,all things fhould futter mixtion and be in confufion, And where confufion is,thete is no order ,nor any thing can be ex actly knowen,either what it is,or what is the nature,and what are the properties thereof. Vnitie there- fore is that which maketh every thing to be'that which itis . Boeriws fayth ¥ ery aptly : num guedg, ideo ef quiaGunm numero of that is enery thing therfore is(that is,therefore hath his being in nature,andis that it is) for thatitis on in nomber. According whereunte /ordane (in that moft excellentand abfow lute worke of Arithmeticke which he wrote)defineth vnitie after this maner. 3 Vnstas,eft res per fe diferesto: that is,ynitie is properly,and of it felfe the difference of any thirig, Thae is,ynitie is that whereby cuery thing doth properly and effentially differ , and isan other thing from allothers.Certainely 2 very apt definition and it maketh playne the definition here fet of Euclsde. 2 Numbers amutltitude compofed of Dnities. As the number of threeisa multitude compofed and made of three ynities . Likewife the number of fiue is nothing ells ,bur the compofition & putting together of fiue vnities. Although as was before fayde , betwenc a poyntin magnitude , and ynitie in multitude , there is greatagreement and many thinges are common to them both, ( forasa poyntis the beginning of magnitude , fois vnitie the be= ginning of nomber. And asa poynt in magnitude is indiuifible , fo is alfo vnitie in number indiuifible) yetin this they differ and difagree.There1s no line or magnitude made of pointes,as of his’partes. So that although a point be the beginning ofa lyne,yetis it no part therof. But vnitie, as itis the begin- ning of number,fo isit alfo a part therof, which is fomewhat more manifeftly fet of Beetivs in an other diffaition of number which he geueth in his Arithmetike,which is thus. | Numerus,ett quantstates acernus ex Gnitatibus profufes, thatis. Number isa mafle or heape of quanti- ties produced of ynities: which diffinition in fubitance is all one with the firft,wherin is Gid moft plain ly thar the heape or mafle;that is,the whole fubftance of the quantitie of number is produced & made of ynities.So that ynitie is as it were the very matter of number. As foure ynities added together are the matter wherof the number 4. is made, 8 eche of thefe vnities is a part of the number foure,name- jy, fourth part,ora quarter.Vanto this diffinition agreeth alfo the definition geuen of Jordane,which is thus. Number ssa quantitie which gathereth together thinges fenered «fonder. As fine men beyng in them- felues feuered aud diftinGte,are by the nunrber fine brought together,as it were into one mafle,and fo of others. And although vnitie be no number,yet it contayneth in it the vertue and power of all nume bers,and is fet and taken forthem. | In this place (forthe farther elucidation of thinges,partly before fer,and chiefly hereafter to be fet becaufe Emc/de here doth make mention of divers kyndes of numbers, and alfo defineth the fame ) is to be noted, that number may be confidered three maner of wayes. Firlt, number may be confidered abfolutely without comparyng it to any other number,or without applieng it to any other thing,one- ly vewing and payfing what it 1s ia it felfe,and in his owne nature onely, and what partes ithath, and what proprieties and paffions.As this number fixe, may be confidered abfolutely inhis owne nature, that itis an even number,and that it isa perfect number,and hath many moconditionsand proprie- ties. And fo conceiue-ye of all other numbers, whatfoeuer,of 9. 12, and fo forth. aid: ~ An other way number may be cofidered by way of cOparifon,and in refpeét offome other number eitheras equall to jt felfe,or as greater thé it felfe,or as lefle thé itfelfe, As 12,may be cofidered,as d= pared to 12. which is equall ynto it,or as to 24. whichis greater then it, for 12 1s the halfe thereof, or as:to 6, which is lefle then it,as beyng the double therof, And of this confideration of numbers arifeth and fpringeth all kyndes and varieties of proportié: as hath before bene declared in the explanation of the principles of the fift booke,fo that of that matter it is needeleffe any more to be fayd in this place. The third way may numbers be confidered as they are applied to formes arid figures of Geometry. And numbers fo confidered are not reprefented by figures orcharactes of number com monly vfed in Arithmetique,butare fignified by certayne pointes or prickes, which reprefent the vnities which they contayne: which,accordyng to the diuerfitie of the difpofition and placing of them,may reprefent di- uers formes and figures of Geometry + arid. accordyng to the nature of the figure ‘which it reprefen- teth,it taketh his name,and is called a trianguler number, afquarenumber,acube . 4 nuniberior after any other figure, As if the figure of 10. be fo defcribed by his ymi- -**"****** ues — ~ 4 ao ° tang <%., - ee * of Euchdes Elementes. Fol.184.. ties that chey be orderly fet ina ftraight courfe,fo that they reprefent the fortrie of 3 line, then'is the aumber 1e called a lineal! number.And ifthe fame number ro bef deferibed by his vnities, that it fhew forth the forme of a trian le,then itis called 2 trianguler niiber: as yé here fee. Sr Likewife if r2 be in fuch fort defcribed by his vnities, chatit reprefeteth tharformeor figure — , . which in Geometry is called a figure on the one fide longer: then fhall the number rz be . called a number hauyng the one fide longer,and fo may yom conceaue of all others. Thus much of this for the declaration of the thinges following. tees e+ve 3 Apart ale fe numberin comparifon to the greater when the le/se mea/ureth the greater. The third defhs milion, Asthe number 3 compared to thé number 12. 18 2 part. For 3 isa lefie number chen is r2.and Mores euer it meafureth 12 the greater number. For 3 taken Cor added to it felfe).certayne times (namely, 4 tymes) maketh rz. For3 folire tymes is 12. Likewife isi part of 8 : 2 is leffe then §,and taken 4 tymies i¢maketh 8. For the better vnderitandyng of this diffinition, and how this worde Parte, is diuerfly ta- ken in Arithmetique and in Geomety, read the declaration of the firf diffinition ofthe s. booke, ' : . : A The fourth das 4 Partes area lefSe number in re[pec# of the creater, when the lefSe meafus Siiisnen reth not the greater, As the number 3-compared tosis partes of 5 and.not 2 part. For the number 3.18 leffe then the nit ber 5, and doth not meafure 5. For taken once it maketh but 3. once 3 is 3, which is leffe then 5.and 3 taken twife maketh 6, which is more then 5. Wherforeitisno part of 5 but partes,namely, three fifth partes of §. For in the number 3 there are 3 Vitities,and euery vnitie is the fifth part of. Wherfore 3 is three fiftiy partes of s,and fo of others. | 5 Multiplex is a greater number in comparifon of the le/Se, when the lefSe The fh def Li sO8, meafureth the greater. ; ) sox As 9 compared to 3 is multiplex,the number gis greater then the number3. And moreoter 3 the letie number meafureth 9 the greater number;For 3 taken certaine tymes,namely, 3 tymes maketh ‘9; three tymes three is 9.Forthe more ample and full knowledge of this definition, read what is fayd in the explanation of the fecond definition of the 5 booke, where multiplex-is {uficiently entreated of with all his kyndes. 6 Aneuennamberis that which may be deuided into two equalpartes, The fixrbidefe ELEC. * owe As the number 6 may be deutided into 3 and 3 whichare his partes,and they are equall,the one not exceeding the other. This definition of Euclideis to be vnderfiand of two fach equalhpartes whichioy- ned together,make the whole number : as 3 and 3 (the equall partes of 6):ioyned together,make 6,fot otherwife many numbers both euen and odde may be deuided into many equal partes,as intoa. $6, onnmio,and therfore intoz. As 9 may be deuided into 3 and 3 which are his partes, and are alfo‘equall; for the one of them excedeth not the other: yetisnot therfore this number 9.an even number,for that 3-and 3 (thefe.equall partes of9) added together makenorg bur dnely 6. Likewife taking the definiti+ on fo generally, euery number whatfoeuer.fhould be an-even hum ber : forin thas fort of vnderftading there is no number,buetharit may be deuided into twoequall partes : as thisnumber7 may be deni dednto 3 partéssnamely,3.:2. ands; of which two,namely,3 and; are equahyctisnorzan even num- ber, becaufe 3 and 3.added together make not 7.Boermus theifore inthe firtt booke ofthis Arithetike; forthe more playnes,after this maner defineth an euen number : Nudachwe pareft;: gut potef? ix equalia Rud Asusds,Gno medio mors sntercedente,thatis’: rd ‘ acs ha) 3] al a:Anjeuen number is that which may be deuided into two equall partes;without an vnitie comming 247 other defr- betwene them. Ass 8 is deuided inte 4and 4 two equall partes maitbond an vnitie comming betwene pirion of an ewe them ,which added together,make 8, fo that the fence of thisdefnition is;that'aneuen numbeiisthar which is deuided into two fuch equall partes, which are his two halfe partes. tot. risa > Hére1s alife to be hetedthata partis taken in this definition and in certaine definitions folliewing, _. notin thatfignification as it was ffore defined, namely ,for.fuch a partas :meafureth thewholernm=- “ef: ber, but for any-pare which helpeth:-to the making of the wholaland into whichothe whole may he:irex folucd,(o arejand s-partes of § in this fence but notin the.other fence. For neyther 3 non 5. rbeafureth $.Pithagor asand his {chiolers gaue an other definition ofan euen number(which definition Bustins abe, Pit hagerats < fo hath after thismaner. --~ * ) OU tat Beetsns. nisi ber. i, An is ae ga Meng ~ wr RT ee et _— —.\. Pee Fr * = a =" = = — ae = SSS Sn se = Sa — <= = Sas >t

”, mst IO, T he fenenth Booke Aaeuensumber sa that which i one and the fame dissfion. ss demded sntothe greateft andinto the leaf : sa ro the greateft as touching [pace,and sutathe leaft as touching quantitie. As io,1s deuided.into 5-& 5. tr ave his greateft partes (which greatnes of partes he calieth {pace) andin the fame diuifion the number 10. is deuided but into two partes ; but into lefle thé two partes nothing can be deuided which thing he calfeth quantitic, fo thac 10 denidedinto.s ands. in thatone deuifion,is deuided into the greatetl, namely,into his halues,and into the leaft,namely,into two partes,and no mo. 7 3 There is alfo another definition moreauncient,which is thus. An exen number is that which may be densded inte two equall partes,and inte two Gnequal partes but im ney~ ther diusfien (to the confistucton of the whole ) to the ewen part ss added the edde,westher to the odde is added the euen, As 8 may be deuided diuerfly ,partly into euen partes,as into 4and 4. likewife into6 and2z. and partly into odde partes,as into 5 and 3, alfo inte7 and 1. In all which deuifions, ye fee no odde parte ioyned to an even,nor an euen partioyned to an odde : but if the one be euen,the other is euen, and if the one be odde,the other is odde. In the two firft deuifions,both partes were cuen,and in the two lath poe partes were odde.Itis to be confidered that the two partes added together muft make the whole. This diffinition is gencralland common to alleuen numbers, except to the number 2. which can not be deuided into two vnequall partes,but onely into two ynities, which are equall. There is yet gené an other diffinition of aneuen numberjnamely,thus. 4x even wumber s3 that which onely by an Gritse esther nboue it or Gnder it differeth from an odde. A$ 8 being an euen number differeth from 9 an odde number, being aboue it but by one. And alfo from 7. vnder it, it differeth likewifle by one,and fo of others. ; 7 Anodde number is that which cannot be denided into two equal partes: or that which onely by an ynitie differeth from aneuen number. “As the number § can be by no meanes denided into two equall partes,namely, fuch two which ad- ded cogether,fhall make 5. Or by the fecond definition,s an odde.number, differeth from 6 an euen number abouc it,by 1, And the fame 5 differeth from 4 an euen number ynder it likewife by 1. ; Boetins defineth an odde number after this maner. 5. ies he. Ae edde number is that whith cannot be déenided tate two equall partes, but that un Gnitie foal be betwene rhem.As if ye deuide 5 into 2 and s. which are two equall partes,there remaineth one or an ynitie be- ewene them to make the whole number s, There is yet an other definition ofan odde number. 4x odde number ss that, which being densded inte swe Greg¢all partes how/oewer the one ss euer euen; and the otherodde. As if 9 be denided into two partes, which added together,;maketh the wholesnamely,into 4 and 5, which are vnequall:: yedee the one 1s eudn;namelys4-and the otheris odde,namely, 5.{0 ifye deuide g into6 and 3, oriito Sand s.the one partiseuereuen,and the other odde.. « | | 3 A number enenly enen( called in latine pariter par) is that number, aphich anenen number meafureth by an enen number. Ag 8isa numberenenly.euen.For 4an‘enennumber meéafureths by 2,whichis alfo an euen nums bet‘Ehis definition tiath eauch treubledimany,and feemeth nota true definition, for that there are many numbers which!ewen numbersdo meaftre,and that by even numbers, which yet are not euéns ly ené-numbers, aftermof mens:minds:as24. which's an euénumber doth meafure by foure, which is alforan enennumber,and yetas they thinkeis not 24an euenly even number;for that 8an euen.né ber-doth meafure alfo 24 by: 3.. an odde number. Wherfore Campane to make his fentence plaine, after thisunaner fetteth forth this definidon.Parster par off quews cundi pares enm numerantes , paribus Gitibus numerant thatis: ar 7 | 4 _coAn euenly.cuen numberis,when allthe euen numbers which meafureit, doomeafure it by:euen times,that is, by euen numbers,as 16, Allthe euen numbers whith meafure 16, asare 8. 4- ands, do meafureit by.cuen numbers.As 8 by 2;twife:8 is:16:4 by 4, foure times 4is r62and 2 by 8, 8 times zis 16. Which particle (all ewen numbers ) added by Campane maketh 24.to be no euenly:euen number. For thatfome one cuen number meafureth rt by an oddenumberas 8 by 3, Féufates alfoisplainly of this minde,that Ewclide gaue not this definition in fuch maneras itis by T heen written, forthe oy age: & generalitie theref,for that it extendeth itto infinite numbers Which’ arenot’enenly even as he thins keth, for which caufe in place therof,hegeueth this definitiome © <> iT 7 iw < hnumber enenly eucn,ts that which onelyenem numbers do meajure As t6is meafured of none but ofeven numbers,and-therforeis euenly euen.There is alfo of Boetius geuenan other definition of more facilis tie,including in. itno-doube at all,which is moftcommonly vied of all writers;and is thus.* 0) © ¢" = i Anumber enenlyeuenss that which may be deuided inie two euen partes,and that part apayie inte two even partes,and fo continually dewiding without flay til ye come to Gnitie.: As by example,’ 64, may be denided . i cists soedmepratl Of of Euchdes Elementes: Foli8s, into 32 and 32. And either of thefe partes may be deuided into two eifen partes for; 2 may be deuided into 16 and 16, Againe, 16 may be deuided into 8 and 8 which are euen partes, and 8 into gand 4% A- gaine 4 into 2 and 2, and laftofall may 2 be deuided into one and one, 9 Anumber enenly odde (called in latine patiterimpar) is that which an. e« uen number meafureth by an odde number. | As the number 6 which 2 an euen number meafureth by 7 an odde number,thre times.2iis 6y.Like- wife ro. which 2.an even number meafureth by s an odde number.In this diffinition alfo is found by all the expofitors of Each#e,che {ame want that was found in the difinition next before., And-for. that it extendeth it felfe co large. for there are infinite numbers which euen numbers.do meafure by odde numbers,which yét after their mindes are not euély oddenibers,as for example 12. Fors anenént- ber,meaiureth 12 by 3 an odde number: three times 41S 12. yet isnotizas. they thinke an euenly odde number. Wherfore Campane amendeth irafter his thinking, by adding of this worde #/,as he did in the firit,and defineth it after this maner. A number eucnly odde i3,when all the evex numbers whach met{ure if Ao waonfure tt by, Gneuwen tyes, that ss, by ait tade mumber. As 10.1s'a number euenly odde, for no een number bne onely. 2. meafureth 10. and thariis. by san odde numbez. But not all the euen numbers which meafure 12.do meafureit by odde numbers. For 6 an cuen number meafureth 12 by 2 which is alfo enen: Wherfore 12 is not by this definition .a num- ber euenly odde.Fluffaresalfo offended with the ouer large generalitie of this definition to make the definition acree with the thing defined putteth it after this maner. A number eaenly 6dde, is that which an odde number doth mealure onely b ry am euen wumber, As 14.which 7.an oddé number doth meafure onely by 2.which is an euen number.Thereis alfo an other definition of this kinde of number commonly geuen of more plaines,which is this. A number enenly odde is that which may be deutded into two equall partes,but that part cannot agayne be de- nided into two equall partes:as 6,may be deuided into two equall partes.into.3.and 3. bur neither of chem can be devided ito two equall partes: for that 3.is an odde number and fuffereth no fuch diuifion. 10. A. number oddly enen ( called im lattin in pariter par) is that which an odde number mea/ureth by an euen number. As the number r2:for 3,an odde number meaftiréth 3 2.by 4.Which is an etien number: three times 4.18 12, This definition is not founde in the greeke neither was it doubtles ever in this maner written by Exchdé: which thing the flendetnes and the imperfectiori theredf and theablurdities following alfo of the fame declare moft manifeftly . The definition next before geuen is in fubftance all one with this, For whatnumber foeuer an even; number doth meafure by an odde , the felfé fame number doth an odde number meafure by an euen, As 2,an eué number meafureth 6.by 3.an odde number. Wherfore 3,an odde number doth alfo meafure the famenumber 6 .by 2.4n eué nfiber: Now if thefe two defihiti ons be definitions of two diftin& kindes of num bers,then.is this numberlé, both cuenly cuen,andalfo cuenly odde and fo is contayned vnder two diuers kindes of humbers . Which is dire@ly agaynft the authoriticof Ewclide who playnely proucth here after'in the 9-booke , that eviery nomber whofe halfe isan odde number,is a number euenly odde onely.Fluffates hath here very wellno ted,tharthefe two cuenly odde, and oddely even , were taken of Exclide for.on and the felfe famekinde ofnomber. But the number which here ought.to.haue bene placed is called.of the beft interpreters of Ewelsde, numerus pariter par €9 nupar, thatis a number euély eué,and eucly odde, Yea andit is forcalled of Eaclide him felfe in the 34. propofition of his 9. booke: which kinde of number Campanus and Fluffates in fteade of the infuficient and ynapt definition before geuen,afligne this definition. A number enenly.enén. and chenly oddé)) ti that which an eucn “number derh. mealare fometime by uk euen number,and [ometime by an odde. As the number 12:forz.an euen number,meafareth izuby 6,an euen number:two times:6,is 12.Al- fo'g’an cucn niimber meafureth the fame number ta by 3.4n odde number.And therefore is 12.4 hum- Ger enenly ¢tien and evenly odde, and fo offuch others. 90 bk ads: | Thecaufe why that Canitanus and Flaffates were {o {crupelousin amending (as they fuppofed ) the two definitions before; namely, of a number cucly ewen and of a number evenly odde, the one by ad- ding this word-e/,and the other by adding this word gzely , was for that they were offended.wich the fargeues ahd gereralitie of then.For'that by therm, on iand che felfe faniéhuinber might | fended vnder either definition: And fo ,thefelfe number fhonld be both everily-euen q =e Xr : at? 7 > pa r, ; " . odde ‘which they tooke for an abfurditie. For that theyare two diftiné and¢h Buc all things well and iuflly conceiued were let and written by Ewclide je'compre- sand alfo evenly uers kindes of numbers. , it fhall not be hard nor amiffe to thinke , that thefe definitids infuch forme and mans pasthey are deliuered vate vs by Theoniand Sf. ily, that The ninth definitions Cam PARE Anotherd efi iL 6 0P8. Flufsates, Ain other definstion, An other definitson. The tenth dcfinstion. This definition not foundin the Greeke. san other defi» nition Campane and FlufSateswsth other im An ers ror, Wb i i Bh fy if es % a” ert, \ STS ee “4 =~ = —~ - ‘ae ee : Ra a - = ~~ noes io .] j . J —_ Se fe _— -s =~ = = = mn - " =o my OE es aes “ Sh ate F . —- = — 7 = {re ns > zt -_ ons - - - — =~ —--= _—— ——— — —— = a == nt + a ei Ze - ; = =~ wt A ~ —— mh ae se ’ — ~~ o= — SS - ae es —~ : _ = _— - — ead < — = 3 : = : - a = SSS a Se <.SF® or ca a SMe a — —_—— +S : =~ - oes = = : a ai iD te ? a - = Seer ots Se 5 ae s —_— . = C — = = Sa Ps ee one oa a = — : a — - ES as ON ae SO RS a eS as a a = - eae ae - 2 = = = = — —_—— = SS —- — — = —- - + a ee ee ~— —_—-_~= amet eee Saas a a — = a SES = 2 oar : as or a ead Ho a a ——— = > seed . - - 2 : a Se = : = = =< E —= = ~ _ —+-=- - ation a And por -- - a — re aces a > — - St meee 28a a een ~ Loo sce + wees ~ te — —" = _ re ————— —— - - _ -_--— — - — _ 7 . - — = 7 - - ee ee : = : == eo omen re = er = See eee ~ ~~ Loah =e — SSS a Se ee = = aan —_ ———— = — = 3 = ss sins _ errs mies 2 z = =i ——v = . “i = ane’ so te ene 5 eet an he eee neti ng ap . awe One wamber in diners refpeites way beof it~ ssers kewtles of wu mabers. ‘ ° . Poelixs def weh= tion of a&niume ber ewenly eué, anda enemy oa. T he eleuentt definitson, qet sx' F lufsates. An other defi- witton. " The twelfth Prime numbers called sncompen~- fed numbers. he thirtenth definition. BAe The fenenth Booke~ thab they ntede notthefe corrections and amendementes by adding thefe wordes «/ and enely forad- mit thatthey be diftinét kinds of numbers,may notcontraries be attributed in dinerfe refpedies to one thingeMay not one line be fayd to be great and little, compared to diuers > Great in comparifon of & lefie,and lefle in comparifon of a greater? Euen fo one number in diuers refpects may be of divers and contrary kindes of numbers. What are more diuers thema fquare number and a cube number.And yet is 64:0 diners refpectes a number both {quare and cubc . In ref{peét of 8. to be his roote,itis a fyuare number: for 8.times 8.1s 64.and in refpectof 4,to be his roote;itis acube number for 4.times 4 fower times is 64: fo in divers refpectes itis both, without any abfurditie at all. Likewife this number 6. in di« uers refpectes, is anumber on the on fide longer , and alfoa trianguler number : which yetare diuers and diftniét kindes ofnumbers® For 6.defcribed by his vnities refembling a figure of lengthe and breadthe hawing two fides,namely 2.and 3.is a plaine or fuperfici- all number of one fide longer. And ifthe fame'é be fo defcribed by his ynities, that itreprefentéth the figure of a triangle, then is itand beareth it the name ofa trian- gler figure :'as here ye may fee the forme of either. And if ye extende in defcription ; cherof alPhis vnities in length onely,fo is &alfova lineall number. So you fee 6 in di- ¥s uer's re{pectes is a lineall number,a number on the one fide longer, and aifo a trigo- fin nall or trianguler number,and yet therby no inconuenience at all, And why may not likewife.one and the felfe number in diuers refpectes be accompted a number both ove mee euenly euen,and euenly odde ? Yea Zwclde him felfe doth moft manifeftly proue the fame , and in the fame wordes, if it be diligently wayed,in his ninth booke, For he fayth, that all numbers being double in continual! courfe from the number 2 .be ctienly euen numbers only: and agayne all numbers whofe halues are odde ; are euenly odde numbers only : and that number which neither is duple from the number two norhath tohis halfe an odde number ‘isa number euenly euen and a number evenly odde. What in this can be fpoké more playnely?So that by Eaclsde it is no inconuenience thaton num- ber,as 12,for example,in divers refpectes fhould be both a number evenly euen,and alfo anumber e- udniy odde.In refpedt that 6.an euen number meafureth 12 .by 2.a2n euen number,12 is anumber euen- ly euen:and in refpedt that 4,an ever number meafureth 12.by 3.an odde number, 12 .isanumber e- wenty odde: And thus iudge ye ofall others like, ‘Theres alfo an other definition geuen of this Kinde of number by Boersu#s and others: commonly whith is thes, : A number exenly enen,and ewenly odde 1s that, which may be deusded into two equall partes , and eche of them may agayne be densded tuto twoeqwall partes:and fo forth. But this dinsfion ss at lenghth fea 2yd , and continueth not till st come to Gustse, As for example 48: which may be deuided into two equall partes, namely , into 24.and 24. Agayne24. which is on of the partesmay be denided into two equal! partes 12. and 12. A- gayne 12,into6,and 6.And agayne 6 may be deuided into two equall partes,into 3.and 3: but 3,cannot be deuided into two equall partes. Wherefore the deuifion there ftayeth: and continuethnottill it édine to vaitie as it did in thefé numbers which are euenly euen only. | ra Anumber odly odde is that which an odde number doth meafure by an odde number. | As2s,which s,an odde mumber,meafureth by.an odde number,namely by 5. Fiue times fiue is 25: Likewile at.whom7:an odde number doth meafure by 3; which is likewife an odde number . Three Wines Fsiszti. : ae Fixffatws coueth this definition following of this kinde of number, which is all one in fubftance with the formendefinition, | A nuriber odly odde 6° that which onely an oddé number doth meafure. As.1s5 forno number meafureth 15. but onely 5. and 3: alfo 25: none meafureth it but onely which.is.an odde number, and‘fo of others. veel me 12 A prime (or firft,) number is that; which onely ynitie doth-meafure. ‘As 5 i7.21.19.For io. Aumber meafureth but onely vnitie.For v.ynities make thenumber 5. Sono number meafureth 7,butonély vnitie.2.taken 3.times maketh 6.'whichis lefle then 7: and 2. taken 4, times is 8, which is more then 7 . And fo of rt.13. and fich others. So that al prime numbers, which alfo are called firit numbers,atid numbers vacompofed,haue.no part to meafure thé,but onely ynitie, 13, Numbers prime. the one: tothe other are they, which onely vnitie doth meafure being a common meafure to them. : 44 + As 1s.and 2a.benumbers prime the oe to the other « 19-0f it felfc is no prime number,for aad y —— Se ee of Euchdes Elementes. Fola8e. ly vnitie doth meafureit,but alfo the numbers scand 3, for 3\times 5.19 27) Likéwife 22.is of it felfe'no prime number : for itis meafured by 2,and 11,befides ynitie. For 11 twife,or z.eleuen times,make 22, So that although neither of thefe two numbers ts.and 23. bea ptime or incompofed number, but ey- ther haue partes.of his owne, whereby it may be meafured befide ynitie : yer compared together, they are prime the one to the other : for no one number doth asa common meafure,meafure both of them but onely vnitie, which is a common meafure to all numbers . The numbers s.and 3. Which meafure ry. will not meafure 22 + againe,the numbers z-ard 1t.which meafure 22,donor meafure 15. om 14 A number compofed, is that which ome onenumber meafureth, A number compofed is not meafured onely by vnitic,as was a prime number, but hath (ome aum- ber which meafureth it. As 15 :-the number 3.meafureth is namely taken s.times. Alfo the number 5, meafureth 15 ,namely,taken 3.times, s.times3,and 3. times 5, 1s 15 . Likewife 18.isa compofed num- ber, itis meafured by thefe numbers 6.3.9.2. and {6 of others. Thefe numbers are alfo called commons ly Second nunmibers, as comtrary to prime or firft numbers. 1s Numbers compofed the one to the other are they x which fome one riumber , being a common meafure to them both, doth meafures As 12.and 8.are two compofed numbers the one to the other. For the number 4, isacommon mea- fure to them both : 4.taken three times maketh 12: and the fame 4.taken twotymes maketh 8. So are 9.and 15 : 3.meafureth them both. Alfo10.and 2§ : for 5.meafurcth both ofthem : and fo infinitely of others. Inthys do numbers compofed the one to the other or fecond numbers, differre from numbers prime the one to the other : for that two numbers being compofed the one to the other, ech of them feuerally is of neceffitie a compofed number . As in the examples before 8.and 12, are compofed num- bers : likewife 9.and 15 : alfo ro.and25 : but if they be two numbers prime the one to the other , ‘it ig not of neceffitie,that ech of them feuerally bea prime number. As 9. and 22.are two numbers prime the one to the orher : no one number meafureth both of them : and vet neither of them in it felfe and in his owne nature is a prime number, but ech of them isa compofed number. For3.meafureth 9,and tr.and 2.meafure 22. 16 Anumberss fayd to multiply anumber,-when the number multiplyed, is Jo oftentimes added to it Jelfe,as there are in the number multiplying Dnie ties : and an other number is produced. In multiplication are ever required two numbers, the one is whereby ye multiply commonly called the multiplier or multiplicant, the other is that which is multiplied. The number by which an. other is multiplied namely the multiplyer, is fayd to m ultiply. As ifye will multiply 4.by 3,then is three fayd to multiply 4 + therefore according to this definition, becaufe in 3. there are three vnities : adde 4536 times to itfelfe, faying 3. times 4 : fo fhall ye bring forth an other number, namely, 12, which is the fummée produced of that multiplication : and fo of all other multiplications. 17 When two numbers multiplying them felues the one the other, produce an other : the number produced is called'a plaine or fuperficiall number.And the numbers which multiply them felues the one by the other;are the fides of that number. As let thefe cwo numbers 3.and ¢,multiply the one the other, faying, 3.times.6,or fixe tymes 3,they fhall produce 18. Thysnnmber 18.thus produced, is called a plaine number, or a fuperficiall number. And the two multiplying numbers which produced 1z,namely,3.and 6, arethe fides of the fame fuperficiall or plaine number,that is,the length and breadth thereof. Likewife if 9 multiply 11.0r eleuén nine,there {hal | : pad 23 99. aplaine number, whofe fides are the two numbers 9« oS oo oe oe ar ee Be and. tr 2 asthe length and breadth of the fame, They iat called plaine : and faperficiall numbers , becaufe being defcribed by their ynitiesona 9 = plaine fuperficies, they reprefent fome fuperficiall forme er figure Geo- metricall,hauinglength and breadth... As ye {ee of this example: and fo of others. And all fich plaine. or fuperficiall numbers do euer re- prefent right angled figures as appeareth in the example. The fourtenth Kehwetion,; The jiftenth d efimit 107%. The fixtenth definition, Twe numbers required in malt ip licatsow. The feuententh definition. Why they are called [superficie al numbers, The esghtenth definstion, wh yy they are | called folsd numbers, The wsinetenth definition. Why st iscalled wt [quare wiie~ ber. aM The rmenteth definition, — a ei--*%! Why it ss called acube number. The twenty one ewes Padi ‘duced,namely,20.1nto 3: which is the third number,fo fhall ye produce ’ #29. whichis acubennmber . And Evclsde calleth T he feuenth Booke 18 When three numbers multiplyed together § one into the other, produce any number the number produced,is called afolide number ;. and the numbers multiplying them felues the one into y other’, are the fides therof. As taking thefe three niibers 3.4.5. multiply the one into the other. Firlt 4.into 5.{aying,foure times 5.is 20: then multiply that number pro- 6o.which is a folide number : andthe: three: numbers whieh produced the number,namely,3.4. and 5. are the fides of the fame . And they are called folide numbers, becaufe being defcribed by their vnities,they.re- prefent folide and bodylicke figures of Geometry,which haue length, breadth, and thicknes. Asye fee this number 40. expreffed here by hys vnities. Whofe length is hys fide s,his bréadth is 3,and thicknes 4.And thus may ye do of all other three niibers multiplying the one the other. r9\Afquare number is that-~which isequallyequall: or that “which 1s contays ned ynder two equall numbers. As multiply two equall numbers the one into the other, Asg:by9.ye hall produce 81,which is afquarenumber, Ewcésde calleth it anumber e- 9 qually equall, becaufe it is: produced of the multiplication of pwo equall ees a ie boat A OE | numbers the one into the other . Which numbers are alfo fayd in the fe- 1-4 es OB ops cond definition to contayne a fquare number’... As in the definitions.of the (tio 9] ‘3 =f fecond booke two lines are fayd to containe a {quare ora parallelogramme b Hoh dolibaduh a figure . [tis called a {quare number, becaufe being defcribed by his ynities « 7 detach fs 1ae et Ct i it reprefenteth the figure of a {quare in Geometry. As ye here fee doth the peg ciated eee number 81 whofe fides, that is to fay,whofe length and breadth,are9.and rye 3 Be 9,equail aumbers + which alfo are fayd to contayne the {quare number 81: et and fo of others. ' 20° Atube number is that which is equally equall equally:or that which ts cone tayned ynder three equall numbers. . As multiply three equall numbers the one into the other,as 9,9, and 9.: firit 9.by 9, fo shall ye haue $1: which agayne multiply by 9,fofhall ye produce icanumber equally equall equally ,becaufe itis pro- duced of the multiplication of three equall num- bers the one into the other: which three numbers are fayd in the fecond definition (wherein he fpea- 9 keth more applying to, Geometry ) to contayne'the cube number. It is therefore called a cube number, becaufe being deferibed by hys vnities, it reprefen- teth the forme ofa cube in Geometry, whofe fides, that isto fay; whofe length, breadth, and thicknes; are the three equall numbers 9, 9, and 9, of which he was produced : which three fides alfo are fayd to containe the cube:number 729 < beholde here the defoription:therof. -21 Numbers proportional are, when the Wie is to the fecond i nemultiplex: | as the third is to the fourth, or-thefelfe fame part or Be fet Here he defineth which numbers are called proportionall, that is, What numbers haue.one and th felfe {ame proportion, For example’e.to3 > and 4.t02z, arenumbets proportionall, and haue oneane che {elf fame proportion: for ¢.the firft is to 3.the fecod equemultplex.as 4.the third is co.2.the ar - re q i * « all,rationall,and irrationall : and therefore was conftrayned to geue'the definition by the exceffe,equa- litie or want of their equemultiplices,and that generally onely : for that irrational! quantities haueé no" certayne part or common mea{ure to be meafured by orknowenyneyther can they be exprefied by any cértayne numbers. But here in this place becaufe in numbers there are no irrational! quantities, bural are certaynly knowen,fo that both they.and the proportions betwene them may be exprefled bynum- bers certayne and knowen, by reafon of their partes certayne, and for that they haue fome common meafure to meafure them(at the left vnitie which is acommen meafure to all ny mobers)he geucth here this definition of proportionall numbers, by that the one is like equemultiplex to the other,or the fame part,or the fame partes : which definition is niuch eafier then was the fet ‘andisnotfo large, as is the other,neither extendeth it felf generally to all kinde of quantities rational! and irrational but con- tayneth it felfe within the limites and bondes of rational! quantitie and numbers, 22 Like plame numbers and like folide numbers are fuch which haue their fides proportional. Before he fhewed that a plaine number hath two fides, anda folide number three fides. Now he fheweéth by this definition which plaine numbers, and which folidé numbers: are like the one to the other, The likenes of which numbers dependeth altogether of the proportions.of the fides of thefe, numbers, So that if the two fides of one plaine number,haue the fame propertion the one to the o- ther that the two fides of the other plaine number haue the one to theother,then.are fuch two plaine numbers like.Foran example 6 and 24 be two plaine numbers, the fides of 6 be 2 and 3, two fymes 3 make 6: the fides of 24 be 4 and «; foure times 6 makes 24,Againe the fame proportion that is betwene 3 and 2 the fides of 6, is alfo betwene 6 and 4 the fides of 24. Wherfore24 and 6 betwo like plaine-and fuperficiall numbers. And fo ofother plaine numbers. After the fame manner is it in folide num- bers, If three fides of the one be in like proportion together,as are the three fides of the other,then is the one folide number like to the orher. As 24 and 192 be folide numbers, the fides of24.ate2.3. and 4, two tymes three taken.4 times ate:24, the fides of 192 areg:6. and 8:-forfonre tymes 6, 8 imes:make 192. Againe the proportion of to 3 is fefquitercia,the proportion of 3 to 2 fs fefguialtera, which are the proportions of the fides of the onefolide nunrber,namely,of 24: the proportion ‘berwene 8 and 6 is fefquetercia, the proportion betwene.¢ and 4 is fefquialtera, whieh are the proportions of the fides: of the other folide number,namely,of 192, And they are one and the famewith'the proportions ofthe; fides of the-orner,wherfore theferwo folide numbers.24 and 192 be likejandfo of other folide nibers,. The fwenl? 23 Aperfeéénumber is that ,which is equallto all his partes. 19 12d As the partes of 6 are 1. 2.3. three is the halfe of 6, two the third part,and r, the fixth part, and mo partes s hath net: which three partes'r2.3.. added together,make« the whole number, whofe partes ' they are. Wherfores isa perfect number. So likewife is 28 a perfe& number, the.partes whereof are thefe numbers 14.7.2 and 1: 14 is the halfe therof, 7 is the quarter, is the feuenth part, 2 is a four- tenth part,and 1 an 28 part,and thefe are all the partes of 28. all which, namely, 1,2, 4, 7 and 14 added together;makeiuftly without more or lefle 28, Wherfore 28 is 4 perfeét number, and fo of others the like. This kinde ofnumbersis veryrare and feldome found; From 1 to. 10,there is but one perfeat num ber,namely,6.From 10 to an yoo,there is alfo but one,that is,28, Alfo from 100.t0 1000 there is but one Which is 496.'From 1060 to roooo hkewife® but one. So that betwene every flay in num bring which is euer in the tenth place, there is found but one perfect number And for their rarenes and great perfec- tion,they are of maruelous vfe in magike,and in the fecret part of philofophy. «- This kinde of number is called perfect, in refpect of other. numbers which.aie umperfect, For as the nature of a perfeét number flandeth in thts , that al his partes added together are equall to the whole: and make the whole: fo in an imperfect niiber all the parts added together are hot equal to the: whole, nor make the whole, but make either more or lefle . Wherefore of imperfect numbers there are two kindes,the oneis.called absedans or abuading the other dumsmtus,or Wanting. ' A number abundingis that whofe partes beingall added together make more then the whole number whofe partes they arejas 12.16 an abundant number. For all the partes of 12. namely; 6,4-3 2. and W hy the ofa nition cf pranws’ portional mag < situdes 13 Gi - Lrke to the ae fre witge of Propers, tienak negne écrs.- The twenty kwo depitttignt, three definite. ~\ _ LX Perfect num- bersrare €§ of great Sle sn mragthe EF 1 fi ecret philofo- phys la what re[peE a Winker tS perfed = Two kinds of sonperfed numbers, ; ie abe ix ~~ ieee iit ii .. we \. or 7 we ort ' -* tt ia 2. See ~ ; . SE Ee oe aa yt “eee = = Wo ee ———————————————— i 5.438 <—o = is _ - . ~ + Seo 5 ~~ +--+ = — - “ = —— : ae ve =_ = — 2 ~ —s wv 2 re or - a ween k mes om -~ _—~- - ——e i -— = oes - = a ey —— . =. np: eee ° — a —_ ——— _ ee eee = Se ie a 2 ze 5 ~ = = ae : eS. ee Soe Se ae . ee ee we - “ aiies aT Fe a > . — -- > a i a e - - A sumber ware Hag Contasen fen- fomess.: Bir eonumoon fe Ontense. Sécond commen fentexee, Third commen fentence, Fourth common [entence, ue» ae *s. Ss 3 Fifth Comes f- wi emCe. The fenenth Booke. . and 1.added together makes eswhich are more then ts.Likewife:28,is a number abunding, all his parts namely,9.6.3.2.and 1,added together make 2o.which are more then 18:and fo of others. * Anumber diminute,or wanting is that whole partes being all added together, make leffe then the whole, or number whofe partes they are. As 9.is a diminute,or wantiog number,for all his partes,namely,3 and 1.(moe partes he hath not) added together make onely 4: which areleffe then.9, Alfo 26.is a diminute nfiber,all his partes, namo- ly.13,2.1-added together make onely 16: whichis amumber much leffe then a6,And fo oF fach like. tere Ampane and Flufates here adde certayne common fentences , fome of which, fr\o. for that they are in thefethree bookes following fomtimes alledged,1 thought Gh good here to annexe. 1 Ihe leffe part 1s that which hath the greater denomination : and the greater part is that which hath the leffe denomination. » “As the numbers 6,and 8.are either of them a part of the number 2 4:6,isa fourth art,4.times 6,is 24:and 8.is a third part,3.times 8.is 24.Now forafmuch as 4(which enominateth what part 6.is of 24) is greater then 3. (which denominateth what part 8,is of 24.)thereforeis 6.alefle partof24,thenis 8,and fois 8.a greaterpartof 24, thé 6.is.And fo in others. | 2 Whatfoener numbers are equemultiplices to one o the felfe fame niber, or to equall numbers are alfo equall the one tothe other. Asif vnto thenumber 3 be taken two numbers containing the fame number foure times; thatis being equemultiplices to the fame number three : the fayd two numbers fhalbe équall.For 4,times 3.will euer be 12,So alfo will it be if vnto the two equal num- bers 3. & 3,be taken two numbers,the one cotaining the one number 3.foure times, the other containing the other number 3.alfo foure times,that is, being equemultipli- ces tothe equall numbers 3..and 3. “3 Thofenumbers towhome one and the felfe fame number is equmultie ‘oplex,or whofe equemultiplices areequall: are alfo equall the onto the other. As ifthe number 18.be equemultiplex to any two numbers , thatis ,contayne any ¢wo numbers twile,thrife, fower times &c? As for example 3 .times: then are the fayd. two numbers equall. For 18,deuided by 3.will euer bring forth 6.So that that diuifion made twife will bring forth 6,and 6,two equall numbers. So alfo would it follow if the two numbers had equall equemultiplices,namely, if 18.and 18. which are equall nume - bers contayned any two numbers 3,times, sseqgiod fanumber meafure the whole and a part taken a'way:it fhall alfomene ~.. fure the refidue. . a, Asif from 24. be taken away 9,there remaineth 15.And foras much asthe number 3 meafureth the whole number 24,& alfo the number také away, namely,9. ithhall al- fo meafure the refidue,which is 15.For 3.meafureth 15 by fiue,fiuetiges 3.181 5. And fo of others, : 4 hauohst same ale | : ‘5 Ifanumber meafure any numbersit alfomeafureth ewery number that the fayd number meafurethe 0.0. 0 oe ie bas As the number 6,meafuring the number 12. fhall alfo meafure all che numbers that 12,meafureth-as the numbers 24,36,48,60.and {0 forth:which the number 13. doth of Euclides Elementes. Fol.88. - meafure by the numbers.2.3.4.and 5.Ahdforas muchas thenuinber t 2.doth meafure the numbers 24.36,.48.and 60:And the naber 6,doth meafure the number 1 2.(namely by 2.) Itfolloweth by this comm fentence,that the number 6.meafureth eche of thefe nunibers 24,36.48,and 60,And fo of others, | ‘6 Tf anumber medfure to numbers, it hall alfo meafure§ number come i*** ommen [extence, pofed.of them. As. the number 3 meafureth thefetwo numbers 6,and 9:it meafureth é.by 2,and 9, | by 3. And therefore by this common fentence it meafureth the number 1 5 which is compofed of the numbers 6,and 9:namelyit meafurethit by 5, 7 Ifin numbers there be proportions how manyfoeuer equall or the felfe . oe Jametoone proportion: they {hall alfo be equall or the felfe Jame the one ti men fentence, the other. As yfthe proportion of the number 6, to the numbet 3, be as the proportion of the number 8,to thenumber 4, ifalfo the proportion of the number 10, to the num= ber 5.be asthe proportion of the number 8,to the number 4:then fhall the proportion of the number. $, to the number 3,be as the proportion of the number 10, is to the number 5: namely,eche proportion is duple,And fo of others , Euciide in his 5. booke the 11.propofition demontitrated this alfo in continual! quantitie : which although as touching that kinde of quantitie it might have bene put alfo as a principle (asin num- bers he taketh it)yet for that in all magnitudes theyr proportion can not be expreffled, (as hath before bene noted & fhalbe afterward in the tenth booke more at large madé manifeft: )therefore he demonftrateth it there in that place, and proueth that it is true as touching all proportions generally whither they be rational or irrational. q Thefirft Propofition. Thefirft Theoreme. If there be geuen two ynequall numbers and if in taking the le{se continus ally from thegreater, the number remayning do not meafure the number going before, dntillit fhall come to ‘vnitie: then are thofe numbers ‘which were at the beginning geuen prime the one tothe other. V-ppofé that there be two unequal numbers A B the greater, and CD the lefe, and from AB the greater, take away CD the lefse as often as you can leaning F A, and fromC D take away Comfrudven, By}! F Aas often as you can,leauing the number GC. And from fey F A take away G C as oftenas you \! Gan, and {0 do continually tillthere. E.. ‘|| remayne onely vnitie, which letbe D...G..C - iy HA. Then Lfay that no number B..... F:. HOA — eafureth the numbers AB and aa CD. For if it be pofible let {ome number meafure them,and let the famebeE. Now CD abfurditie, meafurin@ABleaueth a lee number then it felfe,which let be F A. And ¥ A meafuring }).C leaneth alfo a leffe then it felfe,namely, GC. And G C meafuring F A leaneth o- yitichi A, encima as the number E meafurethD C, and the number C D mea- fureth the number B ¥ therfore the number E alfo meafurtth BF, and it meafureth the whole rt S 2 aman Peer < ~_ pl . rea 2. rs — Rene aa The connerfe of shes propofitsen. eae : oa) 5 w . > ~ - > —» m © os = " 2 - . tore : a * — : 5 -|. ~s | ~ : * y - —— -- _ — — a = x ’ aa ~. < =e a =a = eee = > “3 = =< ss - — : . —- ~ 7 — = * ~ ——~ = ~we ee” oe — a ——— ~ -~ == x Sar = ee mo a = = ey — ~ z - = > — — = = ad oe - = —_ » —< — _ “ = | — i ~~ = . * . = = = = - - w : - ; co = “ > - = S * oe = ~ a < 7.2 E- oe Pinal — > _ ——— ee + aaa = : — ++ ia . ' ~ = : = = = == — et atin — a —-- — ————— ~~ == Z -- —- a : = == 7 - = = . Ge - = + ————— — > ——_~ =. ¥ ——- = : — - = - : - = : ~ i 6 eta - oe - 3 an = ~ = - = = - - - — ~~ —~- —_—_—— = —— } —T ee pw 2 ~ 7 =e ae = ——S—= — =~ : —— = ae eee —— ss 3 F . _ - ==> ; <= <— == “ enn - be prime the oné — $e —— (a < a = . = = ati & > — = ~ : ss err, The fecond cafe. The feuenth Booke white number BA, wherfore it allo meafureth that which remaywerh imately the mamber EA (by the 4.comen fentence of the fenenth) But the number £ F mealureth the number D'G,wherfore E alfo meafureth D G. And it meafureth alfo the wholeD'C swherforett allo meafureth that which remayneth,namely,the number G C (by the [ame tommon fen- tence):but G C meafureth the number F Hswherfore allo E meafureth E H, and it mes- \ fareth rhe whole number ¥ A, wherfore (by the former common fentence) it allo mealu- reth that which remaynethH A,which is unitie,it felfe being a number, which ts tmpof]i- . ble. Wherfore no number doth meafure the numbers AB and © D, wherfore the numbers A Bund C D avé prime numbers the ome to'the others whith was required to be proued. . The conuerfe of this propofition atter Campane. ‘Andif thé two tian bers j namely A B and CD be prime the one'to the other. Then the leffe being conunually taken trom. thegreater there fhalbeno flay of chat {uftration,till chat you come to ynitic. Forif in the continwall fubitraction there-be a ftay»before yell come to ynitie. Suppofe that H A be the number whereat the flayis made, whichalfobeing DAlsei oe fubtrahed out of G C leaueth nothing. Wherfore H A meafurethG Cwher- s B.... e . FH.a fore alfo it meafurech F.H by thes ;common fentence ofthe feuenthpAndfor ‘as mich .as it alfo meafureth it felfe , therefore it alfo meafureth the whole AF by.the fixthcommog fentence of the feyenth , wherfore alfo it meafureth D G by the s. common fentence. But itis before protied thatiemeafureth GC, wherfore it meafuteth the whole CD , by the fixchcommon fentence ofthefeuenth :wherfore ALL it meafureth BF by the 5 common fentence of the fenenth:And it is alf proued thaticmeafureth F A,whertore alfoit meafureth:thewholenumber A B by the fixth common fentence of the feveath. New foras much as thenurber: KeA meafureth the numbers A-B andiC:D; therfore the numbers AB. and CD are numbers.compofed; wherforethey are not prime the oneto the other: which is contrary to the fuppofition.. ‘nthe And by this propofitionif there be two numbers, geuen.It is eafy to finde ont, whether they be prime the one to the other or no. For if by {uch continual {ubftraGion of the leffe from the greatér,yon come at the lengrh to ynitie. Then are thof¢numbers geuen prime the one to the other,Bat if there bea {tay before you come to Ynitie, then are the numbers geuen,numbers compofed the one to the other. q Ibe 1. Probleme. The 2. Propofition. ' LT wonumbers being ceuen not prime the one to,the other, to finde out their _- greate/t common meafure. nes WANT. V ppofe the two numbers ceuen not prime tht bueto the other to be AB and C BING? PY D. It & required to finde out the greateft common meafure of the {aid numbers A Band CD. Now the number C D gither mealureth== oF se 2g the number A B or not. If C D meafure AB thallo meas An. i BO [ureth it felfe Wherefore C D.is acommon mealureto thenumbers C.D CDand AB. And it « manifes alfo that itis the greate/t common i meafure : for thereisno number greater then CD that will meafure CD. : 2 ae: N -fererre- But if C D donot meafure AT, then if of the numbers AB. Aci, « Bivens. Bo and.C WD, theleffe be continually taken away from the greatel, C...F-i++D A there will before you come to unitie,be left 2 number which wil mealurethe number coing before(by the-1.of thefeuenth ) . For if there (hould not , then fhenld the numbers A Bap ~~ ¥* se SVE Lee mt ee ESE ks 2 a. RSS oe eee 7” & : CD. ef ag the one to the other which 1s. contrary to ae ton Lett he fayd n umb r lefi.by the continuall [ubftraction of the'lelfe number out of the greater beE C So that le the number CD meafuring AB, and [ubtrahed out of tt as often as you can lewe aleffe umber then it [elfe, namely AE. And let A E.mealuring C D, aud [ubtrahed out G i of Euklides Elemontes, Fol.189. 4s often as youcan leaue a leffethedit felfe vienel. GE And fuppofe that GF do [0 meds fare KE that there.remayne nothing Thenl fay that CF isa onion, mealure tothe numbers A Band CDFor foralwnichasC emeafyreth AE; and AE mealureth DK, therefore C F alfa meafurethD Buf ty the fifth. common fentence of the feuenth ).and it i:kewwife mecafureth it felfe, wherforedt alfa meafureth the whole GD by the fixth common fentence of 1 he feuenth.) but CD wneafureth BE | wherefore CF alfo meafureth BE (by the fifte common fentence of the fewenth)s And it meafureth Allok As wherefore it alfo meafureth the whole BA ( by the fixth common fentence of the feuenth):and it allo me it. xeth GS Deas we haue before proued : wherefore the number CF mealureth the numbers AB G Ruwhenfore the numbir C.E-is a commis meafure tothe numbers ABe» GD. I fay alfo.thatitis the greateft common meafure. Bor if CE be not the greateft commp: meafure to A Bitnd GD, let'theré be a number eveateethem > C E,which méafureth AB and GD: which tét be G. ANE » A wick forafmuch us & meafureth CD and CD meafirthB Es Gees therefore G allo meafureth BEY by the jift common féntence < C.avF.ic..D of the fetenth And ‘it meafureth the whole-A. Bwhere- 7 fore alfott meafureth the refidue,, namely, A E( a) the 4. common fentence of the fea wenth) But AE meafureth DE swherefore G alfo.meafurethT) FE ( by the forefayd 5 coms mon {entence of the feuenth) «And it metafureth the whole CD, Wherefore it alfo meafin reth the refidue FC :namely,the greater number the leffe:whichisimpofsible. No number therefore greater then C F hall yncafure thofe numbers A Band - wherefore. © F is the greate/t common meafuréto ABand CD:which was required to be done. Corro lary. Hereby it is manifet .that if a number meafure two numbers it fhall alfa _ meafure their greateft common meafure. Eorit it meafure the whole & the part taken away,itthall alwayes meafure therefiduealfo 5 Which refidueis.at the length, the greateft common meafureof the two numbers geuen. Gg The 2.Probleme. Th 2. Propofition. Uhre numbers being geu? not prime the one to the other:to finde out their Lreate/t common meafure. LahaV ppofethe three number's ceven not prime the one to the other wA.> > ay HA be ASB; C. Nomis required wato the fayd numbers B... : C D E {les }A,B,C to finde out the Rreatefl common Ineafure; Take the - greate/t common meafure of the two numbers A and B (by the 2 of the feuenth) which let beD : which number D either meafureth the num- ber C or not. 4 VIE iy Firfi let D meafureC . And it alfo meafureth the numbers A and B, wherforeD mealureth the numbers A,B, C. Wherefore D-4s.a-goamon mealure onto the numbers A,B,C. Then J fay alfo,that it is the createlt common meafure unto them. For if D be not the greateft- common meafire vnto the numbers A, B,C; let fome number greater then D mealure the'numbers A,B,C: And let the fame number be E.N ow foralmuch as E meat veth the numbers A,B,C, it meafureth allo the numbers A, B. Wherefore vt mneafureth alfo | saPositk Tt. the Demon rari? of the fecoze tafe. That C Fis 4 COMMON Hie an jure to the. 3 numbers AB. andCD. LhatC F ig the greateft COMMON mea= fureto AB axa CD, Two tafes in this Propofi~ 110%. The firft tafe m4 i 1 1g . 9 7 Wy 4 ’ : i = -. —> r 5 = a < os 2 - . ae Pree oe = — = ~d - ee SSS acai = ———. = —— eS ee : — ; = = ——-. = aS Oat ied oe - ; - ———- — . . = SE —— — = = —— a = oe ee a aa ‘Geto~ a a ss ; = = tu bes A === = = 3 — = “2 - SSS => <= So Sc sao wr = a a ie So ese _ —— eee — , —— ee SS se - —" a oe > —— = > . — ‘ —_ * = 3 “ = Se wwe 2 oe = - —~ -—~——> = — — a ne = ore — Se ™ 2-4. Saget + et Se a $ = : The feuenth Booke shegreatest common mesfure ofthe numbers A,B (by theCox Ase evceviee ve eens vollary of the fecond of the feuenth). But thegreatelt common’ B..... mr a meafure of the numbers A, B, is the number D (by Com C.....5. firuétion) . Wherefore the number Emeafureth the number -D...... D, namely the greater the leffe: whichis impofible . Wheres’ E.. foreino number greater the_D meafureth the nubers A;B,C. -F..- Wherefore D is the greateft common meafure to ithe numbers A,B;C. The fecona But now (uppoferthat D do not meafure C . Firft [ay that D & Care not prime num- Cafes vn, bers theone to theother...F or forafmuch asthe numbers A, B,C, are not prime the one to the other (by fuppofition) [c.ne one number will meafure them: but that number that mea- fureth the numbers A,B,C, fhall alfo meafure the numbers A;B,and hall likewife meafure the gréateft comon meafure of A B, namely 1) (by the Corollary of the fecond of the feueth). And the fayd number meafureth alf oC. Wherfore fome one number meafureth the num- bers D and C.Wherefore D and C arenot prime the one to the other. Now then let there be taken ( by the 2.0f the feuenth) the greate/t common meafure vnto she numbers Dand G3which let be the number ©. cAndforafmuch as E-meafureth D, and Dimeafureth the numbers AB; therefore E alfo eofivtth the numbers A, B( by the fixeconmmon fentice): and it meafureth alfo C. Wherfore ©: meafureth the nubers A,B,C. Wherefore Eis 4 common meafure unto the numbers AsB;C. I fay allo that it is the grea- testi. Por if E. be not the greateft common eae untothe numbers A,B,C, let there be ‘ome number zreater then E,which meafureth the nitbers AsB;C . And let the [ame num- ber be F. And fora{much as F meafureth the numbers A,B,C : it meafureth alfo the num- bers A,B. Wherefore alfo it mea{ureth the greatest common mea ifure of the numbers A,B (by the Corollary of the 2.0f the feuenth) But the greatest common meafure of the numbers A\B, is D. Wherefore E mea ifureth D . And it meafureth alfo the number C. Wherefore F wreafureth the numbers DC. Wherefore alfo ( by the fame. Corollary ) it meafureth the greatest common meafure of the numbers D; C . But the ereate(t common meafure of the eumbers;C, is: Wherfore E meafureth E, namely the creater number the lefe: which is impoffible: Wherefore no number greater then ©. {hall meafure the nubers A,B,C.Wher- fore isthe greatest common meafure tothe numbers A,B,C + which was required to be done. 5 | Sat | qo oralary. Wherefore it is manifeft that if a number meafure three numbers, it [hall alfo meafure their greate{t common meafure. And in like fort morenume bers being geué not prime'the one to the other may be found out theirgreds te/t common meafure and the Corollary will followe. > q The 2. Uheoreme. T he 4. Propofition. ovlesBnery leffenumber ts of euery greater number, either apart or partes. Pio Na Tos te AV; pofe there be two numbers A and BC of which let BC be the lee . T bent — naetr§ : yan BC is either apart or partes of A. For the numbers A and BC areét- tion. Sets ther prime theone to the other sor not . F irft let A-and BC be prime the one to the Thefieftcafe. other. And denide thenumber BC into thofe vnities which are in st. Now entry one A he of Euclites E:lewtentes. Foligo, she unitiesmbichare in BC is fome centainepartof A: WhereforeBC wrépartes of A220 nea ow fi uppofethat the numbers -A.and B.C benét pritme the one to. the oth hi | Bc eather mealureth A. or nat mecalareth it . UBC teafure - aa rs Thee AA, then is BC a part of A Butaf ndhy take ('by the. 2.0f the «co fexentir) the greateft common megfure tf A-and B.C) and Jet Poke the fame be D. And let BC hedewided into as many partes PS : asit hath eguall unto-D, thats, into.BE, E Fy and FE C Bok. . And fora [weugh as D meafureth A, therefore D isa part of 3 A. But Dis equall unto euery oneof thefe partes BEE Fj. 3... | and F C.Wherfore alfa cucry one of thefe partes BEs EF, and EC, isa art of A Wher. Sore the number B G FAY partes of. A: Wherefor cleuery lofee lasieaelices.is sebch of . Where either apart or partes: which was required to ae proned). "ey Sr cater Abang q Dhe 3. Theoremes 0. The 5. Propofition, Ifa number be a part a ophaok ec a a ae nun yf iets f 3 iber, and an other niaber the Jelfe fame part hi ” "i ver number, then both the numbers added together fhall be the elfe fame part of both thé numbers added fhdether >phich -) i er which one number of one number. | att. ber Was the number Ato ber part of the number B Cand let | ly,D,be the felfe fame part of an other vumber, pithatly, PE pes yp hat the numbers A and D added together, are the Selfe fame sea of the Z bers B Cand EF ‘added together that. Ais of BC: For Pratooucs it : mber Asisof the nimber:B:C,the felfe fame part Gk is the. number WD of the number BF; therfore how yeany num. - f i 4 ai i 7 ft : 7’? ‘a ¥ : i ' me rs er ; i i le nt. nH’ ' hy ‘ oly : ; tf iy . Hi i it Pile ! : } } i . “ : 7 ’ } 7 i} i ee aad ee es “ So i: ssp 2S os > ee - o> -= a= ass Conftruttion, Demonitra- LION This propop- tion andthe next following. jn difciret. quae be of GC : And for that what part A Ew ofC F;the WG, 0s QU OES. FAD fame part EB of CG, therefurewharpat A Ew | pee sitie anfive- reth to uy ifth propofitso ia fifth boke in conts- nuall quatitys Conftruttione Demoustra- $30%. The fenenth Booke> LENG v-ppole thatthe number A-B be pares of the number C, and let an other number, AN <5 y namely, D E be the felfe [ame partes of an other number namely,of F. Then I fay, NONG thot the numbers ABand D-Eaddid togetbersare of the numbers C and F added together the { elfe-[ame partes that A-B of C.F or foralmuch as what partes ABsfC, the felfe fame'paries is DEOPES CA. GLB therefore how many partes of Cthereure in AB; fomany c partes of F are therein D E. Devide A Bintcthe partes of C3 that is, into A Gand GB, and likewife TD E into the partes D....H....E of F that is, into D H and HE. Now then the multitude of p thefeAG and GB is equallto the multitude of thefe’ DH BATES x and Ec And forafmuch as what part AG is of C; the'felfe fame partis DF of F>. therefore what part A Gis of C thefelfe fame part iS&G and D added together of C and ¥ added together .. And by the fame reajon alfo what part G Bis of C, the felfe fame partis GB and HE added together of C ind F added together . Wherefore what partes A Bis of C, the felfe fame partes are A Band D Evidded together of C ard F added toge- _ ther : which was required to be demonftrated. ee 2 @ een q The s. E-heoreme: The 7.Propofition. Ifa number be the felfe fame partof a number ,that a part taken away ts of a part taken away: then fhall.the efidue be the felfefame part of the refte due, that the whole was of the wlole. , 5 V ppofethat the number AB be of the number © D the felfe fame part, that the aaa part taken away AE 40 the part tiken away CF. Then I fay; that the refidue is 2 EB us of therefidue PD the felf fame part that the whole A B 1s of thewhole CD. What partAE of CF, the felfe fame pari let EB | | of C F, the fame part ( by thes. of the feuent’s)is ABys of F G . But what part AE woof CF; the fame part by fuppofition) is ABof C D.Wher- forewhat part AB ws of F G,the elf [ame part ws A Buf C D.Wherefore AB one ee the {afefame part of both.thefe numbers’G Frard © Di Wherefore G F ts equall unto CD (by the fecond common [entence of the fenenth) .T. ake away F which is common to them both. Wherefore the refidueG Cw equall yato therefidtie D. And forafmuch as what part AE a of CF; the fame part rs E. BofGC: but G © wequallunto FD: therefore what part A Ets of FC, the felfe fame parts EB of FD... But what part AE wof CF, the famepartis ABofCD. Wherefore whit part E. Bis of F D, the fame part AB of CD. Wherefore therefidue EB wof the rifidue ¥ D: the felfe fame part that the whole ABs of the whole CD: which was requirid to be demonftrated . qd he 6. I beoreme. T he a. Propofition: If anumber be of anumber the felfe fame partes, that a part taken away sofa parttaken away,thevefidue alfo [hall be of the refidue the felfe fame . partes that the-whole is of the wholes } Suppofe @ of Euchdes Elemente. Fol.101. +P Peele itabiioe numbet ABdbe of trenumber.C D thefedfe fame partes. that the fs Bf, art taken an "ay AE tof the part tiken. CB oT hen I fapatinges rer efids CE Bis of: ee rhe bevel aue¥ PD the felfe fame parts that thewhole A Bis Sof tf the whole CI)..Knto .B put Al te ‘pitber GHA herefore rebar pc tes GH isofG D sthe felfef« (am, ¢ pal tes SAE of CE. Denide GH tte thepartesof G LD, that ts, GK,andoK Hy, and lekewife A E into the partes of C F , that is; inta.Ala.avd ma Nom thep them valtituae of thefe GiKundKH is aged vee the multitude of thee AA, a Je dae forafmouch a tehat part GK is fC D sthe felfe fame j pariis AL of GP: but CD és creater then CF. Wherefore G Kas Gr ter then AN. Rita to A-Lian equall 7 nuinberM G. Wher refane ewhat part GK ise. GD; tbe aime Be ae ees Bae partis GM0f GP. Wherefore ibe aiaes M Kk. 25 (by the ee 70h Pehe fenenitr) of the refidue ED, the (lfefame part that Con/lur tion, Demon sira- £307» the whole G Kis of th 9 whole GD. ah Fhe Ste cat ps ae ll. Kiera Nii what part Kk I 75 of CD, the felfe fame part is ELofC F: but C D is gre ater th ce CH. Wherefore HK zs greater thew EL. Putvnto E Laze- qual number KN . Wherefore what part XH is of C D, the felfe fame partis KN of Lo: WE 1 Mec thexelicug alfa. INH és ( & the z, ‘of the feuenth) ofthe refidue ED, the ep fi [ane Ke Arh that the whole } KH és of thewhole D.C. Wherefore both thefe MK and N Haiddedti petiber Up e( by (29. bbe >. of the [euesth) of DF the felf ‘Tapre partes that the whole HG is of thew hole Cc Dp But both thefe U KandN H ‘added toaether, are equall unto EB. 4ed H Gis equal uatoB A .Wheiefore the refidue EBis of the refidue F D the self e i abae oe that the whole X Bis of bhewwbrole OC D: which Was nah tobe PPE ql An other demonieatide after Flufeates. Suippofé hate the ves 3 AB be. of the numb:z.C D.the Felfe fame partes chat the part taken away A. Eis of the part taken aw ay CE. -Lhen I fay,tha:the refidue EB is ofthe re Tidue FD thé felfe fame partes that the wholeA B is of tht whol'€ Dike EB be of C I thé felf fame partes that AB isof CD, orA E of C E,. Now forafmuch as BBryis of Cl tiefelfefame partés than Ackis fC F; therefore both oe A EandEB aided toge ther.areofboth thet:C.RandCladded \. 7>,, ogether (that is, the whole AB. ts of Fthe whole FI.) thefelfefame -° . -B = § Oath’ + partes that: A fis of C FY by thefixtof this booke) S But what partes D zs by | oy A Ejs of C F, the felfe {ame partesus themmber A B ofthie number.) Fi K&S MY Leary C Diby frppofition) Wherefore, what partes the numberAB is of the agberk L,the felfe fame partes is the fame number, AB-of the number CD. Wlerefore the numbers Eland CD are equall. Take a- Way chenaniBePE ws whic is comimon to’ thenr both ° Wherefore the nimbéts Fembyning C band HD-are eqnall sW jidrefote what patresithe muri bet EB is of theniumbemO\h the felfe fame partesis 1e-Jame num her BB. ofthenumber,F Dy But, whit partesEB is of CL, the felfe fame.partes (by cone rpction) is AB OF ED . Wherefore what partes she refidue EB is of the refidue A D, the felfe fame a tes wee Whele AB of the whole € D:: yachyas reguiet to be proved salu hon ad Ags meni St Lacks wT ig Fhe! fi “Fheviime. Fhe 9. Prapopea aba: 1 a Ifa numberke part of a nnmber, and: ifamether number be the Self fame vaxt of an other uuber : then alternately swbat part or partes the firft is of "The i th Lee the ue [amep 1g t or cee wis the fecond be oft tbe es th. NOG & err, Te QS See yumbers An otherden monfiration after Flufiae~ teh. | Con Fraubtion, SE - e ey — Sare tgee = oh E ~ = —S—=— ——Sa $a i - = ~~: —- = = ra =? — — ~ = Lhe ee ae - eer ri - y ? 7 = = = — - - - —. ae = ~ < ss * — =a -t o_O - peta Es lc ———_ ——— a = = 4 — a = + - Le = ————————— ——- aes ~ 2 — ee 2 -= Tam —% = < - = - —- =. the ~ > - yy = ae = - = Cs 2 =~ S - = es — — ~ eter ae = = ~— —- oo - > —- — a — <-—--- >: _ > oo - = = - 3 —— 2 ro _ - ~ = = = x o~ ~~ = = > : | = ——-—- ~ =~ 5 = - - — =e — —s = = = = ae = = — = 4 a —— a = ‘ —— > — : — - - = ~- — - — - —- -- = : z - — = <3 ———~ = = _ 5 - ~ _ — = —— : a —-- nae SRE 5 ate Fp eal i LOPE TELE AE I a ~~ - ~ ee == —— = = = == = $$ ———— as S————— = =———.-— = —— == ts = - — - — ~~ —— — = - = — - - -- -_—- — —— — - — oo — — ——--— —— J - ~~ > = -— - —— —-— —- — ~ oe = - -—_- —_ ~ ~ —_——— _ — —_—— = ~ = - ~ — —— —_ = J ae « * ~ <— = a Sr a SSR oe = See 7 ~—— y= = = - —_e er == = o- ~ ——~ meets 2 i SP cx = - — _ eter ee Ss 1 = =——=es -+- Se = - — c2Es* — > = =—~.- ——<——— — — _--__ no " - —* ~ Demoutra- bins Conkirntion. The fenenthBooke numcersthereare in BC equall vnto A, fo many are therein Cmiay: E Fequallunto D . Denide BC into the numbers equall vnto A, that is, into BG & GC : and likewife E F into the num- bers equall unto D,that is, into EH and H F. Now then the Dv woltitude of thefe B G and G C, is equall unto the multitude of thefe E H ¢& H F . And fora{much as the numbers BG and GC are equall the one to the other, the numbers alfo E H and H F areequall the one tothe other + and the multitude of thefe B G & GC is equall unto the multitude of thefe E H and H F. Wherefore what part or partes BG is of E H,the {elf fame part or partesis G C of H F. Wherefore what part or partes BG is of E H, the felfe fame part or partes (by the fift Cr fixt of the feuenth) are B Gand G C added together, of E H and H-F added together. But BG ts equall unto A, and E Hvato D. Wherefore what part or partes A is of D, the felfe fame part or partes is BC of E F : which was required to be demonftratea. q L be 8. T-heoreme. The 10. Propofition. If a number be partes of anumber, and an other niber the felf fame partes . of an other number, then alternately what partes or part the firftis of the »\thirdsthe felfe fame partes or part 1s the fecond of the fourth. i - SS ppalethat the number A Bbeof the number G the felfe (ame partes, that ar (Gy) repent’ D Eas of an other _ F,andlet A yee A D E. Thex LW (Si fay,that alternately alfowhat part or partes AB is of DE, the felfe fame SES partes or partis C of F. Forafwach as what partes A Bis of C,the felfe fame partesis DE of F : therefore how many partes of C there are | in AB,fomany-partes of F alfo are thereinD E.Deuide AB GC itias . | A sntothe partes of C, that is, into A Gand GB. And likewife DE into the partes of F, thatis, DH and WE. Nowthen~ the multitude of thee A G and GB, as equall unto the mullti- Dis ies dit & india once tude of thefe’D H and HE. And foralmuch. as: mhatpart Demonjira- AGBYC? the felfe frome part is D Hof E, therefore alte’ bars sane an Beret tion. nately allo (bythe former ) what part or partes A. Gis of DH, the [elfe fame part or partes is Qof BuiAnd by the fame reafon alfo what part or partes G Bis of HE, the fame part or partesis CofE Wherefore what part or partes A G isof DH, the felfe fame part or partesis AB of D E (by.the 6,0f thefenenth) . But what part or partes AG. isof DH, the felfe [ame part or partes is it proued that C is of ¥ . Wherefore what partes or partAB is of DE, the felfefame partesor partis C of F .whichwas required to be proued. ll bean, Propofitionysn OY q Ehe..9. F-heoreme. ene if, the hole be to the whole, asa part taken away ts to a part taken away: then [hall the refidue be ynto the refidue, as the wholes tothe whole. This propoh- pers, yppofe thatthe whole number AB be umtorhewhoke number CD; asthe part take sapioks iferet BN away.A_E sis to the parttaké away C F.THEL [ay that therefidue E B ss ta the refte sh ore < S22! due F D,as the whole A Bis to the whole oe hp as,A BistoC D, as ninth propofi- A Eis toC F: therfore what part or partes AB is of CD;t eye a sid.af the fifth [ame part or partes is. E of CF. Wherfore allo the refidue EB OC’... FSD boxe six conts- js of the refidue F D (by the 8-of the feuenth) the felfe fameparte AE BY RETA ne or partes nus! Quatitie, — of EucltdesElementess. Fol.192. on partes that ABis of C D. Wherefore alfo (by the 21. definiteon of this booke)as E B is to FD, fo is ABtoC D s which was required to be proned. q The ve Eheoreme: The ia Propofition. If there be a multitude of numbers how many foeuer. proportionall : as one of the antecedentes is to one of the confequentes, fo are all the antecedentes to.all the confequentes. | } ; V ppofe that there bea multitude of nubers how many foewer proportional, name. Ey, A,B,C, D, fo that as AistoBfolet C be to D: Then I fay that as one of the = Alantecedentes,namely, Ais to.one of the con{equentes namely,to By oras Cisto Ds f o are all the antecedentes:namely,A and C to all the confequentes, namely,to B and D:. F or forafnich as(by fuppofition) as Azs to B, «A. fois Cto D, therfore what parte'or partes Ais of B, the felfe fame BS. part or partes zs C of D (by the 21. definition of this booke)where- °C ..... ; fore alternately what part or partes A ts of C thefelfe [ame parte or’ * Do). ... 6. artesis B of D (by the ninth and tenth of the feuenth) wherefore both thefe numbers added together’, A and'C,areé of both thefe numbers Band D added to- gether, the felfe fame part or partes that A is of B (by the s.and 6.of the feuenth) - wherfore (by the 23 definition of the feuenth) as one of the antecedents namely,A 18 toone of the con- fequentes,namel),to B,fo are all the antecedentes' A and C'to all the comfequentes BG D. Which was required to be proued. q The x1: T heoreme... »... A be.13. Propofition. Demon fi ré- tion. This tn diferee quatity anj- wereth to the twelfle prope’ [stion of thz Sifth in cottie nual Guat Demunft ra~ $20Ne If there be foure unmbers proportional: then alternately alfo they fhall be proportional, : (ehlood ai Ly Sy : V ppoe that there be foure numbers proportional, AsB,G,D..Jo that 4s Ais to, . I folet ChetoD. Then I fay that alternately alfo they fhalbe proportional, that is. We as AistoC, fois BtoD . For forafmiphas:(by fuppofition as AistoB; fois C Here is to be noted, that although in the forefayd example and ‘demonftration the number A be fuppofed to be leffe then the number B, and fo the number C isleffe then the number D : yet will the fame ferne alfo though Abe wigs to be-grdater then’ B: ; Wherby allo C ‘fhall be greater then D, as in tig example here put « Forfortiit(by fuppofitiony as°A t5't6 B, fos be greatér théa Byand C'¢greaterthen D> eee bv A be’. definition of this *” Booke) how multiplex Ais to B, (6 tiitiplexis C to D; atid therefore what pare’ Aw... 5s or partes B: is of A.,the felfe fame part or partes is D of é. Wherefore alternately °° “Bo.. what part or partes B is of D, the felfe fame part or partesis AofC,andtherefore C.... aor by the fame definition, BistolD,asAistoC. And fo muftpou vaderttand'of the... D 7 former Propofition next going before. , - % ? Pie 2s bis in defor crete quantity anfwereth to the fixtenel propofition of the fifth bouke sn continual Quantities Notes dfition}as°A i$'t6 B, fois C to Djand’A is fuppofed to | n oo pee : 4 ry c& “4 \ an 0 t . — ‘ft ~ ; % ” P. . tree ¥} x e 9:5 : £ y as c¥3 2 LT er “ae re, o> Sc! a Py wer 1y ti OT 2! : T° ; 7 s a oS S, =" é of Y. : lt \ } 4 } ’ HT | 6 1k f h : i | Haly f Han | if a » +) | i i) ' } } { | : i ‘ | ay i } , » Fae 16 {eon if i | ‘i } ‘ { a \ ¥ ‘7 4 ih i | a el ' ‘ ai hl) 7 y 4 ; a ih i ry U : 5 AE ih ft } 4 ‘ ; yy 4 ay { % | Mf ¥ i" | } , id ; | f ) i A . M ae) cel — _ i — : - 5 ya 8 = = < = = eg —_— ie mc Reet aon he eae yes ss il re i lees —s ee Sats or Se ee " aie, ay aalhs Codeine = = y T—— + : = .~ : : eh ie ol ee: se = 3ST bs x= a “ ge — a 0 = “=a = = inane = —— ew an — _~ < Se 43 ; bier ° trevor) aa as tos A one ee 4 ° > —s = x Z a ~ — vrs =, ' ages se te =* Same 2 Be oe SS ee a — = See a5 : ~e —— - a = ee “3 a ~ = See This in dif- erete quantity anfiwereth to De twety one Pe he ee propafit t708 of | the jifth boake x i taal Guaucities v Demonftiae Tome a : 53 a hee Certaine ad- ditions of Campan e, Two cafes int ahis proposition _ 4, Tbe fir cafe, en dfn SE : aici: nED GITV od A393 aH re a. 4 +an% st cae Wis e444 \4 Yo O84 | 9% mas | LAA ale aged ful ol. bp ee eSSANG wah, | Frobovtsows- hisy compofide Euerfe pro pawtsaicdtey. Ti exe hee we e eT he L2. T heorehie. | rhe 14. Propofiti tion. wart rk. e If there be a multitude of numbers i many foener, and alt ihe mums bers equall yntothemm multitude , whichbeing compared tivo and two -—avein one and the e fame pr re: they hg a ufo of. cee bei in one and Sti0 ehe fame a sabia 4 40) EX Bt 5 ppofe shat sev bea iibalbiede of len ae many Sater: aeihla ad A,B,C, Sef : ae and let the other numbers equall unto them in multitude bé D,E,F : eaoich be. RS (5 Ing ORE pared two.and.two, let bein ane and. thefane proportion +. thatis,asA Se to Bf fa ea beto Ex : ands Bis t0C3 ole E be tok . bee feet ae ties.4s A 18-100: of o4 1SD.t0F .. - or. forafmuch as by fuppofit: ti. ee on as Ais to B , fois D to.E.:. therefire alternately, alfa. (by the....Andereanesens 13.0f the fenenth) as A isto D fois Btok. Aguingfonthatas... Bs. «x } BistoC foi E to Fz therfore alternately alfo(by the Self fame), C+ as Bisto E, [ois Go E...Butas, Bisto B,foss A toD.VV-her- i fore (by. the esfemenh commonfentence.of the fenenth) as Aisto.- Devs cua+ : “\ D,fois C to F. inte ec (by the.r3:0f the enenth) fee tee Be ds beibt0€ le D..to.F Ieee d Re SINE Ee to bedemon-.\ Fi...) p jews 3} aat . y ? FAI ER) GS a. tucks: 5 hte ér ‘itrapslivon ¢ emepane sdernqolcaneth i in, anambers thefe: fonre kindes of ~ TOsErconaae namely proportion conuerle,compofed,denided,ahd cuerfe : which were in continual quantitie,demontftrated in the 4.17.18. and 19. propofitions of the fift booke. And firfthe ab ig a oa conuerfe ignite in this maner. Suppofe tha ie AE Oy yt be hoe humber B, as AaaS C is to oe aan number D . Then I fay, that conuerfedly B is to A,as DistoC.Forif Abeleffe © Assescoce iB Califo {hall beleiic then D, and what. part or partes Ais of B,the felfe » Boweve. “aa pattor partesis€ ofD. Whietefore Bis eqnemultiplex to A, as Dis toC. °C Wherefore b y the 21.definition of this booke) asBisto A, foisDtoC. \ynieay Dx: ot Butif A be aya then B, C alfo is greaterthen D : and ‘what part or ii B isofA; the elf fame art or partes is D, of C. Ree (by the fame rT fe as Bi is 60 As {i D% © whieh Wal required orbe proved -Prgeemate ss mas ee | 23 — eat! tao: ee vols is" unt rhe stead ene RIA. Th BYS0 74 a TP leat denied is vs deminfted. us tna | ‘ e Suppofe that the rere, B sie cathe eather By as pees is to oui number D.. ve { fay,that de uided alfo,as Aci 4s to B,fo i is C to D.. For for {sto wae foisCDtéD? thétefore allernarely (Byiche 14 t4:of this ay pte oe aes . A. Wwe ¢ CD,fo isB to D. Whetefore (bythe pista booke) agi 4G) odad De ada ed) fois AtoC. Wheaeforeae iste? (i \ F0\Cx and Souths as rere 0° 1 os rr. ee ee ee a ee Se ae ee no Se Ot ee ees Pe ee VS eA wh Ls t » «i? Pe Ga ot C, fo is Bto D, therefore eee isto B, fois C to D, . ; ° t . am : . ~-e Ams an . S3 M9 AN 33 w 4 Dye > a he@ i yh ty nen tte oe. 4 ¢ & “ax Lroportianalitie compofed ed, is Bd demonfirated, cur 3 on at ai 3170813 if 161] paar ne dot eT = 719% Seis Ili W 5 Bit 5 {Fi Fo or: Ae ¢ a3) 2 Bes) Dgqut i. » aA be'vanto.B.as C.iste tos then hall AB beta base ¢ Disab For al-, , igus og) sms) otéinately-A iste C,as Bis to.D.; W herefor by.che:13 of beokelist B,»., Basie sn. ary Sits sui namely;all the antecedentesare, t0.C Dyname Osa onfequentes, fois, |. rsa *8sco9ngd B to D,namely,one.of la mt Wher» esleilum yott(sdoad fore alternat ey as AB istoB, fois Dito Dy; to Czi zatisg 10 nag 50 Uh: US otlt A toe Reta ‘ stoisieds bne . to A-elesieg torisg¢omsl 5 Als}: Cio 2t & esiige 10 se jusdw a pe opo tie. hee eG QosetAes. ig t Cotrian: b oral gels yd se Gropersibia ies is rlas proved vocal Ob te@erds y seeniet oi tyro that AB WER Ete Dis toD : then fhall AB be to ‘A; as SCD i isto C, nal rr aa Fie oe ———-— = ee: eens ee eee of Euchides Flemetites. Fol.i92, A Bisto CD, a8 isto. Wherefore(By ther; [OFthis book) 'ABIOD) “A... Bey as Aus to Cy Wherefore altetiately A‘Bus'to A;as OW isto: + which was RATS 3 FEN required to be proued, 5 | | gq A proportion here added by Campane. Tf the pro portion of the ‘fir it fo ‘the fecond be as the proportion of the third to the purth:and if the pro- portion of thefift tothe econd be: as the proportion of the fixt to the fiurth then the proportion of the firft and the fifth taken together, fhall be to the fecond yas the proportion of the third and the fit taken togetiver to the fourth. | As if A be vnto B, as C is to D : and ifalfo E be to B,as FistoD. Then fhall A & E taken together, 7 bis propoft be vnto B,as C and F taken together,are ynto D.. For by.conuerfe proportionalitie, Bis toEyasDisto #6% 2% dijeret ‘Fi Wherefore by proportion of equalitie,as quantity an Ais to E,fo is C to F.. Wherefore (by com. PTH SONNE aR a es ee ee: oe Jwereih tothe pofition)as A and’E-areto E, fo are Cand F 2 4.propofitie to F . But (by fuppofition) as Eis to B,fo is EF toD. Wherefore againe by proportion of equalitie, as A of the fifth and E aretoB,foare Cand F to D:; which was req ured to be proued.. i ye 4 as 4 And after the fame maner may you proue the conuerfe of this Propofition.1fB be to A,asD isto C: : .. | . : and ifalfo B bevito E,as Dis to F § Then fhall B be to A E,as Dis to C EF. For by conuerfe proportio- a 7" antitye nalitie;Ais'to B, as Cisto D.. Wherefore of equalitie;A is to E;as Cis to F. Wherefore by compofi- The con Kerfe tion Aand Eare.to.E,asCandF areto FE. Wherefore conuerfedly, Eisto Aand E,asRistoCandk. ofthe /ame But by fuppofition, B is to E, as Dis to F. Wherefore agayne by Proportion of equalitic, Bis to A and pre pofitron. E,as Disto Cand F : which was required to be proued.: Demonftra- | A Corollary. slow: By this alfoit is manifeftthat ifthe proportion of numbers how many foeuer vnto the firlt,be as the proportion of as many other numbers vnto the {6cond,then fhallche 4 Corollary proportion.of the numbers compofed ofallthe numbers that were antecedentes to the folowing firft,be to the-firft,as the number. compofed of all the numbers that were antecedentes thofe propoft- to the fecond 1s to the fecond, Andalfo conuerfedly if the proportion of the firft to nit= tions adaed by bers how many foeuer,be as mepreportion of the fecond to.as many other numbers; Campaues then fhall the proportion of the firft ro the number compoled of all the numbers that wére confequentes to it [elfe, be as the proportion of the fecond to the number compo. fed of all the numbers that were confequentés toit felfe. q Che 13: Theoreme. Lhe 15. Propofition. If phitie meafure any number and an other number do [0 many times meas fare an other number : ynitie alfo [hail alternately) Jo many times meafure the third number, as the fecond doth the fourth, | ~ gh? Ppofe that unitie A do meafure the number BC: and let an other nitber D fo | ; va ‘many times meafure [ome other nuber namely,E F. T hen I fay, that alternate- ra ly ,wnitie A hall [0 many times meafure the number D,as the number BC doth ae Let méeafure the mimber EF. For [orafmuch as vnitie A doth fo wany times mea- fure BC, 4s D doth E F : therefore how many nities there arein BC, fomany numbers are there in E F equall unto D. Denide (I fay) BCin- A. a to the unities which are in it,that is,intoBG,GH,and HC.Anddeuide B. G.H.C likewife EF into the numbers equall unto D, that is, intoE K,K Land -D.. LF .Nowthen the multitude of thefe B G,G H, and HC, is equalunto E..K..LeckF the multitdde:of thefe EK,K L,L FE. And fora{much as thefe vnities Demoniira~ BG,GH,dnd H C,are equall the one to the cther,and thefe numbers EK ,K L,éy LF pare OMe alfo equall the one ta the othéx,and the multitude of the unities BGG Hyand H Care equall unto the multitude of the numbers E K,K Lge LF : therefore as unitie B Gis to the num- ber E K, fois vnitieG Htothé number K L.and allo vnitie HC tothe number L F Wher- fore C onfirultions _ = ———- ss = SS a — == a - = —s = ———— ee . =- k= 5 = = —-—--—- - > = ~ -~- 2 ee = ah: 0 a Se SEE —— = =90 2 = Rit 2 «s aa Se ey eee SS = ee — —_ = a —— SSS See = — = 4 « - S . a = <= = —— = ——— " 3 oe ones = f , - i -- - x = 7 > a =: -=—- — _—- ——- ee - _— ae - — = Se oe Se ee ~ = a seater OS me mye: Pere eERS ~~ ine, —s * - - =~ m — : mas =~ <== >. A z a ~~ —— ry) ‘*, , ‘ Y = eee STS Sass i SE ae ae Nae : ern nn yn ae Sa ey vinee > SE on = ae re ————— SE= 23ers — - = a, Set Serre a Soe . —— was aa . =z - — . : — 2S Rae a ee aS = = - AF y he feuenth Booke,* 8 ej Act fore (by the r2.0f the fenenth) as one, of the antecedentes. iso one of the. ly 00 i £2 confequentes, fo are all the antecedentes tonllthe confequentessiither fore B. G2 iC as unttie BG is tothe number E K, fois the number BC tothe number’ De EF. But unitie BG is equall unto unitie A,and the number EK tothe E..K..L..F number D .VV herefore (by.the 7 common fentence)as unitze A is'to the coe number D, {ois the number BC to the number E F.VV. herefarg vnitie A mealureth the nit- ber D, [0 many times, as B Cmeafureth E F (by the 21, depunion of this booke) : which was required to be proucd, ; | cn ae 4th q Lhe 14. T heoreme. The 16.Propofition. — If two numbers multiplying them felues the one into the other,produce any . anmbers : the numbers produced fhall be equall the one into.the other. | RSS V ppofe that there be two numbers .A and Bs and let ‘A multiplying B produce C, dS ; and let Be multiplying A produce D. Then 1 fay,that the number C is equall un. SOY} ta the number.D.T ake any vnitie,pamely,£. And fora{much asA-multiplying [SKI B produced Cxtherefore B mea{ureth C by the vnities which areinA. And vni- tie’ E mea{ureth the number A by thofe'inities which are in the number. . A VV herefore vnitie E [0 many times a A,asBmeafurecthc. E. VV herefore alternately (by the 15. of the fenenth) vnitie,E: meafureth.: Ans the number Bfo many times. as.A meafurth€.» Again or that B mule Bis és tiplying A produced Dy therefore Ameafureth D by the unities whith Cviceve vs are in B.-And vuitie E meafureth B by the wnittes which are in Be Divieiees VV herefore vnitie E fo many times dif ooeet the number B, as A mea- : [iret D | But vnitie E fo many times meafureth the number B, as A meafureth C. VV here fore A meafureth either of thefe numbers C and D alike. VVherefore ( by the 3.common Jentente of this booke) Cis equall unto D : which was required to be demonfirated, Lhess. T heoreme. I he.17.Propofition.. » _ Tf one number multiply two numbers and produce other numbers,the nume bers produced of them hall be in the feife fame proportion, that the nume -bers multiplied are. Ben es FI] Vppofe that the number A multiplieng two numbers Band C, do produce-the {CSS numbers D and E.Then I fay that as B is te C,fois D to E. Take unitie,name- % : M Ly, F. And fora [much as A multiplieng B produced D, therfore B meafureth D ‘Tes. || by thofe vanities that arein A. And unitie F meafureth A by thofe vanities whih are in.A. Wherfore vnitie F {6 many times meafureth the ae . number A,as B meafureth D. VV herfore as vnitie F is to the. F . number A, {01s the number B to the number D (by ther1. de- A ..s finttion of this booke) And by the fame reafon, asvnitie F isto Bsaee the number A, {0 isthenumber C tothe number E : wherefore _C «+++ alfo ¢by the 7.common fentence of this booke) as BistoD, fois D serreevrvege _ C to E. VV herfore alternately (by the 15. of the fenenth) as ME os ok oe tea cue ens 10°C, fois D to E.1f therfore one number multiply two numbers, aoe und produce other numbers:the numbers roduced of them, fall be in the felfe [ame propore tion, that the numbers multiplied are : obich was required to be proned. ae | Here Sgn - of Enchides Elemientes. Fol.194.. HereFiufates addeth this Gorollary. If tivo numbers hauing one and the fame proportion With two other numbers do multiply the one the other alternately,and produce any.numbers,the numbers produced of them {hall be equall the one Soren to the other. added by FlufSates. Suppofe that there be two numbers and 8,and alfo two other numbers C.and D, hauing the fame proportion that the numbers 4 and 4 haue y and Jet thenumbers 4 and B multiply the numbers ¢ & D alternately that is, let 4 multiplieng D produce F, andlet & multiplieng C PTs Oe EROS produce E.Then I fay that thenumbers )..4 4.4.4. A ee ee Br nin aorgt vn ents. s Oh ‘ produced namely,£ & Fare equall.Let... B.. Praii * FRR , Aand B multiply the one the other in OF sare. g a eRe fuch fort,that let 4 multiplieng & pro- duce G,and let 8. multiplieng 4 produce H. Now then the numbers G and H are equal by the 16.0f this booke. And forafmuch as 4 multiplieng the two numbers 4 and D, produced the numbers G and F, therfore G is to Fas B is to D by this propofition.So likewife 2 multiplieng the two numbers 4and C produced the two numbers H and £. Wherfore by the fame His to £ as 41s to C. Butalternarely (by the 13. ofthis booke)) 4 is to Cas Bis to D,butas 4 is to€ fois Hto £, and as Bis toD, fois GtoF. Wherfore by the feuenth common fentence,as A is to E,fo is G to F.Wherfore a'ternately(by the 13 . of this booke) H isto Gas E isto F. Butit is proued that G & Hare equall: wherfore £ and F( which haue the fame proportion that 4and B haue) are equall. If therefore there be two numbers, &c. Which was required to be proued. q The 16. T heoreme. The 18. Propofition. If two numbers multiply any number ex produce other numbers: the nume bers of them produced, [hall be in the fame proportion that the numbers multiplying are. % V ppofe that two numbers Aand B multiplieng the number C, doo produce the a numbers D and E.Then I fay that as AistoB,foisDtoE. For forafmuch a8 Derg onfire- A muttipliene C produced D therfore C multipli- Oey S=KSMeng A produceth alfo D (by the 16.0f this booke.) A «++. And by the fame reafon C multiplieng B produceth E. Now B..... then one number C multipliengtwo numbers A and By pro- Cee duceth the numbers D-and E.VV herfore by the 17. of the [e- Dasessoeesses uenthas Ais to B,fois D to E : which was required tobede- E v+.sasesereeeee monstrated. ThisPropofition,and the former touching two numbers, may be extended to num- This propoft- bers how many foeuer.So that ifone number multiply numbers how many foener,and tion and the produce anynumbers,the proportion of the numbers produced, andofthe numbers former may be multiplied, fhall be one and the {elfe fame, Likewife if numbers how many foeuer mul- extended to tiply one number,and produce any numbers, the proportion ofthe numbers produ- numbers how ced,and of the numbers multiplieng ‘hall be one and the felfe fame : which thing by many foexere this and the former propofition repeted as often asis needefull,is not hard to proue, — —~ = — A rene { ] J ’ j i Hass Al HY : \ wa hay { veRy : } ve : i ] : i np i ) ea ; { ; hh t ‘Wa i 4 n| 4 r - 1) tay Hh may , HY | q a, j ik a i } d} i I 4fy! a ne j } : . ‘ : # q The 17. T heoreme. The 19.Propofition. If there be foure numbers in proportion: the number produced of the firft and the fourth, is equall to that number “which is produced of the fecond and the third. And if the number which is produced of the frft and the fourth be equal to.that-which 1s produced of the fecond ¢ the third: thofe foure numbers fhall be in proportion. Suppofe ' i} ny : ie ¥ | i ; : ; vray 4 } ae 1 it] t + af +) % ih iia , ; } PaaS GI > iar iH i L fey4) ! peat . ie } ¢ t t ti ie ' i: <| ee “E4) ah My}: ae i \ Hy meh jE | $ 1. : | ' a ae : : \ } + tah Pale : Hite 7 : tj ‘ ' \ nt tae i} uy ! 4s] saoal i a ui abi it Me oe Wp | ! ey) ; i rh ] ; her EL eat id a ey abt | i Ha uf a bh et ee 4 hi ey } hi on " sf x + all ' { hee \ ea T he feuenth Booke (i | V ppofe that there be foure numbers in proportion A,B,C, ID,as A is to B,fo les Se wes tM Vg C be to D. And let A multiplieng D produce E : and let B multiplieng C pro- bers lemons “Ve Sb i duce F.T hen I fay that the number Eis equall omtothe number Fs Let A mul- firateth chats, “Qe! tiplieng C produce G.Now fora{much as'‘A multiplieng C,produceth G,¢y mut which the 16, tiplieng D it produceth E : it ‘followeth that the number A | of ——_ ‘Ynultipliene two numbers C and D, produceth G and Ev A wei. Do “VV her fore by the 17 of thé fenenth, as Cis to D, fo is G to’ B firateth : a 7 pike teaes E. But as C isto D, fois AtoB, wherforeas AistoB, fois C... D 2 bis propose = bie (tnes, snipe Confirutlion, G to E. Acaine,fora{miuch as-A-multiplieng C produced G; Demonfra- and B multipliene C produced F : therfore two numbers A #30% 6 and B,multiplieng one nuber C,do produce G Cp F. VV her- E veseseeveees fore by ther. of the fenenth, as Ais toB,foisGto F. But) F vescacssvens 4s Aisto B, [01s Gto E : wherfore as GistoE, [oisGtd F. G ccscccccccsvnceves KV herfore Ghath toeither of thefeEand F one cy the [ame proportion (But if one number haue to two numbers one and the [ame proportion, the faide tivo numbers [hall be equall). VV herfore E is equall unto F. | ory ; But now againe,fuppofe that E beequall unto F.Then I fay that as A isto B, fois C to asin D. For the fame order of construction remayning ftill,fora{much as A multiplieng Co D propofstion produced G and E therfore by the 17 .of the feuenth, as C is to D fois G to E, but Eis equall unto F (Butiftwonumbers be equall,one number {hallhaue unto them one and the fame ‘which 2s the conuerfeof proportion) wherfore as Gis to E [ois Gto F But as Gis to E, foisC to D. Wherefore as C tive firfte Wt0'Ds [0 is Gto F,but as G 13 to F fois Ato B by the 18 .of the fenenth, wherfore as Ais to =~ ewonfirs- Bp [01s Cto D : whith was required to be prowed, iolt. : An efeunpe Here Campane addeth,that itis needeles to demonftrate,that ifone number haue to je ded P ? two numbers.one and the fame proportion, the {aid twonumbers hall be equall: or Cs pe a . thatif they be equal,one number hath tothem one and the fame proportion.For({aith atone ‘he )ifG hae vntoE.and F one and the fame proportion,théeither,what part or partes G is of E,the fame part or parts is Galfo of For how multiplex Gis to E; fo multiplex ; is G to F (by the 21.definition)And therfore by the 2 and 3 common fentence,the faid numbers fhall be equall.,And fo conuerfedly,if the two numbers Eand F be equal, then fhall the numbers EandF be either the felfe fame parte or partes of the number G, or they thall beequemultiplices vntoit. And therfore by the fame definition the number 7d P y G fhall haue to the numbers E and F one and the fame proportion. q ihe 18. I heoreme. The 20. Propofition. x If there be.three numbers in proportion, the number produced of the exe tremes is equall to the fquare made of the middle number. And if that nite ber which is produced of the extremes ,be equall to the [quare madecof the middle aumber ,thofe three numbers [hall be in proportion. V ppofe there be three numbers in proportion, A,B,C, as A isto B,fo let B be ta spropofi- WARS hat the number produced of A and C is equall to th , This propofi- he 3 C. Then I fay that the number produced of A and C is equall to the {quare num Siene ste sours » SOS x ber which is made of B. Put unto B an equall number D. bevsdemon- KYSER Wherfore as Ais to B, [vis D to C. Wherfore that whichis A ....... S — . ve produced of A into C,is equal unto that which is produced of BimtoD. B ...60. ofthe fixeh * But that which is produced of B intoD is equal to that which is made Cneee demonBracceh of B(for B 2s equal unto D.) wherfore that which is produced of Ainto D...... iss lines. C is equal to that which is made of B. Dessoafiza- ~ But now fuppofe that that which is produced of AintoC, be equall 4 rsugéing dt ™ ‘ * ' ——- Pe? ¥ - “* . se ae ea ee of Euctides Elementés. Fol.195. to bat whsc his wade of B. Then I fay thatas Atsto B,foisBroc. For (the five oder oF const iH tion reviay wing). foruafe much as that whichis produced of wt jntoG,,is eguall ty that whicr is wiace of B, int tha t whichis made of B, is equal to that wlich is produced of Bin- to D.{ F or B and D are by { pppofition equal): therfore that which is produced of Ainto C as equallivsini mnich.isproauced of Binto D swherfore (by the fecoud part of the firmer propofition) 43.43 to Bsa isD to Crbut Dis equal to B. Vi ‘herefore 4s Ais ta B, je isthe e "y , | [are B to C : which was req wixed tobe proued : q Ihe 19. T heoveme: Lhe 21. Propofition. Some) 4s of thefe C G and G Dis equallwnte the multitude of thefe EH HF C..G.eD And forafmuch as CG-and GDarénumbers eqiall the one to the other, EY F and thefe numbers-E H and HF are alfo equall the onéto the othersand As okeoey. the multitude of thefe'C G and GD; is equall to the multitude of thefe\ °B..} 59 8, E Hand H F : therefore as CG tO Hoi GO D'to ARS Wherefore n ath (dy the taiof the fenenth) as one of the antecedéntes is 10° one of the tonfequentes, fo are ak theantecedontes to all the confequentes. Wherefore a3 OG i540 EH \fors OD to EF Wher fue and EH avein the {elfe fame proportion thakC-D and BE ate, beneallo lefe then OD and BF which 1s impofable F or C D and E Fave ‘[uppofed tobe the leaf? that hare Onvund the fame proportion with them . Wherefore CD 1s not partes of A wherefore it ts apart Wherefore EF is of Bethe felfe [ame part; that CD is of A. Wherefore C D fowany minemenfureth AasEF doth B which was reqnived to be demoxwftvited g The 20. I'heoreme. The 22. Propofttion. If there.bethree ntinbersyand other numbers equall pata them multitude, woich being compared two and two are in the felfe fame proportion ,and if Se \alfo the proportion of Mem De periuybatesthen of equdlitie they [hall be in one and the fame proportion. = FOL INOS ) i ppofe that there beshreé numbers A,B andC rand “AN yes. % let the other ni:mbexs equallonto themin. multitude BS, 29° ©) be.D,E,and£. Andlettwo and tivo Compared toge- O38 2 oneand the fame proportion» and lecthe proportion 1 IMT of them be perturbate fo sa asAisto B, foletE betoFyand Des yy ¢, as Bis toC, folet D béto E . Khen Ffay, that ofequalitioas A “Bs 8 \ ppp: ts 50 C, ois D to F... Bor, for that a0 isto B ois Bt0F OFS, AG Neds . therefore Bons od. The fecond part which ig tie conurrfe of the firfi. DémonSira- b30N. Demonflrt- tion leading ta an tn pofsibie tz t Ys This propof- ton in difcrek gnatitie anf= wereth to thé: 23. propoft=* \ t:6 of the fifth bokein couti- nial Juattties out A 7 ni i a - == vin a a ee — 3 le ay eat ~~ ——w 7 a —— > os 2 en ye o- e =. i | | MW i) |! } i bh { ' in Hl wi ie mae thi) fi I | I ii | i } Gi [ i] | : i i Bo lis : fi | | } ith ae. | ; A] | : Th y { ; ) ‘ : } ' Lhe Yr me ' ie ' Hi! i VW | ’ ‘i i) Ie i | 4), { ' t y HT). f , rt i | ‘ iH i ae? | - { : i) i, ay “' ; { a q t : oye } } Pat cay | ae A te att li} 14)" An \ Pale al! ial: i ‘ } a vit ae As 1. bi ii is HHH ; eh P i t Pay shy mi! ‘ey 1 : . We ih 5 mM, " ‘ q if ") ¥ id, t ah i" ait ai, i : \ VAR M4 e pL fi "i ' | 4 \ ca i} hae } he | "4 A At 5 tb} ' ha - tt ie eg ; mT b ‘et : 14 tie : 1 : i ; , on hy ; i if ; i} } : N } | : , i ‘ ) : if 5 } | if J |) ae mt ‘ " ‘ " 4 / G ay y DY } - i qi) H » 4 . mh ra ' y | | | ‘ ' : i! (f ‘ —— — 7 = 5 peers 7 The fenenth Booke therefore that whichis produced of A into Fis (by ther9.0f A the feutth equall to that whichis —— of BintoE.Againe B.... for that as Bis toC,fois D to E,therefore that which is produ- C ced of D into C,is equall to that which 1s produced of B into E. Andit ve ahgcow , that that which is produced of Ainto Fyse- D-..... qual to that which is produced of B into E Wherfore that whith E veccccss is produced of A into F,is equal to that whichis produced of DF into C. Wherefore(by the fecond part of the 1.9. of the fewenth) as AistoC, {ois D ta F : which was required to be proued. The fame may alfo be proved ifin either order be more then three. numbers: as it was proued in the 23.of the fift touching more magnitudes then three. q The 21. 1 heoreme. he. 23. Propofition. Numbers prime the one to the other: arey leaft.of any numbers. that haue one and the fame proportion with them. = AV ppofe that ‘A and B be numbers prime the one to the other.T hen 1 fay that A and apes iB are the leaft of any numbers that hane one and the [ame proportion with them. ftions follow- poheeen For if A.and B be not the leaft of any numbers that haue one and the {ame propor- sng,declare the tion with thern, then are there [ome numbers lefethen A and | pafjions and B, being in the felfefame proportion that Aand B are. Let the nF Dees 68 peat g famebe C and D.Nowforafmuch as the leaft numbersinany.__ B.... ee proportion meafure any other numbers hauing the fame pro- Bs; portion equally. the. greater the greater , and the leffe the leffe’ .. Cvens (by the 21. of the fementh ) thatis,the antecedent the antece-- ) Das times meafureth Aas D.meafureth B How many times C meafureth B,fomany vnitics Demonftra-~ let-there bein E.WherefoxeD. meafureth B by thofe vanities which arein E..And foralmuch tion leading to as © yntafureth A. by thofe unities which are ink, therefore E. alo meafureth A by thofe an impofibi- ayypitieswhich arein G. And by.the {ame reafon E, meafureth B , Ly thofe-vnities which dst ye arein DD) Wherefore E. meafureth Aand B being prime numbers the one tothe other-which ( by the 13. definition of the fenenth,) isimpoffible. Wherefore there are ng other numbers leffe then A and B, which are in the felfe fame proportion that A.and Bare. Wherefore A and Bare the leaf? numbers that haue one and the fame proportion with them : which was required to be demonflrated. dent,and the confequent the confequent:: therefore C fo many q Ll he22. Theoreme. The 24. Propofition. I ke leaft numbers that hane.one and the fame proportion with them:are prime the one to the other. | am Vppofe that Aand B be rhe leaft numbers that hane oneand the fame proportion ar ieseche ch- NG . ther be prime to anyone of them, then alfothofe numbers ceuen at the bee ginning, are prime the one to the other. PO” ppole that thefe two numbers AB and BC being prime numbers be added to- VS RAM cece : Then I /ay,thad both thet ct , v2 6 gether .T 0 I fayst vat both thefe added together, namely, the number ABC, ) x 15 prime to cither of thefe AB, andBC.ForifC dAand AR be not prime the ICE SN one to the other, fome number then [ball meafure them. Let [ome number mes. fure them, and let the fame be D. N ow then foralmuch as D 3 : meafureth the wholeC'A and the part taken away ABit mea- A fureth alfo the refidue CB ¢ by the g.common fentence). And D-..., it meafureth BA. Wherfore D wmealureth thele numbers AB ana BC,being prime the one to the other - which is impofirble (by the 13 definition of the fe- wenth:) Wherefore no number mealureth thee numbersC A and AB. Wherefore C A and AB are prime the one to the other “And by t [ame reafon alfo may it be proued, that GA and BC are prime the one to the other. Wherefore the number A Cs to esther of thefe num- bers AB and BC. prime. | | But now {uppofe that the numbers C Aand AB be prime the one tothe other - Then I [ay,that the numbers A Band BC are allo prime the one to theother . For if AB BC De not prime the one to the other : [ome one number meafureth thee numbers AB and BC: | ¥V.iy. Let DemonBra- tia”; Demonftraa tion of the first part leas ding to an ab- Krditie. ar iad - —— ~ sy tz een = = = - 2 Star a a - - ¥ — =. -- — —— ——— Yama - = = ee - ~ =. a s a ma - = = — “4 - ae hee — - —— = =2 - — - “= - = = te aS a = a SS =—— = é = a = = = = = ————— SS = Se = = xe : = = ~ = wa = = SS - a = : = s ; —— = = = ol , ” oe = = _ i ae =... > —_— ~~ SKiisnwin s - = ~~ —— < *€ es > a — eS -_ ~ - < Se =~ : E : = = ; = - ~ : ; a ee = “ . Sas + 2 : al s eee See Ee -_— es men oe ek ig, - ‘ty a | 3 Demonftra- tion of the fecoud.part sobich as the conuct/e of the fir leacing alfo to an ab- Jurditie. Demonftra- tion leading to . #4 abfurditie. Dewsnflys~ bist. A Covollsry added by Campane, The foubsith Booke Let [ome one number meafare them,and let shefamebeD And pean as.D meafureth either of thefe numbers AB und BC; i shall es mmeafure the whole C A (by the 6. common lemtence) And it allo meafineth A By Wherefore D mealureth thee numbersC A and AB being prime the one to the other “which iximpofible (by the 13. definition of the fenenth). Wherefore no nuniber witnl uPeth thefe mambers A B nnd BC Wherefore A Band BC are prime the one to the other; which was required vo be proved, : ? Tle 31. Propofition. = Tes $ * *" X 7 . 4 % q The 29. T heoreme. es 1. \ one to the vther...F or if Aand.B.be not prime theone.to theathetr; then {ome {| pumber mealureth them.. Letthere bea number that meafureth them, and let Bi es q The 30. T heoreme. The 32. Propofition. If two numbers miiltiplyingthe one the other. produce any number, and if alfo fome prime number meafure that which is produced of them: then so Shall it alfo'meafure one of thofe numbers hich -were put atthe beginning. aA V ppofethat two numbers Anand B multiplying the one to the other da produce the a <8) niiber Cand let fome prime number namely, D mealure C.Lheé I fay that D mea- j APL Gareth one of the[e numbers either A or B.Suppofe that it mealure not A:now D 4s an a prime nunsber.. Wherefore A and Dare prime the one to the- vrber(by the propajition next going before). And how often D.- Es... mcafwreth Cfo many vnities let there bein E.And forafmuch - D-... as D meafureth C by thofe vnities whichare in E,therforeD _C...0++: ee wrultiplying E produceth C: but Aalfo multiplying B produced 22. FOR C wherfore that which is produced of D into E 1s equaltothat A.. whith is produced of A into B.VV herfore ( by the 19.0f the [e- nenth) as Aisto D5; fots E to B:bat D and-A are prime numbers sand therefore the st numbers in. that proportion: but the leaft in.any proportion meaure the numbers hauing t e ele fame proportion with them equally the greater ithe greater,apa the leffe,the leffe,that zs the antécedent,the antecedent, and the confequent the confequent (by the 21.0f the feueth). whirefore the copfequent D meafureth theconfequent B . Tn like fort may we proue that f DS Wvalicre nol B iv menfureth AV herfore D meafureth one of thefe nisbers A or B:which vas required to tobe proued. . yoy hg eles sities Herebyit is nvanifeltthat if 4 number meafure anumber produced.oftwo nibers inultiplied the ore Mtothe othegor be comnrenfurable to the fame, it fhall “2 or a afure of Gachides Elemente. Fol.198. meafure one of the'two numbers raultiplied, or becommenfurabletyith onc of them, I Le. 3.1.F heoreme. The 33. Propofition. Euery compofed number 1s meafured by fome prime number. =F Vppofethat A be acompofid number. ThenT {4y that Ais meafured by [ome g pil prime nanber. Forforalmsich as ‘Ais a compofed number.,.fome number muft LORS 2 weedes meafure it (by the 14, definition of the feuenth) . Let there be a number has | that mae urcth it und let-the [ame beB. Now if Bhea prime number then. ts that manifeft whichaye fecke for -but if ét bea com- \ poled number [ome number. ten byheedes meafurett uoGx (by the felfe [ame defiyition) Let therebeanumber B........- that meafureth it,and let thefame bE Ga And fore « Merde oe rvvir afmuch as C meafureth By and. B meafureth A: : 3 therfore C alfo meafureth Af by thes common fentéce)andif C be a prime number then is teat manifelt whichrwe fought for... But of it bea compoled number [ome number hall mea. fureit:and the like confideration being hdd there fhallat the length be found fome prime numcer which mealareth the mumber going before; which {hall alfo meafure A. For if there be not found any {uch prime aumber then {hall infinite numbers decrefing meafure the fayd number A of which the one is leffe the the other whichis impoffible in numbers. VV. herfore fome prime number {hall at the length be found which [hall meafure the number going before wud which {halt ifo tmeafure the number ‘A (by the 5.common fentence) .Euery copofed niin» per ther ef ore 1s meafured by {ome prime number which was required to be proued. gf An other iy. Suppofe that A bea compofed number. They i [ay that fome prime number meafureth it. For fordfmuch as\_11s.acompofed number, [ome number [hall meafure it (by the t4.de- jinition of the fenenth) Let the least number that mealureth it be B. Then I {ay that Bis 4 prime mimber. For if B be not x priménum. A ber’ fount vorimbir Jhiall meafure it. Let C meafure jt. WhereforeCis B ... leffe then BS And fora{mach BC wneafureth B, and B mMeafureth ~ C., 4, therfore Calf meafireth CA being leffe then B, which by [uppo- . - | Srtionss the leat mamber that meafareth Awhich is ablurd. Vy ‘her fore B is nota compoled number UE a prime number which was required to be proued. : *ee@@eeeeers ST he 32. T heoreme. >< "Fhe 34. Propofition. Euery number is either a prime number zor els fome prime number mea[ue ea uRGhe tha fore: ea Ee fe | > -* ae o. V ppofe that there be anumber A. Then] [ay that Aiseither a prime number ,or els {ome prime number mecafureth it. For if A be a prime number See lithen isthat hid which is required. But if itbea compofed num. 4 | ber, ome prime number hall meafure it ( by the 33 of the feuenth). E uery = Ad number therforess either eprome namber,orels ome prime number mea- fureth it : which was required to be demonftrated. - . 3 ; & e > = if VAY. s uppofe Demonstra- tion leading t@ an impoffibi- kitits An other de- monfiratione Demon ra-~ tion. « ile a ey a nn i ie ie a reer es Two cafes in this Propoft= 80H» The firft cafe. The Second tafe. Demon'hrae $10 Be Demers [irae tion leading to OU Ab ur ditie. Ai Corollary added by Causp ein — wenth). | ‘ , re; A RP : i i a al Te eee peda i) ————S AY " T he feuenth Boke ig The 3. Probleme. T he: 35: Propofition. How many numbers foeuer being genen,to find out the lea/t numbers that hane one and the fame proportion with them. the le(Se the lefre(by the 21.0f the feueth). VV hereore E meafureth G.And fora{much as.A multiplying G and E produced C and D., therejore (by the 1 7 of the fift) as Eis toG, foisC to D. But E meafureth G. VVhereforeC alfo meafureth D,the greater the lee: which is impofuble. Wherefore thofe numbers A and B do not meafure any number leffe then C. Wherefore Cis the le(t number that is mealured by Aand B: whichwas required to be done. w WL \ Ay y WNT \ { LT he 33: T heoreme. The 37. Propofition. If two numbers meafure any number th leaft nuber alfo “which they meas _Juresmeafureth the felfe fame number. Suppofe Two tafes in thy propofitso. The fir cafe. Demon tra- tion leadin 2 tp an abfurdiiie, The fecona cafe, Demonftra- tion leading to aH abfurdsties . ee ~ = : . a = t \ 4 : I | ith ‘s } if f : The feuenth Booke Hee V poofe that there be two numbers gewen A and B:and let them mealure the nuin- etait ne ay bei C D: and let the leaft number that they meafurebeE.. The] fay that Ealfo tion leading to RON mefureth the number CD. For if E. do not mea- an impoffibi~ fureC DE meafuring C D,thatis [ubtrahed out f CD A.. dette. as ofte as you can,as for example,once,leaue a lefSe then it felfe, B.. namely, And let the number [ubtrahed whichE meafu- E...... reth be D, and fora{muchas AandB meafureE,andE C....F... meafureth \) F , therefore A and B al{o meafure DF. And they meafurithe whole C D wherefore by the 4.common fentence of the feuenth they mes. fure allo tha which remayneth CF being le(fe then E:which is impolfible. Wherefore E of necelfitie mefureth C D,which was required to be proued. q Ihe s. Probleme. Ihe 38. Propofition. Three nunbers being geuen,to finde out the leaft number which they meafure. xh V profe that there be three numbers geuen A,B,C. Itis required to finde out the my | [4p number which they meafure.T ake(by the 36.of the feuenth the leaft nnmber Two cafes in whch A and B meafure,and let the fame be D. Now then C either meafureth D this propofiti0- 9» els mealueth it not.F irft let tt mealure it. And the numbers Lhe first cafe, allo A and} meafure D :wherefore A,B, C3meafureD. Now A... shen I fay thit D is the leaft number which they meafure. For B.... Demonfira- if not,let thenumbers A,B,C, meafure fome number leffethen Crrevee tion leadiugto ; D,and let the fame be E. And forafmuch as A,B, C, WENN: Dose njas ewer E therefore fo A andB meafureE, , wherefore ( 04 the 27.0f EB vcs ssive the feuenth the leaft number which thofe numbers A andB mefure {hal al{o meafureE. But the leaft number which A andB mealureis D .Wherfore D meafurets E,the greater the leffe:which is impo/ible Wherefore the[e numbers A,B,C, Shall not meure any number leffe then D. W, herefore D is the leaf number that A,B,C, doo mealure The fecond a [“ppofe that C meafure not 1).And take ( by the 36.0f the feuenth ) the leaft gts number whth thofe two numbers C and D do meafure,and let the fame ce E.. And foraf- much as t ind B meafure D,and D meafureth E, therefore A and B alfo meafureE , and an abfurditic. cafe. C alfo meafireth E, wher efore A, B, C, alfo meafure E.1 fay moreouer that Eis the leak” number whith A,B,C meafure.F or if 1t be not.let there be fome le{fe number then E. which Demonfira- they meafun: , and let the fame be¥ . And foraf- gion leadingto ph a6 A,3, C, meafure F .T herefore NandB A an abfurattic. alfo meafureF + wherefore the leaft number which B thefe numbes A andB do mealure doth allo mea- C .seceeve fure® (by the 37.0f the fenenth) But the UATIOR . Dies ccssiese r ber which 4 andB doo metfureis1) .Wherefore E D meafurets F. And C allo meafurethE.Wherfore F D and C mafureF . Wherefore the leaft number 7 which Cand) doo meafure,fhall alfo ( by the felfe fame) meafure F.But the least number which C&D meafure is E.Wherfore E meafurcth F, namely, the greater ,the lee, which is tmpoffible. Wherefore thefe numbers A, B, C,do not meafure any number leffe then E. Wherefore ¥, is the leaft number which A, B, C, doo meafure * which was re quired to be demonftratel. @eae4aeaoesceaeseganse8e 228208 4 @eseeeeen0e0ee20880808897088286 8 in like naner alfo how many numbers focuer being geucn, may be found ag IDE : of Euchdes Elementes. Fol.200, leaft number which they meafure, For if vnto. the three nambers'A, B,C, be added a forth,then if the fayd forth number meafure the number E » thenis E the leatt number which the fower numbers genen meafure,But.ifitdoo not meafure E, tlé by the 37,0f this booke mutt you finde out the leaft number which Eand the forth nunber meafure. Which thall be the number fought for. And fo likewife if there be fiue, fixe,or how ma- nyfoeuer genen, Corollary. “Hereby it is manifeft that the leaft commé nieafare to numbers hovmanyfoeuer, meafureth cuery number which the fayd numbers how manyfoeuer meaiure. q The 34. Theoreme. Ihe 39. Propofition Ifa number meafure any number :-the number meafured hal haue a part after the denomination of the number meafuring. wt ih {oS Ao hy F pale that there be a number B which let meafure the number 4.Then I [ays that A hath apart taking his denomination of the number B. For how often B meafureth A, fo many vnities let there bein C. And let D be untie.And foraf- Samy much as B meafureth A, by thofe unities which areinC , and unmtie D meal u- veth C by thofe unities which are in C, therefore vnitie D,fo many times mealureth the number C, as B doth meafureA. Aus Wherefore alternately ( by the 15. of the fenenth) vnitie D, of eee Sere many times meafureth B,as C doth meafure A. Wherforewhat C€.. part unite D 1s of the number B, the fame partisC of A.But D. vpitie Dts a part of B hauing his denomination of B .VV her- Sore C alfois a part of Ahauing his denomination of B.VV herfore A hath C aa part taking his denomination of B : which was required to be proucd. : The meaning of this Propofitionis,that if threemeafare any: numberthat number hath a third part, and iffoure meafure any number the fayd number lath a fourth part.. And. fo. forth, : fhes q Fhe 35. LT heoreme. The 40. Propofition. [fanumber haue any part: the number wherof the part taketh his denoe mination {hall meafure it. Veppofethat the number A haue a part, namely, B: and let the pirt Bhaue his denomination of the number ©.T hen I fay ,that C meafureth A. let D be vni- CN t2¢. And foralmuch as B isa part of A ,hasing his denomination of C : and D LEX SS being vunitieisalfoapart of the number C Aauing his denominatii of C : there- fore what part.vnitieD is of the number C,the fame part ts aifo B of A: wherefore-vnitic D {0 many times medfureth the number C, as B meafu- 4 vet A: Wherefore alternately (by the 1$.0f the feuenth) unitie D fo ma- 2 ny tithes mexfurerh the number Bas C meafureth A Wherefore C méa- °C .. fureth A> whith was required to be proued. D This Propofition is the conuerfe of the former : and the meaning themfis, thate- nery number hauin athird part is meafured ofthree, and hauin g a fourth art is mea- fured of foure. And fo forth, /:)~ | | q he A. Corclary, Demonfira- BO. The conuerfe of the former props/itione Demonfirar tien, | — — —: af Z > eee b J 4 q nm piu ; ( i ij | Nah { ue 4 Vou i iF H \ § 5 1 if Y tig ‘5 U Hie i } i] - : } bt 1h i4 I | 7, 3 4 — ee nes eee ee se poems oa +S eee: * a* Sab - = i PT a hw Construction. Demonfstratio de. dingtoan ab furditice wt Corollary added by Carspane, Howto finde outthe feccnde dealt nunzber and the third, andfo forth sufinitly. How tofied out tie leape sumer ton= (40% $9 f. Layiiiig tue pares of parts. The fenenth Booke yl be Probleme. Th 4.1: Propofition.. To finde ont the least number , that containeth the partes genen. ESS Vppofe that the partes geuen be A,B,C,namely,let A be an halfe part,B.a third | 27 part, Ca fourth part.Now it is required to finde out the leaft nuber which cotai- BIRDING neth the partes A,B,C.Let the faid partes A,B,C, hane their denominations of the numbers D,E,E. And takelby the 38. of the feuenth) the leaft number which the numbers D,E,F, meafure, and let the fame be G.uAna foraft much asthe ; = umbers D,E,F meafure the number G,therfore the number G A ef alfe hath partes denominated of the nubers D,E,F ( by the 3.9.0f the Aes ‘Sof feueth) But the parts “A,B,C, haue their denominatio of the num C afourt : bers 2D, EyF Wherfore.G bath thofe partes A,B,C. ue alfothat D.-. it is the lealt number which hath thefe partes.F or if G be not fe ee leaft number which containéth thofe- partes A, B,C, then let there be fome number leffe then G whicss containeth the faide pe Die pe oaee partes A,B,C. And [uppofe the fame to be the number H. And we eds | ‘orifmuch as H hath the [aid partes A,B,C, therfore the numbers that the partes A,B,C, saketheir denomidarions of fhallineafure H (Ly the 40, of the feuenth) But the numbers wher of the partes A,B,C, take their denominations of,aré D,E,F .Wherfore the numbers D,E,F, meafure the unmber H which is leffethen G,which is impofeble. F or G 1s [uppofed to be the teaft nrber that the numbers D,E,F domeafure. W. erfore there 1s no number le([e then Cwhich contzineth thefe partes A,B,C: which was required to be done. a fone . ye “=< Corrolary. Herebyit is manifelt that if there be taken theleaft number, thatnumbers how ma. ny {oeuer dd meafite,the fayd numtber fhall be the leaft which hath the partes denomi-- nated of the fayd numbershow many foener, | Campane after he hath taught to finde out the firtleaft number that Gontayneth the partes geuen,teacheth alfo to finde out the fecond léaft nuniber, thatis; which except the leat of all is leffe then all other,and alfo the third leaft, and the fowrth-tc. The te+ cond is found out by doubling the number G.For the numbers which meafure the ni- ber G thall alf méature the double therof (by the-sscommon fentence ofthe feuenth), But there cannot be geuen a number greater then the number G,& leffe then the dou- bie therof,whom the partes geuen (hall meafure,For forafmuch as the.partes geuen do seafure the whole,namely, whichis lef then the double, and they alfo meafure the part taken away,namely,the number G,they fhould alfo meafure the refidue, namely,a number leffe then G,which is proued to be the leftnumber that they do meafure,which is impoffible - wherefore the fecond number which the faid partes geuendomeafure, muttiexceeding G,needes reache to the double ofG,and the third to the treble, and the fourth to the quadruple, and foinfinitely, for thofe partes can neuer meafure any number leffe then the numberG,. " ee 7 “By this Propofition alfo itis eafie to find out the leaft number contaifling the partes geuen of partes,Asif we would finde out the leaft number which contayneth one third. part-ofan halfe part,and one fourth part of a third part;réduce the faid-diuers fradios: into fimpl¢ fraGion (by, the common tule\of reducing‘of fractions). namely, the third, ofan halfeinto a (ixt part ofanwholejand.the fourthofthitd inte atwelfth part of am, whole.Andthen by this Probleme fearch out theleaft number which contayneth a fixe. part and a twelfth part,and fo haue you done. : $49 The end ofthe fettenth booke . iY of Euclides Elementes, ~The — ; ae Fol.ze1. @ | heeighthebooke of Eu- clides Elementes. = of the proprietiés of Huimbersin senerall, and of certayne WY kindes thereof ndre {pecially ,and of prime and compofed weer numbers with others:now in this eight bookehe profecu- The Argumet ij) 47 tethfarther, and findeth outanddemonftrateth the pro- of the eighs wo” «pertiesand paffions of certayne other kindes of numbers: bovke, (CX as of the leaft numbers fn proportion , and how {uch may B, C, Dy be ix ee eee meee seeeseseeoeeeessbeaes the felfe fem proportion that AB C D are And p edeteiah Denia: ‘foralmuch.as the numbers A, B, CD, are in the C tion leading to ‘felfe [aime proportion that the pimbers E;F3G3H, Pa See CAL an abfurdities are, themultitude of hele nimbers E, is > RE ag 6 eee Gre ‘sequal bo the rhilltitude of thelepumbers AB 3€°D, therefore of equalitie ( by ther 4 of “the feuénth a3. 3540 D, fois EtOH But AandD are prime the one'to the other , yea they “are prime and the len ft that bane the fame proportion with them. But the lea F numbersin a. ‘RY proportion meafire the nimbers that hane the [ame proportion with them equally the ‘antecedent the antecedentiand the confequet the'con sfequent ( by the 21 of the fenenth wher ‘fore A meal ureth E the greater the leffe:which is tmpolfible. Wherefore the numbers LE; ¥, G,H, bene lefethen ABC sD), are not in the fame proportion that A,B,C;D,are,wher- fore A,B; C,D are the lent of allnumbers which bane one ard the fare proportion with Xs. them , | : } qa All Nid } 4 i $ ‘ y a} ’ y ia ' ley ve \ ; i if { ye ik, i] Wy y - t : Conftrnttions Denson ftra- $i0%* T he eighthe Booke shem. which was required to be demonjirated. q The 1. Probleme. T he 2. Propofition. To finde out the leaft numbers in continual proportion as many as [hall be required in any proportion geuen. 1 Vppofe that the proportion genen in the left numbers be Ato B. Itis required to finde out the left numbers in continual proportion,as many As hall be requi- 4 redyin the fame ares that A isto B.. Let there be required foure. And let SLL A multiplying him felfe produce : and multiplying B let it produce D : and likewife let B multiplying him felfe produce E - And moreouer let A multiplying thofe num- bers C;D,E, produce F,G,H : and let B multiplying E produce K . Aad foraiaach as A multiplying him [elfe produced C,and multiplying B produced D, now then the number A multiplying two numbers A and B produced Cee D .Wherefore (by the 17.0f the feuenth) as. Aisto B, fois Cto D. Againe, fora{much as A multiplying B produced D, and B multi- plying him felfe produced E, therefore ech of. thofe numbers A ana B multiplying B, bring- eth forth thefe numbers D and E VV herefore (by the 18 of the feuenth) as AistoB, foD to E 2 But as AistoB,foisC to D .Whereforeas C isto D, [ois D to E. And forafmuch as A multiplying Cand D produced F and G, therefore ( by the 17 .0f the feuenth)as Cis to D, fois F toG. Butas Cis to D, fois AtoB . Wherefore as Ais to B, fois F toG. Againe for- afmuch as A multiplying D and E produced Gand H, therefore (by the.17 .of the feuenth) as D isto E, foisGtoH .Butas D isto E, fois Atos. Wherefore as A is toB, fo is G to H. 7, ara Bai. i Cy. eesee? 8 etetBPCoevoeaeveenonee2 808869 8 eesen ee? © e448 6 Oe F G H cecooeovseeoeeeovesoeeenoe ene eee eo4ueer BEER SUT K ccc e ebb be 6:450 C66 eet pa sag 64600 R2 CRESTS StSe TSOP T UN TONS °8 Oe And forafinuch as thofe numbers Aand B multiplying E produced H and K therefore (by the 18 .of the feuenth) as Ats to B, foisH tok, And itis proued,that as AistoB, fois F to G, and GtoH : whereforeas F is toG, fois GtoH andl to K. Wherefore thefe numbers C,D,E,and F,G,H;K,are proportional in the [ame proportion, that AistoB. Now I fay, that they arealfo the left. For forafmuch as Aand B are the left of all numbers that hane the fame proportion with them » but the left numbers that haue one cy the fame proportion with them are prime the one to the other (by the 24.0f the feuenth) «therefore A and B are prime theonetothe other : and ech of thefe numbers. Ao B multiplying him felfe prececns we numbers C and E, and likewi{e multiplying ech of thefe numbers Cand E.they produced F and K . Wherefore (by the 2.9.0f the feuenth )C,E and F)K, are prime the one tothe other. But if there be numbers incontinuall proportion how many {oeuer, and if their extremes be rime the one to the other ,they.are the left of all nitbers that haue the [ame proportion with them ( by the firft of theeight). Wherefore thee numbers C,D;E,and F,GsH,K are the i of | Euchides Llementess © Fol.161. ofall numbers that bane the fame proportion with Aand B And forafpsnch as (by the 293° of the fenenth) that alwaies ha peneth touching the extremes, namely, that Aand B wmultio phing the numbers produced FuwdK [ball produce*other prime numbers, namely; the ex tremes of fiue numbers in continuall proportion, therefore ( by the first of this booke) all fine are the left of that proportion . And [0 infinitely : which was required to be done. Seas, q Corollary. Flereby it 1s manifeft that if three numbers bein gin continuall proportis on, be the left of allnumbers that haue the fame proportion with them, their extremes are {quares : and if there be foure their extremes are cubes. For the extremes of three are produced of the multiplying of the nfibers A and B into them felues.And the extremes of foure are produced of the mul- tiplying of the rootes A and B into the fquares C and E, whereby are made the cubes Fand K. Ihe 2. Theoreme......T he.3. Propofition. If there be numbers in-continuall proportion how many foener,and if they be the lest of allnumbers that haue one and the Jame proportion with the: their extremes [hall be prime the one to the other. =a P ppofe that the numbers in continual proportion being the leat of all numbers | f~| thathaue the [ante proportion with them be.A,B,C,D. Then I fay that their This propofi~ tp. tion 1s the 6s RS xy extremes Aand D are prime the one ta the other. T ake(by the 2.0f the eight or ant of the phil by the 35.0f the fenenth) the two leaft numbers that arein the [ame proportion oS that A, B,C,D,are,and let the [ame be the numbers E,F.And after that take thre numbers Pee Pe Peers ork ae oe et eS . aaron eeeeee eesreeeeceaeaeaseeee ee s@e@eds sbac i ee ee ee ee ee 2 en ee ee : “ Ww RHP LGU Fas} Wad dep MaviKaene- b> weetees ve crmdemmewerd eda acecece > = 3 . ; : . es EX eas) So Sess ¢ . we st q“\’ NY A. > ~ = Se ee ye er ABE 7 3 te 3 m » Ss ud a eee | G es ee 9 howe we | aA) tS, pnd \s oe Rerwene wee $5 1a. Qu A. ee 400 oe @ . o4 eae Ca “ERE 4 E . : * ESR Ee SE eee te ea lene oe ned. —— 2. so eS ey ~ i" Mawuan cue = e« 26.09 810 @ of @ os 6 <9 5% © Bee Plece twa tal 4. , . : aus Sta <3 LA See ee ee “ee me eee oO hee ed 6 ee wie eee edevewrdeodveo ~ be \ : ~ > . ‘, + h AAS Se eee iyi nce seco \a a 9 Glo we ven dak wee Bawieeees Pere SC ewe wee Cee we sy! HK; and fo alwayes forward in more ( by the former propofition) untill the multitude taken be equall to’ the multitude of the numbers genen A,B,C,D.. And let thofe numbers ~ be L,M,N,0 . Wherfore(by ther. of the feuenth) their extremes LO, are prime the ont Demonfira= £0 the other For fordfmuch ae Band are prime the one to the other and eche of them mul. tio, = Xx. Ye siplieng = = —— = —S — a ne | The eighthe Boo ke i. lieng hsimfelfe produced G and K's likemife ech of shefe G & E multiplieng himfelf pros. ti | ta L & 0, therfore(by the 29 of the feuenth)G o K are prime the one tothe other,er fa. Two cafes in this propofitsd. Demonfira- tion leadzg to ber s in continuall. br opor tion,and in th 4 an abfirditit, £ to 2, re ee ee : The fir cafe. likewife are L and 0 prime the one tothe other And fora{much as A,B,C,D, are the leaft of A ©0686 OOo G4 0.9 O86 OOOO Oe OrF © 28 ‘6 eceoeeveceees eeoeeereeen ese? @eesee *@eeeresemreneseeaeveseee@ 7 seeeenp#eeeeseeee ee eee @ t~ all numbers that bane the [ame proportion with them,and likewife L, M,N, 0, are the least of all numbers that are in the {ame proportion that A,B,C,D,are,and the multitude of thefe nunphers A,BsC,D, is equall to the multitude of thefe L,M,N,O therfore enery one of thefe ABCD, is equall vate euery one of the[e L,iM,N,0.Wherefore Ais equal ~ntoL, and D is equall unto 0, And forafmuch as L and O are prime the one to the other, and L is e. guall unto A,and O is equall unto D : therfore A and D are prime the one to the other: which was required to, be proued. } Wetec se T he 2. Prsbioue. Srey T he 4. Propofition. r@&eteeee © w@ & * Proportions inthe leaft.numbers bom many foener Hong genen, to finde -- > -ont the leaStnumbers m continuall proportion.sn the {aid proportions gene. (EN # V ppofe that the proportions in the least numbers gewen,be Ato B, Cto-D; and x SY % \E to F. Itis required to finde out the least numbers in continuall proportion, in lime proportions that A isto-B, and C to.D, apd then WGK, inthe [ame proportions that A yaebers be NX M.O « And foralenMh ge ARG 5 ns > . « ss By Le é S of Euclites Elementes. Fol.2.03. B, fo ts Nto Xe gaa A add Bane the left but the beat medfiive.. ids. thofe numbers that haue oneand the fame proportion with them B... equally the greater the greater and the lelfe the leflesthat iss the. antecedent ,the antecedent,cr Phe confequent the confequent.-( by oe ee the 21.0f the feuenth) therfore B meafureth X And by the fame "D> realop.C allo megfureth X, wherfore Band C mealure X.Weher- fore theleas: Ramber whig Band C theafire hall alfa by the 37°» E Of the /onbnth weafure X.Butthe last number Whore Bana C -F therfire isG Wherefore G WeafureahRx> the greater, thé lefve, which is impofableV herfore there fhall not be any leffe numbers “HT... then H,G,K,L, in continual] proportion, andin the fame pro- =e : iS L portions that Ais to Band Cto Dsand Eter. | But now fuppofethat E meafure not K: And bj the 36.0f the 2... a heess The fecond feuenth,take the leat number whome E and K meafure, and let ~~ cafe. 4 the fame be MU. And how often K meafureth A, fo. often lef RE | either of thefe G and H meafure either of thefe.N and X. a And how often E mealureth M,fo often let F mealureO. And **M forafrnuch as how often G meafureth No often doth H mea ifure oO X therfore as i1istoG fo iSX toN. But asHistoG fo is Ato B e | Wherfore as-AistoB fo is X to N, And by the [ame reafon AS C as to D fo is taM. ‘A- gaine forafmuch as how often E meafureth Mo often F meafureth O, therfore as E isto F {ois Mto 0,\Wherfore X,N,M,0, arein continual proportion andin the [ame proporti- Uns thar is 10 Bande tO Dana E to F, Tay allothat | | they aré the leastin that proportion. For if ¥;N.M.O, be a mot the leatt tn Continnall proportivn,and iu the lame pro- f saees portions that Ais to Band CtOD: ind Eto P, then fhall there be [ome numbers lefse thea X,N>M,O in continuall C Proportion, andin the fame proportions that AistoB; and. °° “D Cro D,and EteP. Le the fame be the numbers P,R, S, L. Ana for that ts P is toR fois Ato B. and A and B - xtre the leaft but the leat nurabers meafure thofe numbers ~~ F. H G K ope tear Demanitras sks 6 Sco Su : tions that hinne ome the fume proportie with them equally,the greater the greater,and the leffe the lefe,that is the antece- dent the antecedent,and the confequent the confequent b the 21. of the feuenth therfore B mealureth R, And by the fame reafon alfo C mexfurerh ®: Wherefore B and C’mea. [ure R. Wherfore the least number whom B and C meafure xX foal alfa mealureR (by the 37,0 the feuenth). But the leff oof tices number whom B and CT meafure is Gwherfore G meafureth AP agers. eF N Ue R. And as G isto Rois K toS. Wherfore K meafureth S. is ekee vers yoo 72 Eo 2 *) 2 os eS 6 a. 8 oe Wherfore the leaf? number whom E and & meafure, {hall Posi f by the felfe fame) meafure S. But the least number whom. Ro Land K meafure i: 0, Wherefore M meafureth S the S Kreater the lefeswhich is impofable.Wherfore thereareno T., numbers lelethewX Ny! 0,30 continual proportion: er sedrig 17 the [ame proportions that Ais to Band C to D and Eto F Wherfore X. V,M,0, are the dealt numbers sn continuall proportionsand inthe fame proportions that A isto Band G:te Dand E toF which was required tobe done.’ 208. | And E alfo.meafureth 8. Wherefore E and Xk meafure: S. eee eee aengeang Xx. ty. The ! 1 } | | | | ’ : it ii | } : i) | ne | i Di win: t | } Wi nil | { { ie ' : nay i" i a } 1 ; ‘ } ; i : i! ; iq : 1 i Hy it : th Wee + Hh | Hi “ine Tl we i} iat eat | ‘ ; ib cee. rn } } : MH ¥ i Bal eee ae | i ma) : ne sin | 1) ett ee | Walk aia | gc ae oy ith ee | ea iy - | Bf ae i| nt tal a Wt Me ee i iy" \ ig ait i 11 Ag BY ‘ he i! 1 ht aed 4 i A hit ne ' Hi i Ml T his propoft- fion in num- bers anfive- reth to the of the fixth touching parel delogrammics. Conftruttion. Demonfira- £102. An other de- msonflratio af- ter Campanes ) E, fo let G be to, as D | py T he eighthe Booke | g The 32T heoreme.» | Thees:Propofition: Playne or fuperfitiall numbers ave m that proportion the one to the other which is compofed of the fides. - | hd V ppofe th at A and B be playne or fuperfictall numbers , and let the fides of we QB A be thé numbers C it D , and let the fides of B,. be the numbers E. and ane) F . Then I fay that A isto B in that proportion that is compofed of the fides. Take (by the fourthof the 4 | eight) the leaft numbers in ey Ply err Oey pre continuall proportion and * L vs.sesverne ernest: be BES | in the fame proportions that, ~ Bs ciarttanevsensene peaemrecbeAts > C istoE,and DtoF. And. let the fame be the numbers... G,H, K: fo that as C is to a. i: se @e e848 is toF , fo let H be to K. Wherefore thofe mumber§ —_ G veeeene oa G, HK, Aane the eens | Ge ee Fane ons of ihe fides: but the pro- i ae LE ReaD portion of G to K 1s con po- ete | fed ofthat whith G hath+oH and of that which H hath toK ; wherefore G is unto K ix shat proportion which is compofed of the fides. Now I [2y that as Ais to B fois G to K.For let D multiplying E produce L. And foralmuch as D multiplying C producea A,and mul- tiplying E producéd L:therefore(by the ¥7-0f the feuenth)as C is to E.foisAtoL..Butas C isto E, fois G to H wherefore as G is to Hfoss A toL . Agaynefora{muchas Emal- tiplying D produced Lier multiplying F produced B: therefore( by the 17.0f the fenenth as D isto F, fois L to B.But as D ts to E,foss H to K wherefore as His to K foes Lito B. And it is proued that as G is H,fo is Ato L. Wherefore of equalitie (by the t4.of shefenenth) as GistoK, fo is Ato B. But G is-unto Kin that proportion which is compofea of the fides, wherefore A is unto B in that proportion which is compofed of she fides which mas required to be demonftrated. e um A ) | | « Ai other demonftration of the fame after Campane. Suppofe that A and B bé laine atimbers:and let the fides of A be the numbers C and D ; andlet the numbers E and F be the fides of the nimberB . And let D multiplying E produce the number G; Thé I fay that the proporti6of ; Ato B is cépofed of the propor Be cadin ame Uo nace t-ew'’ ‘ C ans tids of CtoE & Dto Fthatis, GF. eee aco e sss 0330 BRts Ree Deseesee of the fides of the fuperficial ni | Bs ae ee Ars AES EY ERE ore i ber A tothe fides of the fupéf- TB cccaveee ficial] number B.For forafmueh | i < . as D multiplying E produced G,and multiplying € it produced A,therefore by(the 17.0f the feuenth) Aisto GasC isto E : agaynéforafmuch as E multiplying D produced G and multiplying F it produ- ceth B,therefore by the Bak GS isto Bas Disto P: Wherefore the proportions of the fides namely ; of C to Eand of D to F are one and the fame, with the proportions ofA to G and G co B.But(by the fifth definition of thefixth ) the proportion of the exttemes A toB 1s ae 4 ae of the proportions of the #ianessnamely;of A to G and G to B,which are proued to be one and the fame with the proportio of thie fides € t0 Band D to FE. Wherefore the proportion of the fuperficiall numbers A toB is c6 of the eee of the fides C to E,and D to F. Wherefore plerne. &c. which was tequited to, Be proue . qT be ; ¥ . of Stctiiles Elementes. Fol.205. gL he z.T heoreme. Thee. Propofition. If there be numbers in continnall proportion: bow many foewer and if the firft meafare not the fecond neither ‘[ballany one of the other meafnre any one of the other. | ISRO d {uppofe that A mealure not B. They I [ay that ne- Fels for A mealureth not B. Now I [4 that neither Jhall any other of them meafure any 0- the TS of the [enenth) and let the fame be F,G,H. And ‘forafmauch 45 F,G,H ure in the elfe famte | Demiuitra<% Oe vcs os bn ee ; . , ¥ : sion. © 0 6 ff ©. 00.0 8 6 0.6 4.8 OKO @ 0 HO 2636'S SOS O39 2 OO Oe Se Ue Oe Oe Oe ese Ok Bie ee Ne narnia tah at tie hla ae aki ee proportion that A,B,C,are: and the multitude of thefe numbers A,B,C 15 equallto the mul. titude of thofe numbers F ,G,H, therefore of equalitie (by the 14. of the feuenth) as Aisto C, [ois F toH «And for that as Ais to B ,[0is F toG, but A meafureth not B, therefore ne ther doth F meafure G . Wherefore F is not vnitie . For if F were vnitie, it Should meafure any number . But F and H are prime the one to the other(by the 3 of the eight) , Wherefore F meafureth nit Hig as FistoH, fois AtoC, wherefore neither doth A meafure C.In like [ort may me prowe that neither (hall any other of the numbers A,B,G,D,E, mealire any bother of thé numbers A,BLC,D,E : which was required to be demonftrated, Ihe 5. Theoreme. Lhe 1, Propofition, If there be numbers in continual proportion how many foeuer, and if the frst meafure the laft, it fall alfo meafure the Second, SSS Sead” ppofe that therebe nmmltitade of numbers in continyall cep | , proportion,namely, A,B, LONG C;D . And let A the firft meafiive D the laff. | eee: 1 hen I ay, that A the first mealireth Bthe A... | Demon/fira~ fetond. Por af A do not med[wre B,neither oallany eo. B ..- tah oe ce ‘5 ther meafire any other (by the 7 of the tight): which ©... °° 3 : litie. — (by [uppofition)is not true.F or Ais [uppofed Tee | nr —— D .Now then A meal uring D, jhall alfo meafwre B: syicl was required to be proued. ee ih Al Th ” fe ry) tid Hi Hit eu i \ 9 apes . — ‘ — > — . ma. - 2 pik . 2. — Sc . — Sate ee “Sta 2! an Ss - = —_—-— — oe oaks : ‘ : = — ee — “ =. 2 om 2 - = = ee ion - ; > - : — =e : + - = = —— ——-— 7 = = . . =e = — — St ee =e *: are Cae <= 3 : ee - - a= GY Pf = ee ee — = = = = = : = = = -—-— at crws =" = = = : = = ~ —a =a : = | - - - —s% = ap cacaeeeeion z ——> Se ==> ee = — _ —__ a \ = = = — ~~ ——= r ———= — - - — —- ~—-———— 7 — — = -- - = —< = —_ = = = — — - —— eo 7= = — = =—— a = : = = : = . a RPA aR 0 5 7 ~~ . _ ——__—- - -— + — ~ - — - z —S—————— = - ox = —— —— —_—— = = =~ = = es = > gg er = ~ < ; ; . — - _ — == ; =——= Seems = = — - z- —— ~ S ~ - — —_ — = = = 3 i te = — = — —_ — oa ~ ee 4 Corollary added by Flufeste s, os WA T be eighthe Booke . Glbhe 6. T heovene. bbe sd. Propofition. icy, Uf betwene two numbers there fall numbers in continual proportion: how _Inany nunvers fall betwene them, fo many.alfo fhall fallin continual proe portion betwene other numbers “which hane the felfe fame proportion. V-ppofe that berwene the two numbers A and B, do fallin continuall proporti- onthe numbers C and D . And as Ais to B,foletE betoF . Then 1 fay, that how many numbers1a continnall proportion do fall betwene A and B, fp many LESSEN nnmbers alo in continual proportion fhall there fall betwene E and F. How many “A,B;C;Dyare in multitude, take by the 3 5-of the feuenth fo many of the leaft num. bers that bane one and the fame proportion with A,B,C,D, and let the fame be G,H,K,L. Wherefore their extremes G and L are prime the one to the other. (by the 3.of the eight) .And foralmnch as Aand Cand D'and B,arein the (elfe fame proportion that G cy Hand K and L are, and the multitude of thefe numbers A,C, D, B, is equallto the multitude of thefe numbers G,H,K,L: therefore of equalitie (by the 14. of thé feuenth) as Ais tab, fois Re Oats ia ace D eseenee*te*eoeeeweeoeneeoeesethteeorteegeee se @ es B Seeeeeezpgceeegeees @eeeetseeet eos see eeecasorea seeerace eeeaecoae @ecagtatceaereecesrcredsead ge : fi n K o *@ ee ta eeesvese @ e@eepepeer aeehecepe . ict ak i ~ 4A} \. B. we aioe ““a'e oré © : TEL) ese see se *@ Xs . be - - > - = &- ve = SS = ee ee me ee ee ee ae ew eh a ce eor wate e ees ®eeeteveoouvaesaerPepeeeeevee epee evz een e os Grol Bur as -AistoB, fois Eto F . Wherefore as Gis to L, fois Eto F. Bat Gand L are prime the one tothe orbers-yea they art prime and the least’. But the leaft numbers meafure thofe numbers that haue the [ame proportion with them equally,the greater the greater, and the lefse the lefe (by the 2 r.of the fenenth) that is, the antecedent ,the antecedent, > the con- fequent, the confequent . Wherefore how many times G meafureth E, [o many times L mea {ur eth E-nH ow often Gmeafureth E, fo often let H meafiire Mand K meafure NR, Where- fore thefe numbers C,H3K,L, equally. mmeafure thefenumbers.E,M,N,F «Wherefore (by the 18 .of the feutth) thefe nuber$G,H,K,L,are in the felfe fame proportion that E,M,N,F, are . But G,H,K,L, arein the felfe [ame proportion that A, C,D, B, are; mbherefore thofe wnmbersA,C,D,B are in the felfe ame proportion that E,M,N,F ,are, But A,€,D,B,are in continuall proportion : wherefore alfo E,M,N,F ,are in continual proportion Wherfore how many numbers in continual! proportion fall betwene.A and B,fo many allo.in continwe all proportion fall there betwene E and F : which was required to.ke demonftrated: : eA Corillary added by Flufeates Betwene to numbers whofe proportion is fuperparticular or fuperbipactient there falieth we lmeanse propertian wall. For the lealt numbers of that proportion differ the one from the other onely by ynitie or igi ey ut ‘y De oe $23 5 sy % €i% a." * of Euclides Elementes. : Fol.206. But if betwene the greater numbers of that proportiom there fhould fall a meane proportionall; then fhould there fall aif a meane proportionall berwene the lealt numbers which haue the fame proporti- on by this Propofition .But betwene aumbers differing onely by ynitie or by two, there falleth no meane proporuonalla.s Yocox ) 9, eA” | | I he 7.J heoreme. Ihe. 9. Propafition, If two numbers be prime the one'to the other and if betwene them fhall fall numbers in continuall proportion: bow many numbers in continuall pros portion fall betwene them , fomany.allo {ball fall in.continuall. proportion betwene either of thofe numbers and vnitie. | al ppofe that there be two numbers prime the one tothe other A and B: and let SPY" therefall betwene them in continuall proportion thefe numbers Cand D: and \ DSI let E be unitie . ThenL fay, that how many numbers in continuall proportion a ff, Ce fall betwene A and B,fo many alfo fhall fall in coutinuall proportion betwene A and unitie E : and likewife betwene B and Uuitie E.T ake (by the 35.0f the feuenth)the two leaft numbers that are in the [ame proportion that A,C,D,B,are: and let the fame be F and G:and then take three of the leaft wubers.that are in the {amé proportion that A,C,D,B,are: and let the [aime be A,K,L : and [0 alwaies in order one more, untill the multitude of them be equall to the multitude of thefe numbers A,C,D,B : and thofe being fotaken let them be M,NjX;0. Now itis manifest; that F multiphing bimfelfe produced Hand meultiplying Wirnieties: H produced M.A pd G multiplying bim felfe produced Ly and multiplying L produced o: tiews : And forafnuch as M,N X50; are( by [uppofition)theteaft of all numbers that haue the fame proportion with G,F xand A,G;D5B, are (by the firfh of the eight) the leaft of all numbers what hane thefameproportionwith GF: and the multitude of thefe numbers M,N,X,0, isequall ta themmuttitude of thefenumbers A,C,D,B: therefore ensry.one of thefe numbers M;N,X,0; is equalltoeuery one of thefe numbers ASC,D;B. Wherefore M is equallunto Con Sruttion. ¢ M27 A 27 dx een whi 04 Be, ot Fa Bi wnetle FG shi edd QD ER ; Martius Gsealy | X48 D. 48 + eS i he Reanard 0 awe | O 64 B 64 Ay and Owsequall upto B. And foralmnch as F multiplying him felfe produced H: therfore Fi meafureth Faby thofe vinties which ave in-F + and-vnitieE meafureth F by thofe vnities whicharein'E + wherfore-Chy the.ssiof the fenenth) vnitie E, fo many times meafureth the nuyinber Fak F medfureth H enbrefore asunitie Eistothe number F fois F toH Againe forafaiuch as-F niultiphing H produced M, therfore H-meafureth M by thofe-vuities which arein F.. And vunitie E meafurethF by tholewnities which are in F ¢ wherefore (by the felf fame) initieE fo many times meafurethF as mealuneth M..Wherefore asvnitie E is te theuumberF, [06s H. to. But tt is proued;thatasunitieE isto the number F , fo is F to H : wherefore as unitie E isto the number F, fois F toH, and H to M . But Mis equall unto A - wherefore as-uniticE isto thenumber F, fois F toH,¢> H to.A. And by the fame reafon as Unitie# istothe namber G; [ois G to Land L'to B. Wherefore how many num- bers fallin contipuall 41 hia betwene A and B : {0 many numbersal(o in continuall pro- portion fall there Ceiweneonivie Einid the number A, and lrkewife betwene vnitie x nw) an This propofi- tion ss the con - nerfe of the former. C, oniBructioni Demonftra- tion, The erohthe Booke Aw dnd the number B: whith was required to be demonfirated’.° °° | q The 3. Theoreme. Th te Propofition. If bet-wene two numbers and pnitie fallinumbers in continuall proportion: bow many numbers in continuall proportion fal bet wene either of them 7 ynstie fo many alfo fhall there fall in continuall proportion betwenethem. LAN V ppofe that betwene the two numbers A;B,and vnitieC, do fall thefe numbers in ’ Dy, ‘begee) 4 . . : ‘ . : K G y continuall proportion D,E,and F,G.Then I iy that how many numbers tn con- Bis the \G tinnall proportion there are betwene either of thefe A,B, and unitieC , (0 many Ao2z Bk wvak, sa ep sD Dy vai & ay K36 © wvuitie FF Os DARD: | ‘ Z 4s Foss GOI B94 B64, numbers alfoin continiall proportion hall there fallbetwene AandB. Let D wmeultiply- ing ¥ produce. H-,and let D multiplying A produce K ; and like wife let F multiplying Hi produce L. And for that by {uppafition.as unitie G isto the number TD), foi D to E, there» Sorehow wiairy times tonitie CS meafureth the number Dsfomany timesdith D meafure F. Bet upstie C. meafurethD by thofe vanities which arein D «wherefore DimeafurethE by thafe-unities whicharein Wherefore D multiphine himfelfeproducehE . Againe for thatas vustie Qiste the number D fers Eto A, thereforehow many tines unitie © mer fureth the number D.fo many times E. meafureth A. But unitie C meafurethD, by thofe unities which are in \), therefore E meafureth A by thofe vnities which arein D . Where- fore D multilying E produced A . And by the {ame Peafon F multiplying himfelfe produced Gand multiplying G produced B. And forafmuch as D- multiplying himfelfe produced E,and multiplying F produced H,therefore¢ by the ¥7.0f the feuenth as Disto F5 fois E to H. Ani by the fame réafon as D is to F,fois H to G. Wherefore'as E. is toH , foisH toG. Agayne forafmuch as D multiplying E produced A,And multiplying H produced K, theres fore Gy the17 .of the feutth)as Eis toH, fois AtoK . But as Eis toH, fois D to F,there- fore as Dis toB, fois AtoK . Againe fora{much as D multiplying H produced K,and F multiplying Vi produced 1, therefore (bythe 17. of the:fewenth,) as TXas toF ; fois K to Dy ButasD 1s to B fois Ato K:, wherforeas:A 3s to Ky fo as Ko Es Againe forafmuch atB multiplying 1 produced Land multiphingG produced B, therefore ( by therpoof the fenenth)as Histo Gifois Lite BBitasHisteG fou DoF whereforeas Dis toF fois LtoB. And itis proued thatas Dis to Py fois A. t0K jad ¥. to L: and L toB:Wherfore. the numbers AK, L3B are continiall proportion . Wherefore how mamy numbers. in conti> nuall proportion,fall betwene either of thefe numbers ts Bye onitie@ fo many alfoin cons tinuall proportion fall there betwene the numbers Aand BY whith was required tobe proneds oe ff Lhe 9. T heoreme. Me ~ cd he.11. Propofition. = » Betwene.twofquare numbers.there is one meane proportional numberdAnd = a [quare of Exielidles Elemente’: Fol.207, afquare number tod |quare is trdonble proportion of that which the fide ‘\\ Sof the onte #8 t0 the fide of the other: AP ppofethat there bé two fquare numbers A and B, and let the fide of A be C, & ler she Vide of Bbe D:Then I fay that betwene'thelefgurre numbers A ind B, ) there is one meane proportionall number, and alfothat AisvntoB in double ha! proportion of that which C sto D.LetC multiplieng D produce E. Andforaf- i 7 much as-A is afauare nuber cy the fide thereof. is.C,. .. ay, | = fn . : therfore C multiplieng himfelfe produced A.And by... . Pricnh aden series ) fition demon= she fanze reafon D multipliieng himfelfe produced. B,. 4 Prated. Now forafneuch as C multiplieng C produced A, and D ah ok hae oti multiplieng D produced E,therfore (by the 17.0f the ms i feuenth) asCistoD,fois AtoE, Againeforafmuch = = © "tr as C multipliong D produced E, and D multiplieng himfelfe produced B, therefore thefe two numbers C and D multiplieng one number pamely,D produce E and BW herfore (by the 18. of the fenenth) as C is to D fois E to B. But as Cis to D, fois ALoE. Wherefore as A isto E,fo1s E to B.Wherefore betwene thefe{quare numbers Aand B, thereis one meane proportionallnumber,namely,E. Now alfo l fay that Ais unto B in double proportion of The fecond shat which Cts to D. For fora{much as there are three numbers in continuall proportion, A, part demon E.B, therfore (by the 10. definition of the fift) A is unto B in double proportio of that which _frated. Ais to£. Butas Ais to E, [0isCto D. Wherefore Ais untoB in double proportion of that which the ide Cis vito the fide D : which was required to be proued. q Ube 10. T heoreme. Ihe 12. Propofition. Betwene two cube numbers there are two meane proportional numbers. \eAndthe one cubess-to the other cube in.treble proportion of that which the \oftde' of the oneis tothe fide of the other. Pts 7 ¥ ppofe that there be two cube numbers A and B, and let the fide of AbeC, and let the fide of B be D.Then I {[y that betwene thofe cube numbers A and B, there are two meant proportional numbers,and that A1s unto Bintreble proportion of that : which C is to D.Let C multiplieng himfelfe produce E,and multiplieng D let st produce F, Confirutlions aad let D multiplieng himfelfe produce G. AndletCwultipliong F produce H, and let D — B. peer es * *.@ eee ¢€248 ** 26.02 H es @eseen Peeseweseeese ®@eseeeewevee% peas aee ee : K. ees"eeeeseee8 ‘% *®eee @eaeees eee eeeeeeae ea eee eeed e es. > ° * B eseeeeeeses eeoreersesseaeseees eetos0euveee ba 6 btce Ke eseoaosvepese oops OF om, : ne Sofa 3 \ SN 8 Das... ‘ be XA ' “am ees e é€48 8 6 Tae ; > ot ° ' 7 ee 8 © 86 © B62 ; ue . 2 x! me ; A . ‘ 7 ‘ Goh ; wanitiplieng F prodiceK And, forafmsuch as A isacube namber,and thefidetheroftCs@ snot C roubtiplieng bimfelfe producesh E,therfore C multiplieng E produceth A.And by the fame y is demona reajon ftrated. Thevighthe Boke. realon for thar D multiplieng himfclfe,produced G therfore D multiplieng G produceth B. And forafmuch as C multipheng C and D produced E and F : therfore by the 17.0f the fift, as‘Cisto D,fois E to F.And by the fameveafon alfoas Cis to D.fois F toG. Againe foraf- much asC multiplieng E.and-F produced A and H, therfore as Eis to F fois. toH.But-as EistoF fois C to D. Wherfore as C isto D-fois A to H, Againe foralmuch as eche of thefe eee oe Pv enene a ®*eeesvespeeeset ©e eee ese eaten neae aumbers C and D multiplieng F produced H and K, therfore (by the 18 of the feuenth) as Casto D.fois H to. K .Againe fora{much as D multipheng F and G produced K cy B : ther- fore (by the 17.of the fenenth) as F is to G fois K toB.But as F is to G.fo isC to D, where- fore as Cis to D,fois K to B. And it 1s proued that as Cis to D,fois Ato Hand to K,and K to B : wherfore betwene thefe cube numbers A and B, there are two meane preportionall numbers,thati#sH and k. ~°~* -° = pious fs al{o I fay,that A is unto B in treble proportion of that which C is to D. For foraf- she i ond ere are foure numbers proportional A,H,K 3B; therfore (by the10. definition of emons Y sibee, beginnin rtey ale ae ad V ppofe that there be a multitude of nubers in cotinuall proportia, psa 4 aN 4s Ais to B,fo let B beto C. And let A,B,C, multiplying ech himfelf bring forth ; 7s the nubers D,E,F, cy multiplying the nubers D,E,F, let thé bring forth the nit bers G,H,K.ThEI fay thatD,E,F are in cotinuall proportio, and alfo that.G,H,K are in cotinuall proportia.F or it is manife(t that the niubers D,E,F, are[quare number:, ch that the niibers GJH,K, are cube nisbers. Let A multiplying B produce L.dnd let A & Brnul- tiplying L produce M and N.And againe let Bmu iplying cS produce X: and letBandC ninltiplying X produce Q and Px. Naw by the difcourfe of the propofition going gy Conftrnfiion. Demon fra MOM, 1°: of Euilides Elenientess Fol208. woay prowe thar DT, E, amdaifaCiyNt\ «Wag cli Pies NH ,arein continuall propeatien andin the fame proportion that A istoB: and D4 dikewife that E,X,F und alfo H,O.P.XK,... . Lg are.in continuall proportion and.in the £16 Jame proportion that B 15 t0 C.ButasA x 32 15 to. B,foisBtoC..WhereforeD, L, E, ie rein oneand the fame proportion with 8 i, x n F ana MEDC OUEL G M ? N yd AVE in G . one and the fame proportion. withH. O, | M 16 wi P, K-44 the multitude of thefe numbers NV 32 D,L,E,7s equal to the multitude of thefe HE 64 pumbersE,X, Fiand likewife the multi- 0.128 tude of thefe numbers G, M, N, H, is en P 256 qual to the multitude of thefe numbers KF 512 H,O,P,K wherefore of equality (by the 14.0f the fewenth as D is to E, fois E to FiAnd as G is to H foisH to K-which was required to be preued. | | gq Lhe sz. Theoreme. The 14. Propofition. Ifa fquare n umber meafure a /quare number, the fide alfa of the one [hall measure the fide of the other. And if the fide of the one meafure the ide of the other , the '/quare number a H/o fhall meafure the ‘/quare number, V ppofe that there be two [quare numbers A and B, and let the fides of them be xi Cand D- and let A mealureB. Whereforec alfa fhall meafure D. Let C mal. . RDA ephing D produce E. Wherefore ( by the 17.and 18.if the fenenth, ands ziof The frlt pare ASSAYS the eight) thefe numbers A, KE, Byarein continuall Proportion, and arein the - fi ee Jame proportion that CistaD. And forafmuch as A,E,B,avein continysl] proportion,aind : A meafureih B; therefore ( by the 7 .0f the ccht) A.meafureth E. But as Ais to E, fois Cte ‘D: wherefare C meafureth D. PPV POA ES E oreo eee e see eee eee eosd esenseesses B Be er rent eee 4s 60906 st TURN ede eeT INEST C e®i1@e8 D's Ves * But now (uppofe that the ‘fide C do mealure the fideD: Then t aysthat the [quaresumber Tp, Second ‘A:allo mea{srcth the {quarenumber.B. For the fame order of cinftruttion YEMAYNING,WE part isthe way in like jort proue, that the numbers A,E »B are in continuall proportion ey in the [ame conuerfe of she Proportion, that C is to D. And for that 4sC isto D, fo ISAteoE but & meafureth D: ther hirfte | fore A meafsreth E : and A, BB apeip continual proportion: wherefore A mesfureth B, ij f therefore a fanare number mealures [quare bumber the fide alfo of the one {hall meafure tne fide of the other. Aid if the fide of the one meafitre the fide of the other, the ‘{quare num. ber alfo fhalimealare the [quare number : whith was required to be demonftrated. g Ibe 13. I'heoreme, Ihe 15. Propofition. If acube number medfureacube number, the fide alfo of the one [ball meade Lj. fore 4, ‘4 ” ‘ 1% i i i Se Soa ¢ ng SS z ——~. ——- Po = ‘ a : — = ee ie ee SE A = lal, ) are ~ P ‘+ 82 > he | fare the fel of tbe cbr. Auf the fide oft oe miei tba fide ofthe other, the cube number alfojhgll meafurethe cube numbers so: a of this propo- rhofenumbers E,F ,Gagpagl/o A H,K,B,are in'continuall proportion, cm the [ame pro- fiton. ortion that Cisto D . edi ora{much as A,H;K,B; are in continuall proportion, ana A > ry V2s 4 A eoeeaeeoee et gr? ow *.\ a a. ees ° “&° 2 eeeene#ee . sf ot | aa Ke? = S & svi c= Wo + *& + wh “+ pice oe eae 68 6 WSS 6 8 8 Oe. Oo 8 eo © @ 8 2 24% @ . »~ 47¢ ee | Rea a ee, Sy Gee See L. BS OS Re ee ea SES ee ees & . Shae f yt g " ~ C t WOU S 2 \ 3 \ as e tT) ~ - ._aerh * + O55 % be AS re se a” ey + x. es ee“ evee ere . ++ “ Ls ty esoecvcseonwveeem eee ’ — ¥ ws me , - as ck 3 : . The Pee de Bit now fuppofe that the fide Cdo meafure the fide D: T hen I fay, 7 that the cube number ‘ssthe , wi alfomeafareth thé-chbevuimber BF or the [ame order of conftruction being kept,in like conuer/e of the fortomay we proucsthat ASHK;Barein continua ll proportion, and in the {ame proportson firfte “that Cisto Ds. And forafmuch as G meafureth D, butas Cis to D [ois AtoH; therefore A wien H - wherefore A.alfo meafureth B. If therefore a cube number meafure acabe number, the fide alfo of the es meafure the fide of the other . And if the fide of the one meafure the fide of the other, the cube number alfo {hall meafure the cube number : which was required to be proved. q Ihe 14. T heoreme. The 16. Propofition. - . I fafquare number mea/ure not a [quare number net ther hall -the. ide of uel tt wiendhoone nheaflre the fide-ofithe other And if the fide of the one meafurenot sateen wonpheyide of the other, neitherfhall the: /quare number meafure the quare Fa r-ppole that Aand B betmo [quare “nuaabersandlee. Aa. eesenes proparteon. BS aft he fideof Abe C: and let the fide af. BbeD,...And be 2B wrsssrhycseeeess i ee part poyeeealit that A meafureth not BT. ben Lay, that neither ~ | ts % ie propo- thal C meafuréD.. For if © do meafure D3 then (bythe 14.0f. Cors> the eight) A alfo meafureth B.But Ab uppofition meafureth D.... not B : wherefore meitherdotlG meafureD: 8. A megat: ne The fecond . SS aagany : ‘hoy fecond | But now againe fuppofe that the fideC meafure not the fide D . Then I [ay, shat neither he co- wag, Aa . ; ee. j the Pralithe{quarenumberA meafure thefquare nnrmber-B Forif Adomeafure By we, fir. Sis" i. 9 —— & of Euchides Elementes, Q(by thei4.6f the eight meafure Dx But (by {uppofition) C mcealureth not D , Wherefore a / wetther deh A useafure Bs which wasrequired ta.be proned | q TL heors: I heoreme. [facube number meafirenot acubenumber, neither fhall the fide of the “ont meafuye the (ide of the other, Andif the fide of the one mealure not the fi = 4° ; ‘ , fide of the other neither [hall the cube nuber meafure the cube number. ntl Pafition) A meafureth not Bs wherefore nei- ther hal LC wre Af ure D, But now fuppefethat the fide Cmeafure not the fideD . Then I [ay, that neither hall the a) “4 fae ee of Se Se +h, b s wf cnlonnimcer dd weeufuve the. cubenumber B. for if A domeafure B, then aifo ( by the x5. Gf the eight) fhall G meafure D.But (by [uppo- ~ ta proued. gq The 16. 2 heoreme. Bet wene two like plaine or ‘[uperficiall numbers there is one meane propore tionallnumber. And the one-like plainenumber ts to the other like plaine number in double proportion of that which the fide of like proportion, is to the fide of like proportion. | As 45 one meane proportionall number, and that A 1s unto Bindoxble proporti 0 of that which Cis unto E , or of that which D is vatoF, that is, of thatwhich fide of like proportion intafide of iske proportion . For for that asC is taD fais E to F therefore alternately ( by the 13.0f the fenenth ) as Cis to£,fo is D to Fi. And foralssach as A is a plaine or [uperfi- eich awmaber , and the fides thereof are G and D. : therefore D multiplying C produced A. Axd by the fame realou alfo E multiplying F pr oanceaB. Let D-weult iplying Ep roduce G. And forafeasch as D meultipiyine C produ ced A, and multiplying E produced G , er epies (cy the ¥7.0fthe feuenth)as C is to E,fois Ato G,But as Cis to E,fois Dto F, whereforeas P is to F fois At0G, Again eforafmuch as E multiplying D produced G , and multiphing F pro duced B, therefore(by the 17.0f the fenenth ) as Dis to F , foisGtoB. Butit > aad HK V ppofe that the cube nuomber Ado not meafure the cube number B: and let the fide A M8 tine i hag yf 4 TT ‘ ; OPONtione KN os ¥ of A beC: ana the fide of B beD .ThenI {2y> tpatC hall not meafure D. For if ait cote Pah’ DS +o. cS ph I. * = 7 = . x . | z fitton) Crseafureth not D . Wherefore neither fhall.A meafure B : which was required to be 1a.) ppofe that there be two like plaine or fuperficiall numsbers-A cy B. And let the fides ash of A be the nubers C,D:and the fides of Bhe the numbers E, F.And foralmuch as ete ike plaine numbers are thole which heir lid rtivmall( by the 22.defini- Demonflvac set lake prasne numbers are thojfe which naue their fides proportional by the 22.defini- Demonftres tion of the feuenth) therefore as C is toD,fais E to R.Then I [ay that betwene A andB there tion of chefari part of thas A propofitren, a gh ee Oe eer a ee Gay FERAL Hb + Bane Fiat neces Fol.209¢ Lhe-17. Propofition. The firk part of thes propos . = PR See frisou, ‘ B era eaeevpeneenteraeeanese anon theaaag y's | The fecond C partis the cg * werfectthe — a... A fiji. = — — =S. - -——— — = ee ee Seer et yes : - ee ees : — See ie SS SS wees - - 7 - — ens =" = =<“ er ent tS = =a Nn ee ee * _— * = —— — = _ — no i : —— I he 12. Propofstion. 4 } | i | i i : Wl nt o 7.4. shat = The eighthe Boke shit asD is to F fois A 10.G? wherfore ns Ais 10 G;f0 is Gt0 BWherefore thele numbers A; G,B, arein continuall prorortionsWerefore betwene Aunad B there is one micane proportio= nall number. ae | ) _ seg Now alfo Iyfay that A isUnto B in dooble proportio of that which fideof like proporti- ; fecond part, #5 to fide of like proportion , that is,0 that which C is unto E yor of that which D is unto EiRor feral much as A, G,Byare in continual proportions therefore (by the 10. defiaition of the fift) A 1s. vate Bin dguble proportion of that which A ts unto G.'But as A isto G., fois C te E,and D to F: wherefore is unto.B in double proportion of that which C 4s t0 E,or D to F :whith was required to be demonftrated. Fe ee Ftp by; 35 | PE be 17 Eheoreme.* Th'19:Propofition. °° . ci LAURE THR PAI . ‘St al’ WIA 1. AA Sk FSS RS Hs Ag Betwene two'like folide numbers thére ave two meane proportionall nume! bers. And the one like folide number, 1s tothe other like folide number in’ treble proportion of that whych fide of dike proportion.is.to fsde of. lyke proportion. ; eid V ppofe that there be two like folide ninibers\ Land B. And let the fides of the | aN number A, be the numbers€,D,E. And let the fides of the number B,- be the We" wumbers F,G,H. And foralmuch as (by the 22- definition of the feuenth) lyke eee, ‘ OF APRS tek. Caches, (4 ‘i folie numnbers haue ther fides propertig- Aagh' nall,therfore asCistoD,foisFtoG, and cA as DistoE,foisGto H. Then I fay that = Nv... i yess betwene A and BY the¥eaveiwo meéane ae er . | | ortionall numbers. And that AisuvntoB BB .,..... oe uss nee os wa treble proportion oft thatwhith Cista Fy , OL BNW HSE ‘ NS o7.0f that which Disto.G; or nioreouer of At HISECOT that mpich Bis veto. 00) 5 dig er y9d For let C ntultiplieng’D produce K.And ‘4 Demonfirt- let F meultiplieng G,produce L. And for- tion of thefars afmuch as C,D,arein the felf. "(ame propor. Hato tS Go that FG, are, GrofC & Dis produced \ Prepyeret “Ke and of F and Gis produced L; therefore: © | K and L are like plaine numbers. And ther=~* fore betivene thofe numbers K and L, there is one meane proportional number (by the Sry a 18 .of the feaenth) Let the firme be MWher L erie = — . r& : = —_~s a a : : eo - : a > a : a ee 2 , F ey - -—+ ; - a —" - : . 4 - £ ’ - + = = ~~ 2 ws, ES a - — — . _es ee —— a = : = = = = : : = = na Sa mo ee — me: — = o— = — : . ee RO. Le = : -= a 7S aoa eT A? Se Tee ee . - 2g feats Tw st - . — —s <1 To I er aoe “~~ —~~ = = x a — - = < = = ag = yes = Se eet = z - — — = 7 - ae laa “a ind 1. Lede = : - —-—— —s —— ae = = = - Some . on - -- ~- = = = — ——" a ed c= == —~ a — —_ Se . =" — eee ee = ~ = a = —_ ts - . ~_ — _ = Ss - — 2 s —————— —— as nat ~ 7 at .- Pare - ~ _ ~2 und ~ . - - a as ital SSS = = = = = ¥. ~ - ~~ ——o—~ - - -—- = - + ——— = — - ue - = = = =< = - + ———- = Se Sa ee - a+ ee eines aia Pe entre See —— = —= — — v= ~ — 2 ~ = ee : 2 Ed a EE Kk sama = = = — — ae = - = z- = = a i = == = = = = SE === \¢ he. Sa == ee ore =: : = ————<—_ i a = —— = = = aa —— - a eS SSS ——_- === —— —= <== = ; ee — = = hy: — - . = _ —— ——— = —— = =. = = eS ITS —— - Ne —————— SS ———_ = rn ——-~ ———~ - -— — — — - ————$—— _—— — = ——— — —— - — —_-—— — + a «4 Shs Late ei BOT sr fam ahs MEER = 2s ag Sie US tie eee —— ~~ ee ne = : ——- — == — _ — es a = E : == ae ae ——- : ‘ ——— ——— : : = — - 2 ——- ~~ —-—— —— ——_——— er f : oS: Sm 0k er EO ieee An ——— + - a I — =<. — - - =< Se iia aa ~ nee pew es ee et canoe a nS ———— =~ —— ——— ——-- = EE re ee — . . : - = = — ————— —- arn - = — ’ =- > ae == = ~ —- -- > + — ee a rn — ee — - - — + “ ee ane ——_ = . x - . oN —— == ; _ — = 4 —- ==* cone wid neil ~ — = — —_—_ Rees a = all 2 ; ae - Wee “a ~ - _ ~ = . c# eS Ss = = ” r y ial aon -~ ~ es ns a . : - = Sut . - - — - ~ ta oe = he Siw . - 2 > - -——_—---— —e— 2% 2 rit sa a ar oerees ~~ - . = = — ST a eS aa inne Fe —_ — — \ — ~ Vannes = “ ~ inh, Gere. 2c ES ee So a a > ——--- ==: —s aa a = F< TS SEES TEST MES - = —s se . ' { of Euchdes Elementes. Fo/.210, duceth B. And forafmuch as E multiplieng K produced A,and multiplieng M produced N, therfore (by the 17.0f the feutth), asK isto M fois AtoN.Butas KistoM ,[0is C to F ee Did Gand moreoher Eto H stherforeas C is to Fyand D to G, and Eto H, fois AtoN. A. gayne,forafmuch as E mult iplieng M produced Nand H multiplieng M, produced X, ther. fore (by. the 18 of the fenenih) as Eis to H fois N toX But as E isto HfoisC to F, and D MGR herfore as C isto F and BD toG na Eto H fois Ato Nyand NIoX, Againe foraf- mitch as H multiplieng M, proanced X and multiplieng L produced B,therefore ( by the x7. of the feuenth las Mista L . [ 01; X.to B.But as M is te L.fo isC to F,and DtoG na EtoH therfore asCisto Fjand D toG and E.to H fo 8 not onely X to Bibut alfo Ato N, and N to XP he? fore thefe numbers A ofV 4 Bare in continuall proportion, and that in the proportt- ons of the fides.I [ay moreoucr that_A is unto Bin treble proportion of that,which fide of luke proportion ,ts to fide of like proportion,that is, of that which the number C hath to the num. ber F or of that which D hath to-Gyor moreouer of that which E hath toH. For foralmuch as there are foure nuvabers in continual proportion,thatis, AN,X,B, therfore (by the 10.de. finition of tie fift) A is unto Bin treble proportion of that which A is vato N. But as A isto V,f07s it proued that C is to Fyand PD toG 4nd moreouer E to H. Wherefore Ais untoB in treble proportion of that which fide of like proportion is unto fide of like proportion,that is of ree Sa, that wid; A tisnpwready 7 VIsSG Servi muer C isto the number F and ofthat which D is t0G, and moreouer of woe: - Po pe ‘ that whichE ito A: Which was reg nired to be proned, q The 18. Theoreme. The 20. Propofition. If betwene two numbers there be one meane proportionall number : thofe numbers are like plaine numbers. eh nt GAY ppofe that betwene the two numbers A and B there be one meane proportionall 4 f LON f vaeal . fp? , ; IAS"! number, and let the fame be C.Then I ay, that thofe numbers A and B are like | ‘ \ ~ ; AY | S\N" plaine numbers . Take (by the 35.0f the feuenth) two of the least numbers that } Melee haue one cy the [ame proportion with A,C,B : and let the fame be the numbers D, EW ‘her ef ore as DistoE£, fo is A to C, but as Ais to G, fo 1s C to B, wherefore as D ts to E, fois C to B .Wherefore how many times D meafureth A, fo many times doth E mealure C. How many times D meafureth A » [0 many vnities let there bein F. Wherefore F multi- pHing D produceth A, and multiplying E it produceth C : wherefore Ais a plainenumber: A. and the fides therof are D and F( by the 17.de- ~ fauition of the fenenth) . Againe forafmuch as pa ee ae PAARL D and Earethe left wumbers that haue one cy the [ame proportion with C »Btherefore(by the D 21.of the feuenth) how many times D meafu. E reth C, {0 many times doth E meafure B. How often E meafureth B, fo many vanities let there F bein G . Wherefore E meafureth B by thofev. uities whith are in G : wherefore G multiply. ng E producethB : wherefore B isa plaine number ( by the 17. definition of the feuenth), And the fides thereof are E and G . V4 herefore thofe two numbers A and B are tp plaine numbers .1 fay moreouer that they are like . For foralmuch as F multiplying E produced Cs and G multiplying E produced B: therefore (by the x 7.0f the feuenth) as F is to G, foisCte B, but as Casto Bs fois D to E, wherefore as D is to E, fois F to. Wherefore A and Bare eB like plaine numbers, for their Sides are proportional ; which was required to be prone. Yy.ty. — og The a eeeees The fecond parte Thes propoft- 140 ts the cone nerfe of thei8. propofitione ConStrnbion. Demonfira- tion, T be eighthe Booke qT he 19. T heoreme. Lhe 21. Propofition. If betwene tit0 numbers sthere be two meane proportionall numbers, thofe numbers are tke folide numbers. This propofi- = ON AY, ppofe that betwene two numbers A and B, there be two meane proportional tion ts the con- numbersC,D.T hen I fay that A and B are like folide numbers.T ake(by the 35 werfeofthelge }| ‘\ ¢8"| of the fesenth or 2.0f the eight) three of the least numbers that haue one and | propofition. tener Ka the [ameproportion with A,C,D,B and let the [ame be E,F ,G. Wherefore (by | ___ the 3.uf the eight) thir extremes E,G are prime the oneto the other. _And forafmuch as Consivutlion. borwene the numbers & and G there is one meane proportionall number : therfore( by the 20 of the eight) they are ike plaine numbers.S ale fh the fides of E,be H and K. Andlet the fides of G,be L ani M.Now tt s manifest that the{e numbers E,F,G, are ta continual ceeoeoeveoeeoemenveoeoneweeoweevneaewaereoeeanse oe ee 8 St a Sg ag Oe 8,6 0 e OOH, 0) 0S Me OTO-0 @ 0 0 6 0. 6e ee 0.6 * 4 Che oe oe ee tr tPPeoeaeces#e @2# 8 @ Rie proportion,and in the{ame proportion that H is to L,and that K isto M.And ‘fora{much as ts "é- "E.F,G are the least mmbers that haue one and the ame proportion with A,C,D, therefore : of equalitie(by the 14, of the fenenth)as Eis to G,fois Ato D. But E,G,are( by the 3. of the eight) prime number.yea they are prime and the leaft,but the leait numbers (by the 21. of the fenenth) meafurethofe numbers that haue one & the fame proportion with them equal- ly the greater the gret ter and the le(fe the leffe,that isthe antecedent the antecedent, cy the confequent the confecnent: therfore howmany times E meafureth A,fo many times G mea- fureth D. How manytimes E meafureth A,fo many vnties let there be in N. Wherefore N multiplieng E ,produeth A.But Ets produced of the numbers H,K. Wherfore N multiplis eng ae which is produced of H,K, produceth A. Wherefore A ts a folide number, and the fides therof are H,K,N. Agayne,fora{much as E,F,G,are the least numbers that haue one and the {ame proportion with C,D,B, therefore how. many times E meafureth C, [omany times Gmeafureth B How of tétimes G meafureth B,fo many nities let there be in X .Wher- : fore G meafureth B bi thofe unities which are in X.Wherfore X multiplieng G producethB. But G is produced of the numbers L,M. Wherefore X multiplieng that number which is pro» duced of Land M,priduceth B.Wherfore B is a folide number,and the fides therof are L,M x.Wherfore A,B are‘olide numbers.L fay moreouer that they are like folide numbers. For forafmuch as NandX multiplieng E produced A and C: therfore by the 18 .of the feuenth, as -- > of Enclides Elementes. Fol.211, as Nis toX, [ois AtoC,that is Eto F.But as E istoF fois H toLand K to M : therefore asHistoL,fois K to M,and NtoX. AndH,K,N,arethe fides of A, and Lkewife L,M,X, ae the fides of B: wherfore A,B,arelike folide numbers : which was required to be proued. ¢ The 20. T heoreme. The 22. Propofition. If three mumnbers bem continuall proportion , and if the furft be a fquare number the third al/o [hall be a fquarenumber. h EN. V ppofe that there be three numbers in continuall proportion A, B,C, and let the UN a9, firft bea [quare number . T hen fay that Se Dimon lira VK Ne the thirdis alfo.a [auare number . For for- tiow, afmuch as betwene A and C there is one meane pro- Bosse ess Laanasss : portionall number namely B,therefore( by the 20.0f © salesmen ainiin 2 e the eight) Aand C are like playne numbers. But Aisa Square number. Wherefore C alfa is a {quare number: which was required to be proued, q Ihe 21. Theoreme. The 23. Propofition. Tf foure numbers bein continuall proportion, and if the first be a cube nie ber the fourth alfo phall be a cube number. es V ppofe that there be foure numbersin A i ey; continuall proportion A,B;C,D. And» B......... 22 Demon fira- Ace . = ie vant} let Abe a cube number: Ther a) FeRe S Sas LOR $n, D alfois a cube number. For forafmuch as be. D..... We eo tae he we ESSN ; twene A and D there are two mezne proportio- wall numbers BLC.T herfore A,D are like folide numbers (by the 21.0f this booke) But Ais a cube number ,wherfore D alfois acube number : which was required to be demonftrated. q: Lhe 22..T heoreme. The 24. Proposition. If two numbers be in the fame proportié that a /quare number is to a fquare number and if the firft be a fquare number ,the Second alfo fhall be a fquare number. en) V ppofe that two numbers A and B be inthe fame proportion, that the {quare CON x number Cis unto the ,, fquare nuber DAnd~ A... ; DK let A be a fqnare mile 022. ox pane ber. Then I fay that B alfoisa BB ..4s.-... = Square number. For forafnuch © Cc. .ceve.ss dance. a5 C and D are {guare numbers. Rr fe Se T herfore C avd.D are like PLONE .. - Dcvoen te ves RRS TOR ee numbers. Wherfore (by the 18. 0 the eight )betwene C and D there is one meane proportionall number But as Cis to Dfois A to B.Wherfore betwene A and B there is one meane proportional number (by the 8. of the eight) But A is a fquare number Wherfore( by the 22 of uke eight) B allo isa {quare number whith mastequired to be proued. Ty. siij. T he | T he eighthe Booke qT he 23. I heoreme. Lhe 25. Propofition. Iftwo numbers be in the fame proportion the one to the other, that a cube number is to a cube number and if the firft be a cube number, the fecond ae fo fhall be a cube number. | ppofe that two numbers A and B be in the [ame proportio the one to the other, that the cube niber C is unto the cube number D. And let A be acube number. Ry. «| Then I fay that B alfois a cube niber.F or fora{much as C,D, are cube nibers, SAX therfore C,D are like folide numbers, wherfore( by the 19 .of the eight) betwene i, ae Ce pega E e280 0¢8 2808 | . b x Demon fira- biO Re F eaeece ee Ceoeoseanees eee Cand D there are two proportionall numbers.But how many numbers fall in continual pr- portion betwene C and D,fo many (by the’8 .of the eight fal there betwene the numbers that hane the {ame proportion with them. Wherefore betwene A and B there are two meane pro- portionall numbers which let be E and F And forafmuchas there are foure numbers in con tinwall proportion namely,_A,E,F ,B,and Ais a cube number, therefore( by the 23 .0f the eight )B alfois a cube number : which was required to be demonftrated. | + A Corollary added by Fluffates. | Betwene a {quare number and a number thatis not afquare number, falleth not the proportion of one {quare number to an other.For if the firft be a fquare number, the fecond alfo fhould be a fquare number which is contrary tothe fuppofition. Likewile betwene acube number,anda number that is nocube number falleth not the propor- tion ofone cube number to an other.For ifthe firft be a cube number, the fecond alo fhould bea cube number,which is contrary tothe fuppofition, & therfore impoffibls. A Corollary added by FiufSates, q Thez4. beoreme. The 26. Propofition. Like playne numbers are in the fame proportion the one to the other, thit a {quare number is to a {quare number. =A V'ppofe that A and B be like plaine numbers . Then I fay that Aisunto Bin the [ame proportio that a [quare number is toa {quare number.F or -forafmuch as AB, Confirn fin. NP ave like plaine numbers, there- fore betwene A and B there falleth one... A «.sa+ers reese meane proportional nuber(by the 18. of. © sseessseeees EGEEEAT On the eight) Let there fal[uch anumber, .. Bssnvevcccscrsocorererccnvovoeses and let the fame be C.And(by the 35.0f ny the feuenth ) take the three leaft nums <% ayes 3 5 a bers that haue one and the fame propor- are Ye tion with AC, B, and let the fame be : reeaenaevesee*782 8089 D,E,F, = x of EuchdessElementess \: Fol.2 1% DE, F :wherefore ( bythe corollary ofthe 2 OF. the eel) their ether es, that is D, Fy are Sayare numbers.And for that as D is to F fois Ato B, (by the 14.0f the feueth): and D, F, nfquaremuacers L berpere A ts vigte B in that proportion,that a [quare puber is unio a pnw wliraberimbichwvasrepuined tabepromeds:::: scsi), ; : [2 3 OF Lisup Lensom 5 si sual, pea §, de peorense.. ; AL he,27.Propofition... Like [olide niimbers drein the [ame proportion the one tothe other, that a cube number is to a cube number. ENG V ppofe that a Aand B be like folide numbers «Then l fay that Ais unto B,in the NES» [ame proportion, that a cube numbe is to to acube number For forafmuch as A,B, NG are like folide numsbers.T herefore( by the 19 .of the eight) betwene A and B there ‘ . on Pa ; , A : a Se eoeeepeseveeeeepneeaees % 2. 8% .¢ 6 8 O16 0.8 8-0 6. ¥.e\.6. 016.9: 6.6.2 6.6.6.0 6% fal two meane proportional numbers. Let there fallimo {uch numbers, and let the fame be Ciamd D . And take (by the 35.0f the [ewenth )the least ntimabers that hane one and the (ame proortion with A,C,D,B, and equal alfa with thé in multitude, and let the [ame be E,F, G,7.Wherfore( by the corollary of the 2.0f the eiv/t)theix extreames, that is,EH, are cube nunbers. But as E isto Hfois A to BWherefore:Ais vito B in the Jame proportion, that a cule number isto a cube number :whith was required to be demonstrated. ¢ A Corollary added by Flufates. : Tf two manbers be inthe fame proportion the one to the other that a |quare nunaber is to a {guare Constra tien, Demonfire- tion. number : thefé two numbers {hall be like fuperficiall numbers. And if they be in the fame propore AC orollary tron the one to the other that acube number isto acube number, they (hall be like folidenibers, added by Hirft let thenumber A haue ynto the number B the fame proportion,that the — number C hath to tic {quare number D. Then I fay,that A and Bare like fuperficiall nfibers. For forafmuch as betwene the quare numbers € and D there falleth a meane proportionall (by the 11.0f this booke) there fhall COO PORTO O PAPO ETOH EOE OR SETH EOE HECE HOSES EE HERES ESOS SBOE EECS alfobetwene A and B (which haue oneand the fame proportion with C and D) falla meane proportio- nall(by the 8.of this booke). Wherefore A and Bare like fuperficiall nfibers(by the 20.0f this booke). Fut if A be vnto B,as the cube number C, is to the cnbe number-D.Then are A & B like folide num- ber; . For forafmuch as C and D are cube numbers, there falleth betwene them two meane proportio- POC EPO eed gs ee esees reese? nallnumbers (by the 12.0f this booke). And therefore (by the 8.of the fame) betwene A andB (which are n the fame proportion that C is to D) there falleth alfo two meane proportionall numbers. Wher- fore(by the ar. of this booke) A and B are like folide numbers, ~s % An FeufSates. The.eighthe Booke ss. An other Corollary added.alfo by Fluffates.. Ifa number multiplying afquare number produce not a [quare number: the fayd anmber aultiplying Pall be - C 26 + Sr aavzber . Si if Seen bea fyuaré number, then fhould itand the number mulated being oAinother Com jixe fuperficiall numbers (by reafon they are {quarénambers:) haue.a meane proportionall ( by thes. vollary added of thys booke') . And the number produced of the fayd meane fhould be equall to the number by FinfSates. contayned vnder the extremes, whi¢h,are fquare numbers ( by the 20. of the feuenth). Wherefore'the number produced of the extremes being equal ro the fyuare num- ber produced of the meane, fhould be a {quare number. But the fayd number a fuppofition ,:is no fquare number . » Wherefore neither is the “number multiplying the fquare sumber,a{quare number. ‘The firft part of the firft Corollary is the conuerfe of the 26.Propofition . of this booke, and hath fome yfeinthe tenth booke , The fecond . part of the fame alfo is the contuerfe of the 27. Propofi- 3 tion of the fame.” Sa) The end ifthe eighth booke _ of Euclides Elementes. oe, “a ry 4 BA shits — Fol.2. @ 1 heninthbookeetBy- 7 clides Elemefites;> "8s | £058 SI NTH LS NINT Mp0 0 K A) Ewchdecontinneth his } Wo Purpofe touching numbers: partlyproféchting thynges tiomin +4\ hiore fully ;whichwtse before fomewhat (pokenofsasof The i4prmnb¥s ee Rt ae fquareandicubé numbers : and partly fetti: Vi Mima turesand proprietics of fuch kindes of number, as haue booke, i Stig 4 , SKE aS : ah HBR 4rel re Ey OSS pafirons and conditi6ns are in this booke largely tancht, with their compofitions, arid fabdtQions of théone froth q The 1. T heoveme. Lhe-1. Propofition. Tf two like plaine numbers multiplying the one the other produce. any nume bers the number of them produced fhall be a | /quarenumber: | | And fora{much as A,B) arelike plaine numbers, therefore ( by a | the 18: of theteight ) berwene A and B there falleth ameane om proportional number ; But if beiwene two numbers fall num- bers in continuall proportion, how ma- : ; ny numbers fall betwene the i many. Ae... adfo..( 2. the’..of bhe eight) { VALLLDENE-. ‘serena full betmene the nunphers that Laue the \ Bacar se evn afquare number. Wherfore(by the2z2. C ..... Paks CRASS Wee RR ERE oe vs oie of the eight) C alfois a {quare number: which was required to be proned. | = gq Ihe 2. Theoreme. The 2. Propofition. If two numbers multiplying the one the other produce a /quare-number: thofe numbers are like plaine numbers. > ww nee yt Suppofe This propof- Brot ss the com a rie of the former. Demon ra- a AA Corollary a dedby Campane, Denionflrae 830% T he eighthe Booke Ovens Vppofe that two numbers A os... x Aand B agers UIGIeZTs KXOSSAi onethe other ao produce B .....+4+- Ki SONS C 4 (quare number. T. hep. I fay,that A and Bare like plainenum- D ......- ‘endiasn's bers. For let A multiplying bimfelfe sss se eeee EST yi ieee produce D . Wherefore D. is & fyMare GC vesvieeineerecrenseessnecedt ees ceva number . And fora{muchas A mitkti- plying bins [elfe produced D, and multiplying B produced C, therefore (by the 17. of the fe- teri) aA is to B, fois Dt0C. And fora{much as D ts a {quare number, and fo likewifeis C thévefore D and C are like plaine numbers .Wherefore betwene D and C there is ( by the ‘48 of the ight) one meane proportional number. But as D isto C, fois Ato B. Wherefore (by the S.uf the exght) betwene A and B there is one mseane proportionall number. But if be- ‘tmene tivo. nunsbers there be one meane proportionall numsber, thofe sumbers are (by the 20. of ohecight) like plaime numbers Wherefore A and B are lake plaine sambers : whith was required to be proued. as x. A Corollary added by Cawepane. Hereby it is manifest, that twofguare numbers multiplyed the one inte the other do alwayes produce a fquare number. “For they are like fuperficiallna bers, and therefore the nuin- ber produced of them, ts (by the firkt of this booke)a fquare number .. But 4/guare num- “ber malriplyed into a number not {quare, produceth a number not fquare. For if they fhould pro- duce afjuare number, they thould be like fuperficiall numbers (by this Propofition) . But they are not. Wherefore they produce a number not fquare. But if 4 (quare num- ber wultiplyed into an other number producea{guare num ber that other number fhall be a fquare number... For by this Propofition that other number is like vnto the fquare number whith nultiplyeth it, and thereforeisafquarenumber. Bue if a {quare number multiply- bd int an oi ber number produce anumber not [guare, neither foal that orber number alfo be a {quare umber. For ifit hould be a {quare number, then being multiplyed into the {quare muimberitihould produce a (quare nuniber,by the firft part of this Corollary. Lhe 3. T heoreme..... The 3. Propofstion. If. acube number multiplying himfelfe produce anumber the number pros GY | ber B.Then I fay that Bis «cube number.T ake the fide of A,and let the fame be ADE" the number C,and let C multiplieng himfelfe roduce the number D. Now st 1 ple! manife/t that C multiplieng D produceth A(by the 2.0 definition of she feweth) ie === ¥ ppofe that A being cube number multiplieng himfelfe, do produce the num. fo \ ' 1 PACES EE th et ee aie Se eee rPrrrre rete seer yl UU @eeenvneveeaeve eo aeeeeneerereeeeen eee e@eeeeorvevenoeoeaoe 2@ Me Few as 45 C5 Vauitie.. Aad forafmuch as C multiplieng himfelfe produced D therfore € msta[wreth D by shafe ev. 310 L masses Ld 7 a ad - —_—— ao ee . of Euclides Blementesy Fol.2t4.. » : nities mbich arein CBut unitie alfa menfireth Cby thofe unities which are in CU ‘her fore as vnitieis.to.C,fo isC to.D Againe forafmuchas.G multiplienge D proauceth A: therefore D meafureth A by thofe unities which are in C. But vnitie meafureth C by thofe vnities which arein C: wherefore as vnitie is to, C, [ais D to.A. But as vnitieis to C,foisC to D, wherfore as vnitte is toC foisCtoD & Dto A Wherefore betwene vnitie cy _A there are i0 Ineae provortionall numbers namel )0,D. Againe forafmuchas A mul tiplieng him- Jelfe produced B, therefore A vaeafureth B by thofe nities which arein.A.But vnitie alfo waenfureth A by thofevnities which are in AWh erforeas unitieis to.A [0 is Ato B,But be- tipene A and unitie;there are two meane proportionall numbers Wherfore betwene A-and Halfothereare two meane proportionall numbers by the 8. of the eight: But if betwene two numbers, there be two meane proportional numbers, and if the firft bea cube number, the fourth alfo fhall be a cube numbers 4. the 21 .0f the eight. But Ais a cube number wherefore B alfoisa cube number: which was required to be proued. q Ihe 4. I heoreme. The 4. Propofition. Tfa cube. number muftiplieng a cube number, produce any number , the number produced {hall be acube number: iV ppofe thatthe cube number A multiplieng the cube number Bdo produce the nit- i ber-C.T hen 1 fay that Cis a cube number. For let-A multiplieng him[elfe produce , LD rm Vherefore D isa cube number ( by the propofition going before) . And jo ‘afmuch ast muttiplieng | himfelfe produced Ds and mul-) Av. cscv, ' D 64 upline Bit:produced C : ther- wisteialk dd de He 96 nap ¢ bins td; panitned : fore (dy. the x7. of the feuenth) eee eet ere 144 as A 1s to Bais DtoGe And Bowes ceis..: Soralmuch as A and B are cube ae numbers, therfore A and Bare like folide numbers Wherfore betwéne A and B(by the x 9.0f thesight there are two meane proportional numbers Wherefore alfo(by the 8 .of the fame) betwene D and C there are two meane proportionall numbers.But Dis acube number Wher fore C alfo ts acube number (by the 23.0f the eight) which was required to be demonftrated. q The s: Theoreme. Ihe s.Propofition. Tf cube number multiplying any number produce a cube nuber : the nume ber multiplyed is a cube number: ees @eees ee RBar eaeanteeae C 216 Vppofethat the cube number A, multiplying the number B,do produce a cube number wamely,C. Then t fay, that Bis a cubenumber . For let A multiplying (| hing [elfe produce D Wherefore ( by the 3 .of the ninth) Dis a cube niuber. And =e SM forafmuch as‘A multiplying him [elfe produced D,and multiplying Bit produ- bed Cs therefore (by ther 7 -of the fenenth) as ‘Ais to B, fois DtoC. And foralmuch as D : . = —- @*eeees#es o- 0 Se ee vee ES 6 Saree dhe Pes ee 0 bee ee eee and C ure cube numbers, they are alfo like [olide nitbers. Wherefore (by the 19. of the eight) betwee D and C there are two meane proportionall numbers . But as D is toC, fois A to B. AA.i: Wher ° Demonftra- bon. Demon trae biGite 4 - thks ia LS a4 {4 ie) hy - nee —— — ——_——eo— eS ee oe ee aS S=. ~ oe x a mo he — ef Corollary added by Campane. Demon itra- S40 Ne Demonitra- Stor 6 T heninth Booke ; Wherefore (by the 8 .of the eight ) betwene A and B there ave two per Se ser be RUMB- bers . But Ais acube number .Wherefore B alfois a cube number (by the 23. of the eight) < which was required to be promed. « A Corollary added by Campane. Hereby is is manifest, that sf a cubenumber multiply a number not cube,tt fhall produce anumber not cube. Forifit fhould produceacube number, then the number multiplyed thould alfo be acube number(by this Propofition) which is contrary to the fuppofition . For it 1s fuppofed to beno cube num ber. endif acubenumber multiplying a number produce a number not cube, the number multiplyed fhall be no cnbe number. For if the number multi- plied fhould be a cube number,the number produced fhould alfo be acubenumber(by the 4.0fthis booke) : which is contrary to the fuppofition, andimpoffible, ¢ The 6. I heoreme. The 6.Propofition. | Ifa number multiplieng him ifelfe produce a cube number: then is that nume ber alfoa cube number. gq The 7. I beoreme. The 7. Propofition. Ifa compofed nnmber multiplieng any number produce a number:the nite bey producea ball be a folide number. kA ppofe that the compofed number A multiplieng the nuber B,do produce the num. BR: ber C.T hen I fay that C is a folide number.For forafmuch as Ais 4 compofed nis- ; a4 4 ber,therfore {ome number meafureth it (by the 14.definition). Let D mea [ure tke i ope Mke< e%% ee 54h 6 46 eR Aaa aa SN ee . BD useacees And how often D.meaf{ureth A fo many unities let there be in E. Wherefore E multiplieng D produceth A.And forafmucn as two numbers D and E,multiplieng themfelues, produce A, which A againe multuplieng B produceth C : therfore C produced of three numbers mul- tivliene the one the other.namcly,D,E,and B ts (by the 18. definition of the feuenth )a fo. lide number. And the fides ther of are the numbers D,E;B. Iftherefores compofed numsber exc, which was required to be prowed qi hes. I heoreme. T he 8. Propofition. If from ‘vnitie there be numbers in continuall proportion how many foeuer: the third number from bnitie is a fquare number, and fo are all forwarde leaning one betwene, And the fourth number ts 4 cube number, and foare all farward leaning two betwene. And the fenenth is both a cube er Ane of Euchdes Elementes,\~ Fol.215, and alfo a {qguare number and fo are allforward leaning fine betwene. ot sa V ppofe that from onitie Bere be the{enumbers in continual proportion A,B, ks) © the fourth number is acube number,and [aare all forwarde leauyne tivo be-. = hee alte twene. And that F thefenenth mumiber is both a cube number and alfo a {quare number,and fff pare i : 4 Yt by thofe vnitieswhichare in A,wherefore A meafureth B by thofe unities which are in A. And foralmuch as-A-meafureth-B by thofe unities which arein A. ‘herfore A multiplieng himfelfe produceth B Wherfore B is a{quare number. And forafmuch as thefe numbersB,\ V nitie | Now alfol fay that the fourth number from vnitie,that is,C,is-a cube number,and fe are all forward leaning two betwene.For for that as vnitie is tothe number A, [0isB to C, The Second therefore how many times Unitie meafureth the number A, fo many times B meafureth C. part demone But unitic meafureth A by thofe vnities which arein _A, wherfore B meafureth C by thofe firated, dnities whithareta A.Wherfore.A multiplieng B produteth C. And foralmuch as A multi- plieng himfelfe produced Band multipleng Bit produced C; therefore Cis a cube number. And fora{meuch as GD,E5F are in.comtinuall proportion.But Cis acabe number therefore 9 the 23. of the eight) F alfoisacube number... Rae », ,,Andit is proued, that F being the fenenth number from vnitie is al{o.a{quare number. Demonstratio Wherfore Fis both cube number and alfo a [quaré number.In Re may we proue,that of the third eating alwvaies fie betwene,all the reft forwarde, are numbers both cube and alfo (quare: parte which was required to be proued. : 3th : NG The. 9. Theoreme... \ The 9. Propofition. | If from wmutiebe numbers in continuall proportion how many foeuer : and if that number “which followeth next after ynitie be a [quare number then _ all the reft following alfo be [qnare numbers, And if that number ‘which dy d Oloweth next after bnitie bea cube number , then all the reft following hall be cube numbers. = WOO V ppofe that from unitie there be thefe numbers in continuall proportion A,B, . Ce C,D,E,F . And let Awhich followeth next vato unitie be a [quare-number. > % Then I [aj that all the reft following alfo are {quare numbers ya hat the third Demon'tra- SOSA number namely ,B,is a {quare number, [oall forward leaning one betwene,it ton ofthe firft 1s plaine by the Propofition next going before.l [ay al[o that all the reft are \quarenumbers. Fatt of this For for a{mu ch as A,B,C are in continuall proportion, and Aisa [quare number, therfore propofition, (by the 22. of the eight) C alfois a [quave number. Againe fora{much as B,C,D,are in con- et AA. y. tinuall = ‘= tinuall proportion,and Bisa [qnare num- ber,therfore D alfo(by the 22.0f theeight) isa fquare number . In like fort may we prone; that all the reft are [quare numbers. But now [uppofe that A bea cube num- ber .T hen fa y, that all the reft following are cubenumbers T hat the fourth from v- nitie, that is, Cis a cube number,and fo all forward leauing two betwene ,st 45 plaine (by the Propofition going before) . Now I {ay,that all the reft alfo are cube numbers. For, for that as vnitie is to A, fois AtoB: therefore how many times vnitie meafu- reth A, fo many times_A meafureth B. But unitie meafareth A by thofe unities which arein A. Wherefore A allo mea{ureth B by thofe unities which are in A .Wherefore A T he ninth Booke Cubes. - i —————__ / 262144. 32768 Bat ./ 4096 multiplying him felfe produceth B. But Ais a cube number . But if a cube number multiply. ing him felfe produce any number, the number produced, is (by the 3. of the ninth) a cube number. Wherefore B is a cube number . And fora[much as there are foure numbers in con- tinuall proportion A,B, C, D, and Aisa cube number , therefore D alfa (by the 23. of the eight) 15 a cube number . And by the famereafon E alfo'is a cube number, and in like fort areall the res following : which was required to be proued ; The 10. Propofition. ¥ The 10. I’ heoreme. If from bnitie be numbers in continuall proportion how many foener, and if that. number which followeth next after ynitie be not a fquare nume ber; then. isnone of the reft following a fquare number excepting the third from bnitie, and fo all forward leaning one betwene. And if that number which followeth next after bnitie be not a cube number, neither 1s any of . the reft following a cube number, excepting the fourth from ‘bnitie, and fo 4 all forward leaning two betwene. | | ; ze —_ 2g-a1¢. ¥ ppofe that from unitiebe thefe numbers in continuall proporti on A,B,C,D,E,F. ete y dnd let A which followeth next after unitie be nofquare sumber.T hen I fay,that Demonftratt- + WHORE wezther is sf of the reff afquare number excepting the third from unttity® foall sfermard leauing one betwene,namely;B,D,F which are[quare numbers bythe 8.of this “Viitie. aD Sreces CPAs eve ceenneer nes Ode ceo rose eer Ose Par et SFE AGOREKHAP STS SSESEOTASTMAHATKS SECC TO Fe F729 firit part lea- “Gooke) . For if tt be posable, let C be a {quare number. But B alfois a [quare number. Wher- Sore Bis unto C in that proportion that a [quare number is toa ‘{quare number . But as B ss = ee ee of Euclhides Elementes. .. Fol.216, 66 C; fois Sto B Wherefore Ais unto B inthat proportion that afquarée number is toa Square number. But Bisa fquare number. Wherefore A alfoisa Square number (by the 24. of the eight) : which is contrary to the fuppofition. Wherefore Cis nota {quare number. And by the (ame reafon none of all the other ts 2 {quare number, excepting the third from vnitie, ana foall forward leauing.one betwene, But nom fuppofe that Abe novacube number .T hen 1 fay, that none of all thé vest isa cube number excepting the fourth from vunities: foal forward leaning two betwenepame- ly,C,ana F, which ¢ by the 8 of this buoke) are ctbe numbers. For of ite pofsble, let D be a cubenumber . But Caifais d cubeiumber (oy. the 8:0f theninth). For itis the fourth from viitie . But as C15 to.D, fois Beto. Wherefore Bis untoC, in that proportion thata tube number istoa cube dumber. But C isa cube number, Wherefore B alfois acube number (by the 25.0f the eight). And as-unitieisto A, [ois AtoB. But vnitie meafureth A by thofe vanities which arein A Wherefore A meafureth B by thofe unities which arein A Wicerfore A mustiplying him felfe productth B a cuke number. But af anumber multiplying hiss Selfe produce a cube number then is that number alfo acube number ( by the 6.of the ninth); Wherefore A is acube number : which is contrary to the luppofition . Wherefore D js Hot a re cube number . In like fort tay We broue, that neither is any of the ref? a cube number excep ~ ting the fourth from vaitie, and [eal forward leaning two betwene : which was required to be proued ° ¢ The rr. Theoreme. The tr. Propofition. If from vnitie be numbers in continuall proportion how many foener, the lefse meafureth the greater by fome one of them which are before in the jaid proportional numbers, yaa V ppofe that from vnitie A be thefe numbers.in continual proportion B,C,D,E. EN 8 T hen I fay that of thefe numbers B,C,D,E:B being the le(Se,meafureth E the eren therfore how man y tines Unitie_A meafureth B,fo weany times D meafurethE :wherefore EB Fes ete by the 15.0f the fenenth how many times uni Bic 105,03 tie A meafureth the number D » fa iwany times B mea. Gai Jurcth E. But unitie A meafuret D by thofe unitie. Bx ] which avein D. Wherefore B alfo meafureth E by thofe A, viities which arein D. Wherefore B the leffe, ste . veth E the greater by Jome. one of the numbers which went before E in the proportionall Rumbers. And fo likewife may we proue that B meafureth D by fome one of the numbers B; Sears by C._ And [oof the ref. If: therfore from vunitie rc. Which was required te C prowea. | | ms G9 The 12. Theoreme. Lhe 12. Propofition. If from vnitie be numbers in continual proportion how many foeuer, how any prime numbers meafure the lea$t fo many allo fhal mealure the nume ber which followeth next after bnitie. | | AA if. Sappofe . Demon rans tion of the fe- cond part leae ding al/o to an abfurdities Denson fra~ 20M. Demon fraz- tion leaiimg to an abjurdiise. An other de- mon[iratio af= ber Fiufsatese The eivhthe Booke aay V’ppofe that from vnitie be thefe numbers in continual proportion A,B,C,D. The con gl fay that how many prime nibers meafure D,fomany alls do meafure A.Suppofe liseae that [ome prime number namely, E,do menfure D.Thé 1 fay that E alfomea{ureth A,whichis next unto unitie.For if Edo not meafure A,and Eis a prime number but eue- ry number isto enery number which tt meafureth not a prime number (by the 31. of the fe- nenth). Wherefore Aand E are prime numbers the onetothe-other. And fora{much as. E meafurith D,letit meafure Dby the number F. Wherefore E multiplieng F produceth D. A caine foralmuch as.A mea{ureth D by thofevnities which are in C,therefore A multipli- eng C produceth D.ButE alfo multipliene F produced D, wherforethat whichis produced of the numbers A,C is equal to that which is produced of the numbers E,F.Wherfore as A istoE ,f0 is\F to C.But A,E jare prime numbers,yeathey are prime and the leaft.But the left numbers meafure the numbers that haue one and the fame proportion with them equally by the 21.0 the fenenth,namely the antecedent the antecedent, and the confequent the confe- quent Viher fore E meafureth C.Let it meafure it by G. Wherefore E multiplieng G produ ceth Cut Aalfo multipliene B produceth C. Wherfore that which is produced of the num- ’% Cc eeesooseeseeneaneee2eneee@ eeoavneoenpeeoesepeenesBeoaaes ’ CPR OS Pee Sea eee bers A,B,1s equall to that which is produced of the numbers E,G.Wherfore as\A1sto E, 0is G to B.But A,E are prime numbers,yea they are prime and the leaft. But the leaft num- bers (by the 21.0f the fenenth) meafure the numbers that haue one and the fame proportion with th? equally namely the anteceact the antecedét,¢> the cofequet the confeqet.Wherfore EB meafureth B. Let it meafure tt by H. Wherefore E multiplieng H produceth B.But A alfo multipheng him{elfe produceth B,wherfore that which is produced of the numbers E,H, is J _ equall o that which ts produced of the number A.Wherfore as Eisto A,fois AtoH.But A E are prime nitbers,yea thep-are prime co the leaft,but the least numbers( by the 21 .of the fe- uenth meafure the numbers that haueone and the {ame proportion with the equally name- ly, the intecedet the antecedent and the cofequent the confequent Wherfore E meafureth A and it ufo doth not meafurett by fuppofition,whien is impofsible.Wherf ore Aand E arenot rime the one to the other, wherforerhey are compofed But all compo fed numbers are meal u~ Fedof {une prime number,wherfore A and E are meafured by [ome prime number.And for afucuckas£ is {uppofed to be a prime number. But a prime number 1s not (by the definition ) mecafired by any other number but of him felfe Wherfore E menfureth A and E,wherfore E meafurth A,and it alfo meafureth D Wherfore E meafureth thefe numbers A and D.And in like jort may we prove that how many prime numbers meafure D, foman) alfo fhall mea- fore 4: which was required to be proued 3. An other more briefe demonftration after F/u fates. ris i that from vnitie benabers in cOtinuall proportion how many fo ever namely,A,B,C,D. And letfome prime naber,namely, £ meafure the lait ntiber which isD .Thé I fay thar thefame Emea fureth A which is the next number vnto vaitic. For ifEdoonot meafure A, then are they prime, the one te.tie other by the 31 of the feucuth . And forafmuch as A,D,C,D,are proportional from agg eI- of Euclides Elementes. Fol.119, therefore A multiplying him{elfe produceth B. WherforeB and E ate piimes\eWnitie numbers ( by the 27.0f the feuenth )’, And forafmuch as A ae a B A 4 3 procuceth C, therefore C is to Ealfoa prime number by the 26, of the fe B36 9 nenth.Andlikewifeinfimrely A multiplying C produceth D: wherefore D C 6 27 and E are prime numbers the one to the other (by the fame 26. of the fes © OD oes6 St uenth) . Wherefore E meafureth not D as it was {uppofed, which is abfurd, BEB $ 3 wherefore the prime number E meafureth A, whiche is nexte ynto vnitie: which was required to be proued. qf Ube 13. Theoreme. Ihe 13. Propofition. If from bnitie be numbers in continuall proportion how many foener , and af that which followeth next after vnitie be a prime number: then {hall no other number meafure the greate/t number, but thofe onely -which are bes fore in the fayd proportionall numbers. ROS Vppofethat from vnitie be thefe numbers in continuall proportion AB;C,D, 1% K? Nand let that which followeth next after vunitie, that is, A, bea prime sumber. eS SX} T hen fay, that no other number befides thefe numbers A,B,C, mea ifureth the ILS ereate/t number of them whichis D . For if it be pofsble,let E meafureD.And bet E be none of thee numbers A,B,C,D . Nowitis mantfeft that E is not a prime sumber. For if E be a prime number edo alfo meafure D, it fhail likewife meafure A being prime numcer and not being one and the fame with A, by the former Propofition : which is im- pofzble . Wherefore E is nota prime number. Wherefore it 1s a compofed number. Lut euery compofed number (by the 33.of the fewenth) is meafured by [ome prime number. D eeceeceeeeégescesd PHSSOHESHOSHOSHEFESEHe DHHS ES HSHHTESE SERS SSEOOSOAOCOSOSSECEESES S eer eeBe ee? eee Vuitie . ‘Now I fay, that wo other prime nisber befides A fhall meafure E. For if any other prime nuber do meafure E, ey E meafureth D, therfore that number alfo hall Shile- D (by the s.common fentence of the feuenth) Wherfore it fhal ao meafure A (by the propofition next going before) being a prime number and not being one and the fame with. A: whith is im- pofiole . Wherefore onely the prime number 4 meafureth E which meafureth thegreateft number. i. < And foralmuch as E meafureth D, let it meafure it by F . Now I fay, that F isnone of thefe numbers A,B,C . F or if F be owe and the fame with any of thefe numbers A, 1,C, and it meafureth D by E, therefore one of thefe numbers A,B,C, meafureth D by E. But one of thefe numbers A,B,C, meafureth D by fome one of thefe numbers A,B »C, thereforeE is one and the {ame with one of thefe numbers A,B,C : which is contrary to the [uppofitios. Where fore F is not one and the fame with any of thefe numbers A,B,C. _ lualike fort may we proue, that onely the prime number A meafureth F, proning frft that | | AA jij, = = FS Demon flyaa ton leading to an abf{urditics i} A re i) , ay Bt thoy) ME t | ; 7 } ant ae 4) ' 4 i | 1 iby 2 | | as + : Th ae ly mh i | : " al inf hal t | a . 4 ) 4 ae it) Hi He} i) oy } Hh eh ti >. i} 7 th : Ll ? t i Ki, i\) | | - 1) i} : inet ie h pial aie NaS ‘ - ,, a] ‘ By) Hh Ml: a tl! ghia uM i aly eh ‘ neers n Se Tt] : fia 2 Neer a aE ——— ‘ — ———— ee ~— a ~ = ee ae An other de- monfiratio af~ ger Campane, T be ninth Booke F is not a prime number : For if F be a prime number, and it meafureth D, therefore it alfo medfureth A being a prime number and not being one and the [ame with A, by the former Propofition : which is impofiable . Wherefore F is not a prime number: wherefore it is a com- ‘sa number, and therefore [ome prime number hall meafure it . Now I fay, that’ no other prime number befides A {hall meafure 2 F or tf any other prime number do meafure F, and F meafureth D, therefore that number [hall meafure D (by the 5.common fentence of the [e- aenth ). Wherefore it fhall alfo meafure A (by the former Propofition ) being a prime num- ber and not beiag one and the [ame with A : which isimpofible . Wherefore onely the prime number A meafureth F . And fora{much as E meaf{ureth D by F, therefore E multiplying F produceth D . But A alfo multiplying C produceth D , therefore that which is produced D SHCSCSSSLSSSCSSSCHSE SSCS SSHSSSHESCAHEGESHEEG CEES @eeeenn seeeneseansartereseceea esacaecereaeneest eenneeen . tC seweeeesvepeeoeneeweereesaenkntenaeet ee | oe) pare Vuttie . E H G So oes va ‘ of AintoC, is equall to that which is produced of E into F .Wherfore proportionally as Ass to E, fois FtoC.. But A meafureth E . Wherefore F meafureth C. Let F meafure C by G. And in like fort may we proue, that G is not one and the fame with any of thefe numbers A, B,C, and that Gis acompofednumber, and alfo that onely the prime number A meafureth it . And forafmuch as F meafureth C by G,therefore G multiplying F produced C. But A al- fomultiphing B produced C . Wherefore that which is produced of A into B, is equall to that which is produced of F into G . Wherefore proportionally as\_Ais to F ,fois GtoB. But A mealureth F .Wherefore G al[o meafureth B . Let G meafure B by H. Now inlike fort as be- fore may we prone, that H is not one and the fame with A,and that H is a compofed number, and mea{ured onely of the prime number A. And fora{much as G meafureth B by thofe uni- ties which are in H, therfore G multiplying H produced B . But A multiplying him felfe pro- duced B. Wherfore that which is produced of H into Gis equall to the y at number which as produced of A.Wherefore as Histe.A, fois AtoG. But A mealureth G.Wherfore H mea fureth A being a prime number and not being one and the fame with it : which is abfurde. Wherfore no other number befides thefe numbers A,B;C, mealureth the greate/t number D? which was required to be acmeonflrated. err & An other demonftration of the fame after Campane. Suppofe that E not being one and the fame with the numbers.A, B, C,D doo meafure the num- ber Di And lerit meafure it by the number F. And'forafmuch as #being a-prime number meafureth — thenumberD,which is produced of E into F : therefore by the 32. of thefeuéch,A meafureth either or F.Leritmeafure E. Now-forafmuchas D is produced of into C,and alfo of E into F: therefore by the fecond part of the ro.of the feuenth, isto E,as Fisto C.But4 meafureth E: wherefore F meafu- eth C.Letit meafire it by G: Wherefore by the 92. of the feuenth « fhall meafurecither F or G . Let it meafure F. Wherefore as before by the fecond part of therg.of the feuenth G fhall meafure B. Leris, meafure it by He Now then as before it foiloweth by the 32. of the feuenth that «4 fhall meafure either G or H: fuppofe that it meafuré G-Wherefore by the fecond part of the 20.0f the feuenth H fhall mea-~ fare 4(forof 4 into himfelfeis produced By aiid-F H into G alfo is produced B.) If therefore Hbe =a - | | equ of Euclides Elementes..- Fol.218, equall ynto 4,4 fhall be no prime number. Which is contrary to the fuppofition.2ut if it be equal vie to.4,then euery one of thefe numbers G,F,E,fhall be fome one of the numbers'4, 8, C, D, by the x2. Vuitie . A e “See 2 C See eee ee ee ee D Ceceect ees MHSeHoerseeeeseetoverses Beesses CPP PHORHOCSH OSH ADSG EES Os OH OCEeD ES bEeSheRE KE | ee G SaaS H ropofition of the ninth repeted as often asneede requireth . Wherefore Eis noe 2 number diver rom them, but is one and the fame with fome one of them: which is contrary to the fuppofition, wher fore that is manifelt which was required to be proued. | q The 14. I heoreme. | The 14. Propofition. If there be genen the lea/t number whom certayne prime numbers ceuen, f a> do menfure: 10 other prime number ‘hall meafure that nuber, befides thofe prime numbers genen. > ~~ vein Y ppofe that the leaft number whom the[e prime numbers B,C,D,do meafure,be A. IESE T hen I fay that no other prime number bolide B,C,D,meafureth A. For if it be ea pofsible,let E bein C4 prime number méeafure A,and let E be none of thefe numbers B,C,D. \ And forafmuch as E meafu- | | ae | , : Demoufra- veth A,let it meafure it by F. Wherfore Se ee ee ee sion leading to E muktipliene F productth 4 tad Bo po eifurtieres A.Eut if two numbers multipliene the onethe other produce any number. And if [ome prime nimber metfire that which is produced, it {hall alfo meafure | one of thofe numbers which were put at = the beginning (by the 3 2.0f the [enenth) Wherfore thofe numbers B,C,D, meafure one of thee numbers E or P.But they mteafure not E for E 13.4 prime number, and is not one and the fame with any one of the[e numbers B,C. D Whesfer they mes[ure F being leffe then A which is impofible. For Ais [4ppofed to be the leaft whom B,C,D, meafure. Wherefore 0 prime number befides B, spe fie il A. whith was required to be demonftrated. Lt B thefe prime numbers B,C, D,meafure C .., Py E F % A ptopofition added by Campane. Af there be humbers ho many foeuer in continual proportion being the leaft in that proportion: annnber meafiring one of then , thalt be a number not prime'to one of the two leaft numbers in that proportton. SS eS = - = - ~ 7 saat resent: <8 Se — ae ct = —= Sig == == = _ + = . — = . = = a ‘ z — et a = E ~ RN ase =~ - . _— — —_ — ae - = = —<—_- =e = ee . festy* = : - - — —— - = = ; " = —~ aoe - F > ee eS Se = = oe < See : Bd eae ent esi " Creer earn 5 Shaded ry aon a“ = — = — -- ———— . ~ = — we Se ———— == = == = 2 Pt Ee ot ~ 5 oe — canes —~ = eee ——~ oi > ~~ Se = — 3 =a eee eS ee a ie = ace ess = : pe Rr a a es a ED > aes 5 EATS Se PACE EN ee OM dS ae eae = : Dis numbers geuen443ByG3 +> ' eee Neag D,E,Now itis manifett Es) (by the demonttration of the fecond of the eight) that F multiplyed by P,Q,and R produceth K,L, Mr and thar F multiplyed by K,L,M,N produceth 4,2,C,D.4nd forafmuch as H meafureth C: therfore Hiseitherte F ot toM not prime( by the corollary of the 32.of the feuenth added by Capane.jIf it be not-prime-vnto Fi thé is that manifeft which was required to be proued. But if H be not_prime ynto M.Thé fhal it nor be prime either to F orto R(by the fame corollary ). Ifagayne it be not prime vnto F, then is thatproued which was required. Zutif it benot prime ynto R,then(by the fame corollary )ihall icbea number not prime ynto. G( which produceth R by the z.ofthe eight) but G is one ofthe two leaitnumbers For G which are in the proportion of the numbers geuen at the beginning 4,B,.C, D, E .1ftherefore there be aume bers how many foeuer.&c.which was required to be proued. q Lhe 13. I heorene. The 15. Propofttion. Tf three numbers in continuall proportion be tthe leat of allnumbers that haue ong and the fame proportionwith them : enery two. of them added toe gether fhall be prime to the third. LAS TV ppale that there be three numbers in continual! proportion A,B,C, being the left RA Sa49 ¥ of all numbers that haue oneand the fame proportian with them. Then J fay, that OBS etery two of thefe numbers A,B;C,added together, are prime to the third : name- * st «gs @ & fe Now it is manifelt (by the fayd 35.Propofition that D E multiplying him felfe produced A, and multiplying E F produced B, and moreouer EF multiplying hunefelfe produced C.., And fora{much as D E and E F are theleaft in that proportion, they areal{o prime the one to the other (by the 24.0f the feuenth).But WER? Bree Ae af two numbers be prime the one tothe other, — A, wivn area dn: Buranionarcsess then both of thé added together ,fhall. be preme: a de eteeelb ls eh cnctae to either of them (bythe 30, of thefeuenth). Dvvo. Bsevee F Wherefore the whole number D. Fis prime to 22S either of the[enitbers D EG EF. But DE allo is primevato E F.WherforeDF,GD E are prime unto E F Wherfore that which is produced of DF into DE, is(by the 26.07. the fe ueth) prime vito EF But tf two nubers be prime the one to the other ,that which is produced of the one of thé into him(elfe,is prime tothe other (by the 27. of the feweth). Wherfore that which is produced of D F into DE, 1s prime to that whichis produced of E F into himfelfe. But that which is produced of F D into D E, is the (quare niaber whichis produced of D E into himfelfe tocether with that which ts produced of DE into EF( by the 3 .of the [econd). Wherfore the (quare nuber which is produced of DE together with that which is produced. of E into E F 1s prime to that which is produced of EF into himfelf. But that which is produ ced of DE intohimfelfe,is the number As¢e that whichis produced of D E intoE Fys the number B: and thatwhich is produced of EF tnto him/elfe,is the number C. Wherefore the numbers A,B addeg together are prime unto C. ? ‘ Be ese Bn ( | y the . - —- a ‘ ~ of Euchdes Elementes. Fol.219. By the like demonftration alfo AY EB Faw ac sa nneealtetCimes since... cc proue, thatthe numbers. B,C, are prime unto 8 Hee she number A. Now alfo I [ay,that the numbers A,C » Are prime unto the number B, For forafiauchas D.F is prime to either of thefeD Band EF : therefore that which is A ........ CO ccnatee een produced of DF into him felf;ss prime to that B which is produced of DE into EF. Butthat oD... EL ..isF which is produced of D F into him felfe, is e- guall to the {guare numbers which are produced of D-E and E F together with that numbep which is produced of D E into E F, twife (by the 4.0f the fecond). Wherefore the fauare nits bers which are produced of D E and E F together with that which is produced of D E into EP twije ave prime to that which is produced of D E into E F. And y disifion alfo(by the 30.0f tue fenenth) the {quare numbers produced of DE and E F,together with that which 45 produced of D E into E F once are prime to that which is produced of D F into E F'. A. gaine (by the {ame 30.0f the feuenth) the [quare nubers produced of D E and E F are prime to that which is produced of D E into E F. But that which is produced of D E into him felfe 88 A, and that which is produced of E F intv him felfeisC, and that which is produced of D Etnto E F,is B. Wherefore the numbers A,C, added together are prime vato the nuim- ber B : which was required to be demon firated. This latter part of the demonftration which prouveth thatthe numbers A‘& C are prime vato B, is fomewhat obfcurely put of Theon. And therefore I will here make it layner. " Forafeiich as either of the numbers D E,and E F is prime to the whole DF: (as hath before bene tied) therefore that which is produced of D £ into E F(which is the number B ) is prime vnto D F, y the 26.of the feusuth Wherefore by the 27.0f the fame that which is produced of D F into himfelf (whichis the number compofed of AandC and of the double of B by the4.of the fecond ) thall be prime ynto B. Wherefore it followeth that the number compofed of A and C is prime ynto B.For ifa number compofed of twonumbers,be prime toone of the faid two numibers,as here the number com pofed of A and C taken as one number and of the double of B, is prime vnto the double of B-then the two numbers whereof the number is compofed, namely , the number compofed of A and ©, andthe double of B fhall be prime the one tothe other(by the 30 of the feuerith).. And therefore the number compofed of A and C fhall be prinre to B taken once.For if any number fhould meafure the two num- bers,namely the number compofed of A and C,and the number B, it Should alfo meafure the number compofed of A. and C,and the double of B (by the s. common fentence of the feuenth) : whichis not poflible;for that they are protied to be prime numbers, Here hauel added.an other demonftration of the former Propofition after Cam- pane, whichproueth that in nabers how many focuer, whichisthere proued onely tou- ching threenumbers : and the demon {tration feemeth fomwhat more perfpicons then T heons demonftration. And thus he putteth the propofition, Lf numbers how many focuer being in continuall proportion be theleaft that baue one & the fame proportion With them. enery one of them fhalbe to the number compofed of the ret prime, Suppofe that there be numbers jn continuall prepertion how many foeuer, and the leaf in their proportion: namely, 4,.B,C,D.Then I fay that cuery one of them, as for example firftD , is prime to the number com pofed of the reft,namely of 4,B;C.Fot if it be not,let fome number,namely E mea fure Djand the number compofed of 4,8, C, Take the two leaftnumbers in the fame proportion thae 4,8,C, Dare ( by the 35 .of the feuenth: ) which let 5 be F,G.4nd forafinuch as E meafurethone of thefe ; L number 4,B,C, D, the fame E thalbea number not ¥ — RB: prime either to F orto G(by the propofition before ‘7 + added by Campane after the 14, propofition }wher- G Cr fore fome number fhall meafure E and one of thefe ce k sumbers ForG: which ler be H: and forafinuch 3¢ ea Das H meafareth E,iefhallalfo meafure Dwhich num= | er D the number E alfo meafureth ( by thes. come Ricieneny ABC 38 son fentence of the feuenth), Moreouer forafinuch H ABDa ig (by fuppofition ) meafureth one of thefe num~ ig = Demon§tratie cn t0 prone that the nume bers A and ¢ are prime te Be Demonfiras tion leading to aii abfurdite, Demon {tra- 210 Me T he.ninth Booke bers F or G,the fame H fha!i meafure all the meanes betwene.A and D by.the fame cémon fentéce.For either of thefe numbers F or G producethall the meanes by the next numbers in continual! proporti- on and in the fame proportion with them (as by L,1,K) by thefecond of the eight . Agayne foraimuch, as H meafureth E,which(by fuppofition )meafureth the whole A, B, C: the fame H thall alfo meafure, the whole A,B, C(by the forelfayd common fentence) and it meafureth the part taken away , namely, the meanes B,C (as it hath bere proued)wherefore it alfo meafureth the refidue A(by the 4.comnion fentence of the fcuenth) wherefore H meafureth the extreames Dand A\, whiclare prime the one to the other(by the 3 .of the eight) which were abfurd.. Wherefore Disanumber primetothe number compofed of the relt,jamely,of A,B,C. Secondly I-fay that this is fo if-euery one of them:namely that C is a prime number to the num-- ber compofed of A,B,D.Fox ifnot;then as Before let E meafure C,and the num ber compofed of A,B, D: which E fhalbe a number not prime either to F or to G ( by.the former propofition added by Cam- pane )wherefore let H meafure them.And forafmuch as H meafureth £,1t {hall alfo meafure the whole A,B,C,;DwhomE mea2fureth « And forafmuch as H meafureth one ofthefe numbers F or G, it fhall meafure one-of the extreamés A or D: whichvare produced of B or G(by the fecond of the eight) if they be multip! edinto the meanesL or K.And moreouer the fame H {hall meafure the meames, B C ( by, thes.commen fentence of the feyenth )whenas by fuppofition itmeafureth either F or G,which mea- {ure B,C( by the fecond of the eight). But thefameH meafureth the whole A,B,C, D as we haue pro- ved,for that it meafureth E. Wherefore it fhallalfo meafure the refidue,namely,the number compofed of the extreames A and D (by the 4.commonfentence of the feuenth ): And it meafureth one of thefe AorD (for it meafureth one of thefe F or.G which produce A and D.) whereforethefameH fhall meature one ofthefe A-or D and alfo the other of them (by the formercommon fentence )which num bers\A-and:D aré by the 3.of the eight prime the one to the other. Which were abfurd . This may alfo ‘be proued in enery one ofthefénumbers A;B,C,D . Whereforeno number fhall meafure one of thefe numbers A.B,C, D and the number compofed of the reft. Wherefore they are prime the one to the o- ther: If therefore numbers how many foeuer:€¢c: which was required to.be proued. — Hereas.I promifed, Lhane added Campanes demonftrations.of thofe Propofitions in numbers, which Excidein the fecond booke demonitratedin lines . And thatin thys place fomuch the rather, for that T heow as we fee in the demonftration of the 15. Pro- pofition feemeth to.aHedge the 3.& 4.Propofition of the fecond boke: which although they concerne lines onely,yet as we there declared and: proued, are they truealfoin numbers. | @ The firft Propofition added by Campane. That nunsber whichis produced cf the multiplication of one number into numbers bow many fo- ener: is equall to that number Which is produced of the multiplication of the fame number inte thé number compofed of them. This proueth that in numbers which the firft-of the fecond proued. couching lines . Suppofe hat the number A being multiplyed into the number B,and into the number C, and into the number D, doo produce the numbers E, Fand G Then I fay thatthe number produced of A multiplyed into the number compofed of B,Cjand Dis equalito the number compofed of E,F,and G. For by the con- uerfe of the definition of anumber multiplyed 5 what part vnitie is of A’, the felfe fame partis B of E; and C of F,and alfo Dof G. Wherefore | by the 5.of the feuenth what part vnitie mvs is of A , the felfe fame partis thenum- ~“°B ... Ret. D ber compofed of B,C,and D,ofthenum ) rene ts J ber compofed of E,F,and.G. Wherfore.. -- 2255 Oe ee ene bersey, by the definition that which is produ- | ced of A into the number compofed of | ee ee G covceroees B,C,D, is equall to the number compo- penne fed of E, F, G : which was required to 3 giawlee ¢ obiesioh Sees oes be préued, x The fecond’ Propofition. T hat number which is prodaced of the multiplication of unmbers how many focuer into one nit ) bers is equall to that number which is produced of the multiplication of the number compofed of them intothe fame number. en i Thys ww of Euclides Elementets Fol.220. This is the conuerfe of the former,As if the numbers B and C,and D multiplyed into the numbef A doo produce the numbers E and Fand Then the number compofed of B,C,D, mul-_ iB ..., SE Sr tiplyed into the number ~ fhall produce the Aris sumbercompofediof the numbersE,F;G. E..... oi Ce eben pe GT apres cas Which thing is eafly proued by the 16.0f the feuenth and by the former propofition. q The third Propofition. T hat niimber which is produced of the multiplication of numbers how many foener into other siumbers how many foeuer, is equallto that number which is produced of the multiplication of the number compofed of thofe firft numbers, into the number conspofed of thefe latter numbers. As ifthe numbers A,B,C doo multiply the numbers D,E,F,ech one eche other , and if the num- bers produced be added together. Then I fay that the number compofed ofthe numbers produced is equallto A .. Boevd ae the number produced of the number ee aes of the Big te? EE Saves oe ; numbers A, B, C into thenumbercompofed of the num- bers D,E,F . For by the former propofitio that which is produced ofthe number compofed of A,B,C into D is equall to that whichis pendvesd of cuery one of the fayd numbers into D : and by the fame reafon that which is produced of the aumber compofed of A,B,C into E,is equal to that which is pro- duced ofeuery one of the fayd numbers into E:and fo likewife that which is produced of the number compofed of A,B,C into F is = to that which is produced of euery one of the fayd numbers into F. But by the firft of thefe propofitions that which is produced of the number compofed of thefe num bers A,B,C into euery one of thefe numbers D, E,F is equall to that which is produced of the number compofed into the number compofed: wherefore thatis manifeft which was required to be proued. ¢] The fourth Propofition. Tf anumber be deuided into partes how many foener: that nuber whichis produced of the Whole suto him felfe, 1s equall to that uumber Which is produced of the fame number sto all his partes. »: 9. This proueth in numbers that which the fecond of the fecond proued inlines, Asifthe number A,be deuided into the numbers B and C, and D . Then I fay that that which is produced of 4 into himfelfe , is e- Bi eci-ws ieeie quall to that which is produced of 4 into allthefaydnum B., © eas D cece bers B, C, and D. For putting the number E equall to the Bissssevene number A , itis manifeft by ‘the firft of thefe propofitions that chat whichis produced of E into A, is equall to that which is produced of E into all the partes of ABur by the common fentence that which is produced of E into A is equal'to that which is produced Of A into himfelfe; and that which is produced of Einto the partes. of A is equall to that whichis pro« Muced of A into the felfe fame partes. Wherefore that is manifeft which was required to be proued, q The fift Propofition. Ifa number bedeuided into twa partes.that Which ss produced of the whole inte one of the partes; 1s equallto that.Wwhich is produced of the felfe fame part into him felfe,and sto the other part. * This proueth in numbers that which in the 3.of the fecond was proued in lines. For let the num- ber A be deuided into the numbers B and C . Then I fay that that which is ps Re of A into C, is equall to that which is produced of C into him- AES.3 19 felfeand into B. For by the 16.of the feuenth; that which isproduced of A. ~B’..... C... into C is equall to that which is produced of C into A. Now then put the D ies number Dequall tothe number C . Wherefore that whichis produced of ~*~ A into Cis equall to that which is produced of D into A. Butby the firft of thefe propofitions that which is produced of D into A is equall to that which is produced of D into B and of D into C. Wher- fore foraimich as thatwhich is produced of D into A'and into Band into C is equall to that which is produced of C into A.and into B,and into himfelfe , by reafon of the equalitic of C and D; thatis ma- nifeft which Was required to be pro ued. sib BB,j. q The This propoff- Ston ts the ¢o< nerfeofthe — fermer, Demonftra- 540%, This anf wee rcthtothez. of the fecond, Demonfira- 240% This anfwee reth to the 3, of the thirdes 40% a ~ 4 . a ~ . iD 4 me) py \ i 1B 1 : e) i ne } : | f 4 | } “ ze ' ) } } Re } oe) 4 Mt) Lay it } ii wea ia H) Hi mt it We ae en on t : } t} ; oie it \ { ae ' . ws Y 7 | a ’ J in r H ; Pie + j 4 { Y ih! i ie wy i 2) ee 4 ay a it 1h) EE eR wi 0 IBV f ue Y ey iheie ih Mal ee ih ; yeu ie i] \ | 1 i f }| ia eB A ‘ " } 4 } S Wh ; an - lhe i + i] 4 Wl : } } } | ; | 6 ; i yy A Be | , : aay H WT | . ia) eh 4 4th Hee " \ Mid moe | ya: " We i Hine 4 vi HP Gee Be - {i H : hil Bt i ' i i ' 4, rhe i hy " a) | | | 4 H It 14 +H | ') { Ue ea ih ui l nl deli f) it tf yon i 4 i are f it ' Wa + : 1 f a Ae 4 } i ih { at 1 ae Pet) Wh ae 7 } ! f a eh! Ai Dae Lay I : y wi ie Ti nig’ i) : f Sa iy P if ti aah We i ee ; ia} | } | ! Hey’ nike ! ui Hi 1) { a i 7 int : : i i : eet ht hi i ae ; 1H) tf iw {i * ' cey id i 4] Wy oe UGS a 915 GEE } ie ; u rey) th SH ! ay f bh 4 4 ; h 4 ' Ly haat! ' : | i it | i ten.0 we Vieie 1 ti ae { Te. 7 pl! 7 4 14 | : Y Di hh! ih ] bee > 3 sei r i ea : : H ~Th : (BTteee vi } H by } ii) Hitt } 7 a ' Sieh) H j } SPRAYS Be | AE V : ae Tae ee BY ‘ft at ai Ay iH : Ni mee oe } oy hi ii : } ‘ ; a\- ee oe ves ii + : { | ' 4 a | é } Pan ak my Oe bi} Caan tit v tom ft 8) thy | Q i} eat ao : 7} ri Wee 7) ANB aE ni A Bia : ee hy | H Nig v? mt : RT Reet i ! 4 tf x 3 | i ae wy ve : i ; ee) 0 ite Hf ; ‘ i] y ‘ Hh tt i Whe ; ' te " ta ae « } th 7 i it wali YA AY te Kt i eed) by +. i } i ” 5 | } 4 | f f y f } 1 ] : } ye | } hottie t TE i i 14a : i : t t | 'é : " | 1 |i é et 4 i} iv : Wit 7 7 | ne J ie hb - | » 1 - : Pui ia d Hin f y { Wt ine. wrLe § ' Ni] if é | uit} , , #4it " qi) > 4 " . eon : Vari , Fas ; { i z i ; HE , ) ) H Pas a i Wt! f { ; NW He Abs )| 1; us 7 pi ve Ip ai | j i} ur, Dhan ae i 1A Ae a ae , " ; i ’ i} at Br , | Bit t i] ! : ’ ‘ hy ’ ACEH Qi > A © bw ni * ey 4 a d !. " Tain H 1 4, on i mh ee Hh th ee ui, a), : ay ety } i : iim Pe half . teat ed ¥ } :? z Vk Wt {4 my Mi , ; CLE | \ i" i Dany bt t 4 \ ' ‘ aM f 7 HN “ay M4 7 Aant se : i] er iit t ‘ ale i} iPr Ry ' 1 Hy} in : AAAS ni ee rhe } ae mit iy , if : a >i Be : : } i : 5 : } H Ni B Pct i This anf we- reth tothe 4. of the fecond» Demonfira- $t0ne Thts anfivee xeth tothe §. of the fecona, Demenftra@0. LEH e This anfwee veth tothe 6. of the fecond. Denon fra. sions so 9 T be ninth Booke @ The fixt Propofition. If anumber be deu ided into twa partes: that which is produced of the multiplication of the whole anto bins felfe, is equallto that which is produckd of the multiplication of either of the partes inte him felfe, and of the oneinto the other twife. | This proueth in numbers that which thefourth of the fecond proued touching lines. Asif the umber A be deuided into the numbers B and C. Then fay that that which is produced of A into himfelfis equal tothat whichis - » se produced of B into himfelfe,and of C into himfelfe , and of B into Rae bites, © ewife.For by the4 of thefe propofitions,that which is produced of A into himfelfe, is equall to thae whichis produced of A into B , and into.C.But that whichis produced of A intoB., is.equall to that which is produced of B into himfelfe and into C ( by the former propofition). And by the fame that which is pieces of A into C is equall to that which is produced of C into himfelfe and into B . And forafinuch as that which is produced of C into B is equall to that which is produced of B into C by the 16.0f the feuenth, it is manifeft that thatis true which was required to be proued, ¢; The feuenth Propofition. if a number be denuided into two equall partes ee into two unequal partes : that whichis produ- ced of the greater of the vnequall partes into the leffe, together With the fquare niber of the nume ber fet betwene, ss equal tothe {quare number produced of the halfe of the whole. <6 phis prouéth in numbers that which thes. ofthe fecond proued in lines. As if the number A Phe dénided into two equall num bers,which let be A Cand C B:and alfoin two vnequal numbers namely,A D and DB,of whichlet AD bethe greater, and A C m DB the leffe. Then I fay that that which is produced of the - . “~"*" * Pte -- B whole A D into DB together with the {quare number of CD , 18 equall to the aye number of CB. For by the former propofitio the fquare of C B is equall to the f{quare of CD and to the {quare. of DB, and to that which is produced of B Dinto C Dtwife. But that which is produced of 2 D into himfelfe and inte C D is equall to that which is produced of BD into C B by the firft of thefe prs 2 oe therefore vnto that which is produced of 2 D into AC. Wherefore that which is pro uced of B Dinto himfelfe and irito GD twife is equall to that which is produced of BD intoA D. Wherefore by the fame the fquare of C 8 exceedeth that which is produced of B Dinto AD by the fquare of C D: whet- fore that is manifeft which was required to be proued. q The 8.Propofition. Uf anutaber be'devided into two eqnall partes, andif unto st be added an other number : that which is produced of the multiplication of rhe whole being etad into the number added, to@ * gether With the /quare of the halfe, 1s equall to the fquare of the number compofed of the halfe andthe number added. | | This proueth in numbers that which the 6. of the fecond proued touching lines. For fuppofe thag dhe wimber 4 B be deitidedinto equall numbers , which let be #¢ and CB: and ynto it adde the ‘number 8D. Then I fay; that that which is produced of the whole . doonein numbersFor fuppofe thar there be a number wharfoeuer namely 4B. Thea Lay, that ic propofitior — isimpoflible to denide isin uch fortas is required in the propofition.. For fo fhouldit be deuided ace cording to a proportion hauingameane and two ex- | > as - Wes, BB . q A eee ree ofan tas Hi Be 1 aap B treames.But if1t be pofiible, let the number AB befo | Demon/ird= deuided in C.Ancdas ABisto BC,fo let B.C be to.C 4.Wherefore AC fhall beleffe chen CB . Now oj08 leading 0 then takeaway from C Banumber equall.to.A C which ler be C D, And forafmuchas the proportion an impofsizi- of the whole AB to the whole BC xis astheproportion ofthe part taken away from A B, namely,B G bitte. to the part taken away from B C,namely,G Dstherefore the proportion of the refidue of A B, namely, A C,to the refidue of B. C,namely,to B D,.1s as. the proportion of the whole A B;to-the whole B: C.( by the 11-of the feuenth ). Wherefore B Cis to C D,asC Dis to DB. Wherfore CDis greater then DB. Wherefore fubtrahing D E ourof C D,fo that let D E be equallto DB : the proportion of BC toC D is asthe proportié of CD to DE. Wherfore the refidue of C B,namely,D B, thal be to the refidue of C D,namely,to CE,as the proportion of the whole 8 C to the whole C D.Wherfore C E may be {ubtra- hed out of ED: wherforethere fhalbe no end of this fubtraction: which is impoffible. q Lhe 16. EF beoreme: The 16. Propofition. If twonumbers be prime the-one to the other,the fecond fhallnot be to any other number as the fir/tis tothe fecond. Hi V ppafe that thefe two numbers: A and B be prime the onetothe other. Then! 24 (ay that B is not to any other nuber as A is to B.For tf tt be pofsible,as Ais to B, Dement ee | fo let B Leto C.Now A and B are prime nuimvers., a they are prime and the le FET, WoseXoc4 by the 23.of the fenenth. But (by the r1.0f the je ) the leaft meafure the f ° - - ) summers that haue one and the fame'proportion with them equally the antecedent the antecedent,and the confequent the confequent. Where» ~Aaeace fore the antecedent A,meafureth the antecedent B, and it -meafureth B sivisaee alfo it felfe.Wherfore A meafureth thefe numbers A cy B.being prime” Cm the one to the other which ts imposible Wherfore as Ais to B, [ois not B to C : which was required to be proued. q The 17. Theoreme. The 17. Propofition. If there benumbers in continuall proportion how many foeuer,and if theyr 7 extremes of Euchdes Elementées. Fol.222, : extremes be prime the one to the Other:-the lefSefhall not be to any other number as the firltis to the ‘fecond, | their extremes A and D be prime the one to the other.T hey ‘fay that D is not > | £0 any other number as Ais to B.F or if it be poftble,as A ts ta B, f alet D be te KVL NA E Wherfore alteryately by the 13.0f the feuenth,as Ais to DfoisBioE. But A and D are prime, yeathey are prime and ‘% od *) Denorfiraen the leat. But the least numbers{ bythe 21. eS ae tion leading t9 the fe wenth nafure the wummbers that haue > Bvcanee.Y: : ai abfurditie, one and the fame proportio with them Cquil. ~C 2ORNes Spee fs ee ly,the antecedent the antecedent,andthecom D....... Set See aE. : fequent Wherefore the antecedent AmeAfie oR reth the antecedent B: but av isty B fO1P BIOS to C.Wherfore Ballo meafureth G Wherfore A alfe menfureth Cl by the s.comme fentence of the fenenth) and foralmuch as Bis to C,foisC to D but B mealureth C.Wherfore C mea- fureth D But A meafureth C.Wherfore A alfo menfnreth D by the fame common fenténce, and it alfo meafureth it felfe: Wherefore A weafureth thefe numbers A and D being prime the one to the other which is impofable Wherfore D is not to any other number as Ais to B which was required to be prowed. 7 q The 18. Theoreme. Ihe 18. Propofition. I wo numbers being denen > £0 fearche out if it be pofsible a third number Ee i. in proportion with them, . : 2 ot en prime.If they be prime,then. ( by the r6.of the ninth) it is mani: sive ecafee nifeft that it is impoffible to finde outa third niuvsber proportional with them: But now fup- Re u propels pofe that A B be not prime.the ane to the-other. And let B multiplieng:himfelfe produce C: The firf cafes Now A ¢: * SRO ae C,0r meal ureth st not. Firfi,let st meafireit and that by D. Wher- fore A multiplieng D.produceth ° The fecond C.But-Balfomultiplieng himfelf As... | cafey produced C.Wherfore that which BR ».. | | es produced of A into.D, is équadl.. | te that whichés preanned of Bina. <> to himfelfe. W; erefore (by the ‘[e- A cond part of the 19. of the feutth) _ B as Ais to Bois B to D.Wherfore... De unto thefe numbers A,B is found ©... AE SWAN SS Be WHND AIG DA 90 905 out a third number in proportion, namely,D, Fe | But now fuppofethat Adonot B meafure Cx ZT hen I fay that.it is D senpofiableto finde our ashird nike .C 7 1” proportion with thefe num bers.4,B, Por if tbe pofable, let there he found out [uch 2 number, and let the famebe D. Stee so BB.iy. Wherefore The third cafe, Sees ts se & be ere @6 ee | T beninthBooke’ » Wherfarethat which is produced of A intd Dis equall to that which is produced of B into himfelfe,but that whichss produced of B into bionfelfeis C.Whierfore that whichis produced of Aint D is equall unto C. Wherfore A multiplieng D steeree C. Wherefore A meaftt- reth C by D.But it is fuppofed alfo not to meafure it which is impo(ible Wherefore it is not ofible to finde out a third number im proportion with A cy B,whenfoener A meafureth nat Ce whieh WAS required fo be pronea. q Ihe 19. T heoreme. T he 19.'Propofition. Three nambers beyntg geuen ,to fearch out if it be pofsible the fourth nume ber proportionall svith them. | Diners cafes nt this propofie £107» hy (FZ aN ye) FLD ; be pofsrbie a fourth number proportionall with them.N ow A,B;G,are either in | INEZ. | BAY \cozaterazeall proportiou,and their extremes A;€ are prime the one to the-other-or [oases they ane ok 18.COn tinuall pr oportson and thewr.extremes areyet prime the one. to the other: or they avein.continuall propertion,and their extremes are not prime the oneto the other cor they.are weiter in continual proportion, nor their extremes are. prime the one ey ippofe that the three numbers ceuen be A,B,C.Itis required to fearch out if it . 4 | to.the other, Oe gece If A,B,C,be 1n continual proportion and their extremes be prime the one tothe other, St (by the 17 of the ninth) that tt 1s imposible to finde out a fourth number pre- ey ie f. 2 be firje Ca7¢§ 4 . ° _f , J 4t 1S PHAN FE : es r > si at * HAwTTAAA Agee ein Tl-paa ; 3 j J ae. V fe U/ ee Bel seil Mews b PETS 6 7 Bat i ow [uppoje that A,B,C be motincontinuall A eeocsess- dhe fecans proportion, aud yet let their extremes be prime the oneto Bos swresveers safe the other. Then I [ay that [oalfo it is impofGble to finde ~ C svecsscvececvecs out a fourth number proportional witl thé. For if it be 7 pofible,let there be found fuch a number , and let the | le D.Séthakus AistoB,fo let C beta D ‘and as BistoC,folet D beto E. And for that dwne G as AistaB, fois Cto PD, dvd as Bis C,fois D to E therfore of. equallitie(by the 14..0f the fe- and € are prime andthe least: but A vee B Svievewss uenth)as -A isto Csfois€ toE. But A the one te the other yea they-are prime, the hei men[ure.the punshers that hane one<>-the fame proportion wth them. equally,the antecedent) the antes C ser ser vagseveceres cedent,and the confequent the confequent (by the 21 of “D the {ewenth). Wherfore A meafureth Cnamely,the ai EE meee tecedent the antecedent, and it alfo meal ureth. it felfe. ge wWherfore A meafureth thefe numbers. A and Cheing prime the oné to the other, which is Kible Wherfore it is not po/sible to finde out a fourth number proportionall with thefe Z PHIL sd andl | numbers A, BSC. A But new acaine [uppofe that A,B,C, beim .. Aarcevaees The third continuall proportion,and let AandC not be... -Brsevverveors rime the one to the other. Lhen 1 fay that.tt....Cr...6: as poftbleto finde out a fourth number pro- Evsusaudvdvvagensecrereescer’ portionall with them.F or let B multipliengC .. D216 produce D. Now A either meafuretb D, or ‘ — meafureth it not.Rirft let it meafure ityand that by E. wWherfore A multipliong E produced D.But B alfo multiplieng C praduced D.Wherfore that which is produced of AE ss equal f0 that which is produced of BC: wherfore in what proportio AsstoB in y faine is C60 E Wher fore there ts ‘ied buta fosivtl nnameber namely, E proportional: with thefenibers a A J " cafe. ~ of EuclidesElementés. = = Fol.rya, wo But pow fuppofe that A do wot meafire D.T hen fay that it is wot pofible to finde out A fourth number proportionall with thele numbers A; B, C: For if it be pofGble, let there be found fuch a wumber,and let the fame be E. Wherfore that which is produced of A into E as equal to that whichis produced of Binto C. But that which is produced of BintoC is D. Wherfore that whichis produced of A into E is equall unto D. Wherefore A multiplieng F produced D,wherfore A meafureth D but it allo meafureth it not which is smnpofsible. Wher Sore itas inmpofible to finde out a fourth aumber proportionall, with thefe numbers A,ByC, | whenf{oeuer A meafureth not D. ant But now uppefe “The fourth that A,B,C be net- Fe eS ee ate he “s Cafeg ther in continuall EEE Re ee ee proportio,neither al Gh sg =e Cer 00 coe SUNNY Ok E DS OLa. es, fo.their extremes.be.. E prime the onetothe ~ “D 1356 other. And let B mul | tiplieng C produce D.And in like forte may we proue that if A do mealure D,itis poffible to pude out a fourth number proportionall with them.But if st do wot wcafure D,the is it. un~ poffible : which was required to be preued. q The 20. Theoreme. The 20. Propofition. Prime numbers being ceuen how many Joeuer, there may be genen more prime numbers. V ppofe that the prime numbers geucn. be Bee & : 4a she HAT 2 SS Se ACorollary. _ By thysPropofition it is manifeft,that the multitude of prime numbers is infinite, a... pL he. an, Theoreme. > Lhe 21Propofition. pa Tf euen nitbers bow mary foener be added together: the whole hall be ene. B's BB.iiy, Suppofe ‘Demon flra- £08. Ca D emsonfirade- Demonstra- YF @iry Gives? Demon firt- £40. FC The ninth Booke: AV ppofe that thefe even numbers A B,BC;C D,and DE, beudded together. Then 1 fay, that the whole number namely, A E, is an enen number «For foraf- much as enery one of thefenumbers AB, BC, CD; and DE;isan enen unm- West Ka ber cherefore ewery one of them hath anhalfe .Wherefore the whole AE = =_—S : a -* ~~ SVE ~ = aoe = - Se = = - <= in : - = = 2 - —< — - —- - = — — = = a —_ OS 8 Ee EE < = aa = =. 7 SSS = ¥ -- ——— = = -— > ~ —— z — SS ay a oe a a eee : = = —-- : — = — =— _— .see 7 eg ~ = - —— = BER IB ; i Nn Demon stra- 70%, A wPODoftion adde d b y ai ipa tie An other ad- ded by him, Demonfird- tion leading to an abfurditie. T he ninth Booke arene Vopofe that 4 being an odde number multiplieng Bo Csi vecticeees being an exen number,do produce the number C.Then Be... | Lfay that C is an enen number. For forafmuchas A A... SY RX, q The 29. I'heoreme. Ihe 29. Propofition. ~ Ifanodde number multiplying an odde number produce any number , the number produced fhalbe an odde number 'V ppofe that A being an odde number multiphine B being alfo anodde number, SN doo producethe number C.T hen fay that Cis an odde number. For foralmuch as Sexe mmultiphjing Byproduced Ctherefore C is compofed of {0 many numbers equall unto B as there be untties in A.But either of thefe num bers Aand Bis an.odde number -Wherefore C is com- OBS Hb eh tbe oc vs pofed of odde numbers, whofe multitude alfo is odde. Ba Wher fore. by the 23.0f the ninth) Cis anodde nuber: eee a which was requiredto be dem onftrated. + A propofition added by. Campane. Tf an-odde number meafare an enen number, it {hall meafure it by an enen number, For if it fhould méafure it by an odde number, then of an odde number multiplyed into an odde number should be produced an odde number,which by the former propofition isimpoffible. An other propofition added by him. If an odde iiepber mspafurean odde number it fhall meafure tt by an odde number. .. © Foritit fhould meafure it by.an euen anmber., then’ofan oddé number multiplyed into an even number thouldbe produced an odde number which by the 28.of this booke isimpoffible. q Ihe 30. Theoreme. Ihe 30. Propofition. Ff an odde number méafure an euen number it fhall alfo meafure the halfe f hereof. &V ppofe that A being an odde number doo meafure B te Sy ewen number .T hen I ih that it hall mea[ure the halfethereof For forafmuch as Ameafureth B let it p92. mealure it by C.T he I fay that Cis an enen number.F or if not then, if tt be pofjiole let it be odde. And forafmuch as A meafureth B by C: ther fore A multiplying C producéth B.Wherfore Biscompofed A... of odde numbers whofe multitude al{oisodde.WherforeB C ....... is an.odde number ( by the 29.0f this booke ) which is ab-- Bo... ... ces sceevses furd,for it is {uppofed to be enen : wherefore Cis an even | whiner Wherefore A meafureth B by an enen number; and C meafureth B by A. But — | 0 of Euchdes Elemente, Fol.225. of thefe numbers Cand Bhathan halfe part -wherfore as Ciste Bfoisthe halfe to the halfe. But C mealureth B by A. Wherefore the halfe of C meafureth the halfe of B by A: wherfore A multiplying the halfe of C produceth the halfe of BWhet fore A menfurerh the balfe of B: and tt men[ureth it by thebalfeofc. Wherefore A meafureth the halfe of the number B: which was required to be demonfivated, 2 ~ ; | The 31. Theovemes <9 The 34)'Propofition. = If an odde number be prime to an y number yt fhalalfo be prime tothe dows ble thereof. | FMC, } ppofe that A being an odde number be premevuto the number Bs avd let the dow. GAMES b/c of BLeC . Then 1 fay;that A ts prime vntoC For tf Aand C be wot prime the VISIANG one to the other, fone one number mseafureth them both . Let there be fith-a num- bp which meafureth thews both,and let thefamebeD. But Ais anodde number. Wherefore D allo is. an odde ay roteeaae number .( For if D which meafureth Afhould- beam -B ow... euen numbersthen fhould.A alfo bean euen. number (Bey if the 21. of this booke): whichis cotrary to the {uppofite- D on.F or A is fuppofed to be an odde nitber: ¢ therefore a D.alfoas.an.edde number) And forafnuchas D being an.odde number meafureth C, but C isan ent number ( for that it hath an halfe,namely,B) Wherfore( by the Propofition next going before) 1 meafureth the halfe of C . But the hilfe of Cis B. Wherefore D meafureth B : andit al{o meafureth A. Wherefore D meafureth A and B being prime the oneto the o. ther : which is abfurde . Wherefore no number meafureth the numbers A GC. VV. herfore A is a prime number unto ©. VV herefore the[e numbers. ahd ¢ areprime the one ti the other : which was required to be proned. op LD heg2Tpvoremesoxs 08 Efe 32MP ropofitions be | “Binery niiber produced by the donblin Ig of. two ypweird is enenly exen onely. . a | aN foerner; 45 BC, D. Then (a), that BYC3D; ure numbers tnenly euen onely.T hat e- hasee nery one of them is enenly euen je is mansfelb:- for cuery one of thém $s produced by the doubling of two .I [ay alfo, that ewery one of then is enenly euch onely : Take unitieE. And foralmuchas FPO UNE APE Dee eco ek certaine numbers in continuall proportion, eA which OND IS ORS IS . coh wext after nities a prime wnmber therefore by the 30 B a the third) no number meafireth D being the ereateft number? A>. of thefe mimbers A,B,C; D; befieles the felfe fame numbersin’ BE -womitie Proportion. Buteuery one of thefe numbers ABO: 7); euen VV herefore D is euenly euen onely . In like [ort may we prout, that enery one. of thefe unmbers A,B,C, is encily euen onely : which was required to be proned. | MA? Pp oft thar A be he number twos andl From A upward doable wmubers how many te Gt he33.T beoreme. Lhe 33. Propofition, A number-swhofebatfe part is odie is enenly odde-onely, aac ach . e onely. Sao Dewonfrae C0, Demonffrae b30%e { i “ i " , it J Wi t hie K ft) 7 " ) } { qu { 4 ‘ Af ; { Phi NW iteu iM i oh j is | Pith 5 a } Heat { - i 4 } \ ue whi { . ei Bee } { { y 4 | a) a Be hI ( 4 i at i . j " ] i! | ! i ] rl nt a t ite Aah a MI i oy Oe i 4 2 ; 4 a ne 7 4 4 ile ; a bP J ? Ly: ” wae Ut wat iF iy. { : 4H f : } f it) Oe , “« f } ill 4 a} fs : rai} i ; { j ) WG oe Bie! ‘ mea a - ; i" yf ) - 7 43 ‘ + } = t a} E -¢ it ms | D Par | Ky | 4 “| i 1 EW ae | : al ty *» Gia’ 5 e. ¥ 7 4 j ape ht Bt yet 1t i? oho ht : ine ; A r itt iy n - ‘] iy : : ‘4 at . } hi , { hh > rae at tah ha" 3 ei - : f - y } es Paice AMA ! ; i : + oa M Pi) rt ne tally! 7 | Wee a Lae Y -§ y } ‘ : ‘i ‘ Oy) aa z } LS 1 ih : maa i ¥ Wii 4 yarne 7 ' yt ma ath uy ~ i a 1 by q j Hi } y Ze Yl es OF a a : be AY Bt f , n/t i ets 7 / ie} a "4 it he) tl Bet ri 1a i] Mi ; | aa | ‘ } yf if i) oa | bite e : n ? Hi i hh ‘Chat 1 im) | t +) iil * i Bet! hit Da ry an Pah iii ‘e | RPL) = ~* ae 125 — et . -Ite ——- ee = eet — — =e = - — Wwe : = ra — — - — = ite as . YPemonttrae tion leading to wn A0/Wr dite » Jin other de- wsonfiratiomte Demonfira- B80Mte T he ninth Booke IAAT K ppofe that Abea number whofe halfe partis odde. Then 1 fay that Ais enenly Be OH od onely. Thatit is euenly odde itis manifeft : for his halfe being odde meafureth erate) him by an-eue number; namelyyby 2. (by the defini- tion). 1 fay al{o that tt is euenily odde onely. For if A be euen- Assesses ssces saaice 0 ly enen,bis halfe alfo ts ewen.For ( by the definition) aneuen umber mea{ureth him by an euen number . Wherefore that euen number which meafureth him by an enen-nunaber {pall allo meafare the halfe thereof being an odde number by the 4. common fentence of 2 ge which is abfurd.Wherfore A is a number enenly odde onely: which was.nequired to beproned. An other demonftration to proue the fame. Suippofe'that the niimber A have to his halfe an od nfiber,namely, B.Thé I fay chat A is enély od onely, Tharit is euenly odde needethno profe ; forafmuchas the number 2. an even number meafu- reth jt by the halfe thereof whichis an odde number.Let C be the number 2. by which 4 meafureth A ‘for that A isfuppofed to be double vnto.B). And let an euen number,namely,D meafure AC which is poflible for.that A.is.an.euen.numiber by the definition)by F.And forafmuchas that whichis produced of C into B is equall to that which is produced of D into F, therefore by the 19.ofthe fenenth,as C 1s to D,fo is B to F-ButC the number two meafureth D being an euen number: wherfore F alfo meafureth 'B which is the halfe'of AV Wherfore F isan odde number.For if F were an euen number then fhould it make B whome it meafurerh: an oddeiniumber alfo by the 21. of this booke,which is contrary-to the fuppofition And in like maner may we prone thatall the eué naberswhich meafure the number A,do meafure it by odde numbers .. Wherefore A is a number euenly odde onely : which was required to ‘Be'proued. mS | : Mt nes oT be3 4. T beoreme: * T he 34.Propofition. Ifa number be neither doubled from two snor hath to his half part an odde numbersit fhall bet number both enenty enen;ana euenly odde. Avippofe that thenisber Abe anitver neither doubled fro the nitber two, neither eY \al{o let it bane to his halfe'part an odde nuber.T hen 1 faythat Aisa nuber botle DS euenly thensand enenly oddeT hat. Ais euehly ence itis manifeft, for the halfe | Mae ther of isnot odde, and is meafured.by the number 2 which is an euen number. Now I faythat ites encaly, odde alfo..For if we deuide A ipto two equall partes ,and fo conte nuineg flill,we [hall at the length light uponacertaine wer edde number which [ball meafure A by au-gucngumn- A wbe sores sarees ven ber.F or if we fhowld nok light vpon {uch amoade Bi Ad ber,which meafureth A by an-euea number, we fhould at the length come vate the number swo,and [0 fhould Ave one of thofe numbers which aredoubled fromtwo upmard,swhich 6 contrary to the fuppofttion Viherfore Ais ewenly odde.And itis proued thats ts enenly uc? wherfore Ais a number Lotheuenly euenand encnly odde.: whiche was required to be dé- Padnfirated. re os This propofition and the two former man ifeftly declare'that which we noted yp pon the tenth definition of the feuenth-booke namely , that, Campane and Fluffates and diuers other intetpreters of Exchde(onely T heon except did not rightly ynderftand the 8.and 9.definitions of the fame booke concerning a number cuenly cuen, anda num- ber enenly odde. Foreinthe one definition they adde yato Enchides rene > See & Ss £2 so al of Euchides Elementes.. © Fol.226. the Greeke this word onelyCaswe.there noted) and in the othér this word 4/.So that af. tertheir definitions a huniber can not be euenly euen vnleile it be meafured onely.by cuch aunibers likewile a numbercan not be eugnly odde vnleffe all'the enen nnin bers” which doo meafite tt: dag meafure it by an oddé number. The cotitrary whereof in this propofition we manifefthy fee For here Eaciiwe proueth thatohé number may be both eucnlpeuen and cucnly oddé. Andin the two former propofitions he proved that lome numbers are euenly euen onely,and fome eucnly odde onely.: which word onely had benein vaine of him added, if no number cuenly euen could be meafured by an odde number, orifall. thenumbersthat meafure.a nuntber cucnly odde muft needes although it haue nottohis halfe anodde number.Asin this 3 4. propolitié Evchide hath plainly proued. Which thing could by no meanes be truc,ifthe forefayd 32.& 33. pro- politons of this booke thould haue that fence and meaning wherein they takeit. q Lhe 35. T heoreme. The 35. Propofition. If there be numbers in continual proportion how many foeuer and if from the fecond and laft be taken away numbers equall vnto the firft , as.the ex cefje of the fecond is to the firft, fois the exceffe of the laft to all the nitbers going before the laft. = 3 Me . | Xx vuto the firft,named 40.4, and likewile from E F the laft take amay F H e- Ke XS) quall allo unto the * A the firft, fois H E the excefe,toall the numbers D, BC,and Ud, mhich go before the laft numoer, namely, EF. Eoralmuchas EB js the greater, for the fecond is Suppofed greater thee the firft) put.the number FL equall to the-number D.,, and Lhewife.the number ER vito GC, therefore the refidue H K is equall unto the refidue GB. And for that as the mholeF E,istothe whole F Ly fb ts the part taken awhy F L, tothe part taken away FR, RE is tothe whole FE by. OR \ioocg pare (he IN..0f the feuenth.) So. —2 Sauls ukiwife for. that EL is-ta\. & . FKs @BKG5 0 BK: E Gert selon: | | ShalibetoH Kyas the whole kLsisto the whole FR ( bythe ‘[amePropofition ). But as FB i L0F L undas-EeLis tok K pattd EK toF H, fowere F Eto D; andD toB Cand BC tod. Whereforéas DL, E WIOKL, andus K L isto ADK; fois Ds ta BC. Wherefore alter: putely (by the x 3+ Of the feuénsh) as bE iste D f0ts KL to be BCyand as KL 4s to BC; fous K 0A Wherefore alfoas vue of the antecedentes is to ane of the con[eqneptes; ‘ on Seana CC yz, aut WTA A ARWAEGH anlathyy Wolpe Demonffra- 610%, Ths propoft- tion teacbetD bow to finde out a perfelt UU MDET» s Conftructtion. Demonitra- £50%e 8061 eee ee ms ee Oe se The ninth Booke all theantecedentes to all the confequentes . Wherefore a3'K H isto A, fo are HR, KL, ind LE,to D, BC, and A (by the 12. of the feuenth ) . But itis proued, that K H 1s e- guall unto BG . Wherefore as B G,which 4s the excelfe of thefecond , isto A, fois EH the excefse of the last unto the numbers going before D, BC, and A. Wherefore as the excefse of the fecona is unto the firkt, {0.48 the exce(ve of the laft to all the numbers going before the daft : which was required to be proued. q The 16: T heoreiné: The 36. Propofition. If from bnitie be:taken numbers bow many.foener in double proportion continually, vatill the whole added tovether be a prime number, and if the : whole multiplying the laft produce any number, that which is produced ts a perfecte number. : AV ppale that from vnitie be taken thefe numbers A,B,C,D,in double proportion Y\-ontinually, [0 that all thofe numbers A,B C,D co vaitte added together make D> {4 prime number : and let E. be the number compofed of all thofe numbers A ,B; WhonGel c, Dc unitie added together. + and let E multiplying D which isthe last sums ber,produce the number FG. Then I fay, that F G tsa perfect number. Fnitie ’. -« Hommanyin multitude A,B,C, Dyare; fo many i” continsall double proportion take be- ginning at E , which let be the numbers E,HK,L;.andM «WV herefore of equalitie( by the 13. of the fenenth ) as A ts to D, fois EtoM. vi herefore. that which 1s produced of B into D, is equall to that which is produced of Ainto M ey ae _ sa = ma Ee 2 T= a ” = ee —_—— ta _ “a * | Maiyeuck of ner impoflible, to attayne to the rightand full vnderftanding of this booke, Wwith- | se ell dees 0 theayde and helpe of fome otherknowledgé andlearnyng, and chiefly with- Mit nh | ne out the knowledge of that more fecret and {ubtill part of Arithmetike,commonly i shought thar _ called Algebra,which vndoubtedlyfirft well hadand knowne, would geue great HY shisbooke ca \ioht therunto:yet certainly may this booke very well be entred into,and fully yn- i \ cb Pat Pant j derftand without any ftraunge helpe or fuccour, onely by diligent obferuation of ne | withinae MY the order,and courfe of Euclides writinges,So thathe which diligently hath peru- hi : gebra {ed and fully yn derftandeth the 9.bookes goyng before, and marketh alfo earneft- it ly the principles and definitions of this tenth booke,he fhal well perceiue that Eu- ji Theninefor- cide is of himfelfe a fufficient teacher and inftructersand needeth not the helpe of mer bookes @ any.other,and ihall foone fee that this tenth bookeisnotoffuch hardnes and ob- the pen {curitie,as it hath bene hetherto.thought. Yea,L doubt not, but that by the trauell oft . sake and induftry takenin this tranflation,,and by addiciensand emendations gotten floode,this Obothers,there fhallappeare initno hardnes atall,but fhall be as eafieas the reft of booke will not Anis bookesare. | be hard to —— $a Definitions. Thehea i Ma onitsdes commenfurable are fuch ~which one and the felfe fame meae p86, fure doth meafure. | Firft he fheweth what magnitudes are commenfurable one to an other. To the better and wore cleare vnderftanding of this definition, noté that that meafure whereby any magnitude ismeafured, 1s leffe chen'the magnitude which it meafureth, or at leaft e- quail vntoit. Por the greatercan by no‘nieanes meafure the lefle, Farther it behoueth, that-that meafure if it be equallto that which ismeafured, taken once make the mag- nitude which is meafured:if it belefle then oftentimes taken and repeted, it mutt pre- cifely render and makethe magnitude which it meafureth,Which thinginnumbers is eafely ferte,for that (as wasbefore faid) all numbers are commenfurable one toan o- ther. And although Euclide in this definition comprehendeth purpofedly, onely mag- nitudes which ate continuall quantities,as are lines, fuperficieces,and bodies, yet vn~ doubtedly the explication of this and fuch like places,is aptly to be fonght of numbers aswell rational as irrationall, For that allquantities commenfurable have that pro- portion the one to the other,which number hath tonumbers, Imnambers therfore, 9 and 12 atecommenfurable, becaufe thereis one common meafure which meafureth them both,namely,the number 3. Firft it meafureth 12,foritislefle then 12, and be- ing taken certaine times,namely,4 times it maketh exactlyi123 3 times 4is 12, it alfo meafureth 9,forit is leffe then 9. and alfo taken certaine times namely, 3 times, 1t ma- keth precifely 9:3 times 3 is 9.Likewife is itin magnitudes, ifone magnitude meafure twoother magnitudes ,thofe two magnitudes fo meafured,are faid to be commenfuta- ble, As for example,ifthe line C being don- bled,do make the lineB,. and the{ame lyne C tripled,do make the line A, then are. the two lines A and B, lines or magnitudes com tnenfutable. For that one meafure, namely, thé line’C meafureth thé both.Firft, the line © Cislefferhé theline A, and alfoleffe the the line B,alfo the line € taken or repeted certaine times, namely, 3 times maketh precifely the line A,and the fame line C taken alfo certain times,namely,two times maketh pre- cifely the line BSo thatthe line Cis acommonuneafure to them both, and doth mea» fure them both. And therfore are the two lines A and B lines commenturable, And fo ir magine ye of magnitudes of other kyndes,as of fuperficiall figures, and alfo of bodies, I ncoms — _ — — " (te — seer od + = ~ * 2 ee ee oaperanen — = eee ee en ea : es A tnt - =-- QQ eee ° AT e272, % ; . ty j = } ; : 4 ; Wi A ae i id | ; 1 b : , ! y : < | } i Oy FE i i. ii i i ne i) " : fe iii } ' } TMG By a a ‘ 4 RU bad Ha a ‘S at) iti | | aa | +) } t | ’ LE we im " 4 | it ie Pep q! ' 4 ‘ } eB } ! hy h ; 7! i } Pete wee. i ¥ it tiga)! nae Thy it 5 A ll Try “ ALTE N aa HY Te i ee : oR } ' NH ‘ i yi Dh bE ' | q Wi AAD! LVR tel ae Ht a oe ; } t i) in qi ih I} | UE eal atl i DRE it i : BD fer ’ , f 4 ff a ni) | ma] 1, u J ; { Ae) ett \ } : } iW i a | neh i na } i) j ty) ae 5 | } S} ] ) Ah) ee hie eh 1 t haat a a ae f 4 fr, j Hake } . » | it 4 qi Le H wy } : , wea} Pie . - he 4 } i ; | } | 1 ee 4 { a4} }} ab ee if ‘ } he en ye ah ! Po { \ i ia ] ’ ’ i i} ft i, lit " S! { Oe ei Th mitts CPt. 4 i} ; he | fi He ie ip } | ei, I von | th iP } } | ie y HM M aoe 1. EU ‘¥ ’ ' } i a bh : ] te WW ; j i} y a4 : Hi a “a y ai 1 a " ‘= . . : Pe "? 4 wo i nt yan - a ‘nn TT RE ate ' he Bg oie TE tA . Wi i i} Ths id He | 1 se & a . if } ‘ hi iia ’ r re hy H} 4 4 oe) eet: |) We pets h 1 he aie ; 3a) ! aM ahi _ Hi purty ‘tLy > Se 1 eee ae (7 ecaiy 4) a Pa ie yale i Se tt DAL) Ee | kay ' | ¥ } Ts) We of ‘ind tel aon : aaa a Ee Tae) q b || HY ) | Wh viel a 4 } F a ih ie a : ull Ta id a ba : et CtEEre a ; a ‘ ie ; Wh dee f ; i fe q | 4 m Y nf " bY yaa : Beda, bi doe ees Sw Bid ay fl ae i ‘ hh Ie ish BP Mien Pt oe : \- A hy | i Aah ' ca 4 yi i Aid i H eR a: i. ¢ HW aa tei ; : ie ; 4 | at pul t ia. i ' } ct ait eT i) eae Oe r i 4 RCP e } Bh ft 1 ae) at ; Ta nh : iH ae Ue Ae tT ea ea : TA { Y : 1 }} "| beet ratty eed Tea oP Vy ee an 4 ' fj Hy ‘| ey tt) 1” Hi : - \ ; : ’ 7 He Ae : ay ey ‘ 4 iy ANT - } i a " , a } “ 8 thi! : he tH H BI. a Ale : ' a} 3 i We ) 4 - Lith’) } fe A Bi f ea ah Bibi) te } a a j f UN Ta tae i Pf 4 af ty | \ Hite | eae! Wt iit whe a 1 : ar ‘in \ ie . ye 4 -th s} +a) 10 i ht i J + Mu mittee jf A, } aly - 4 a } ‘ ! ih 4p ' ie eb eht |) And i ehie: nts ear We ae , ; hs. aR TAN “ire ai! Hy , fl wae i" 1 ¥ ’ 4 ‘ } t i iN ny | in} ni : ' Piiek i! Tee a a 1 A Pea \ | , Teal . Paar e ye Ny i Ae ; if ; a BR ie hi yi , a Priest at! bee ¥ : | af yo ey ic | wae } it } a W ty Lahti 4 ne \ Bai mih: ew : ‘ if HTH ne 4 ( 4 | i) ae } atl —— = - oy of Euclides Elemtentes. Fol.229, 2 Incommenfurable magnitudes are uch , which no onecommon meafure ry, Roan’ doth meafure. dcfinstson. This diffinition neadeth no explanation at all, Contraryes it is eafely ynderftanded by the diffinition poing made mansfeft before of lines commenfurable . For contraries are Y, pertain kT by thecompa- made manifeft by comparing of the one to the o2 ring of the one ther:asif the line C, or any otherline oftentimes Bo pecee wen eee to the others iterated ,doo not render precifely the line A,nor the line B, thé are the lines A and B-incommenfurable: Alfo ifthe line C ,, or any other line ¢ertayne tines'4 +4 repeted, doo exadtly render the line A, and doo not meafure the line B: or if it meafure the line B, and meafureth notalfo the line A , the lines*A and GS iinsainasag B,are yet linés incOmenf{urable: & fo of other mag. nitudes,as of {uperficieces,and bodyes. B ——+—___, 3 Right lines commenfurable in power are fuch , ‘whofe fquares one and The shirde the felfe fame [uperficies area,or plat doth meafure. definitions To the declaration of this diffinition we muftfirft call to mindewhat is ynder- ftanded & ment by the power of aline : which 4s we haue before in the former bookes noted is nothing ells but the {quare thereof, or any other plaine figure equall to the {quart therof.And fo great power & habilitieis a linefaid to haue,as is the quantitie of the {quare,which itis able to defcribe,or a figure {uperficial equal tothe {quaretherof, What the This is alfo to-be noted that.of lines ,fome are commenfurablein length, theone powerefa to the other,and fome ate comment{urable the oneto theother in power.Oflinescom 4etSe men{urablein length the one to.theother,was getenan examplein the declaration of the firit diffinitiG namely.the lities:A and B,which werecommenturable in length one anid the felfe:meafure,namely,thé line Cmeafured the length of either of them. Of the other kindeiis:geven this diffinition here fet:for the opening of which take this exam- pte. Let there bea certaine lineynaiely,the poilcas : line BC, and Jetithe {quareé of thatline be K D , the {quare B.C. DE. Suppoféalfoan other line; nawiely, thedine FH,&Jet the {quare thereof be the fquare FHIK, and leta certayne fuperficies , namely , the fuperfi- cies A,meafure the fquare BCDE, taken 16,times: which is the number of the litle | areas, {q ai ox glare fuperficieces cétai- < A ned and defcribed within the fayd.{quares | 4 B G ech of which is equal! to the fuperficie: | A } A. Agayne let che me fuperficies Amea. |= | fure the fquare FHIK 9, timestaken,ac- F vee. Bis € cording to the number of the fieldes or fu- | perficieces contayned and defcribed in the fame. Ye fee thé that one and the felfe fame fuperficies,namely,the fuperficies A , isa common meafare to boththefe fquares, and by certayne repeticions thereof , meafureth them both. Wherefore the two lines B C and F Hywhich are the fides or lines producing thefefquares, and whofe powers thefe {quares are,are by this diffinition lines commenfutablein power, =o . ee b + Lines incommen/urable are fuch , whofe fquares no one plat or faperfie pis cies doth meafure, , CC. iiii. This 2 ee - - Hi 1 ery we oe Abe f2fth efile The principles of thts ba2ke o net welt be peifed, for that they are ware firaunce then the prin ciples of the fermer bokes. T be tenth Booke< This diffinitionis cafy to be-vnderftanded. by: that,which was faydin the diffini- tion Jaft fet before this, and neadeth no farther declaration. And thereof take this ex- ample . If neither the fuperficies A, : . nor any other fuperficies doo mea- I | aa - fare the two fquares BCDE, and’ | | EHIK: or if it meafure the one; namely BC D E, and not the other FHIKX, or if it meafure the fquare | EH 1K,and netthefquareB-C DE: che two lines BC and FH , are in power incommenturable, and there. * i foréatfoincommefurablein lengths A B Gio For whatfoeuer lines are incommes furable in power , the fame are alfo incoramenfurable in length.as fhall F mM OB Pa afterward in the 9, propofition of i this booke be proued., And therfore | | ich lines are here defined to be abfolutely incommenfurable . Thefe thinges thus ftanding it may eafely appeare, that ifaline be a{figned and layd before vs , there may Oe innumerableopher lines commenfurable yntoit., and otherincommenfurable vnto sof comimenfaratle linés fome are commenfurable in length and power,and fome in power oncly. @ 5, Andthat right line fo fet forth is called.arationall line. Thefe principles ,diffnitions:and-groundes of this tenth booke ought well to be payfed, they are fomewhat mere ftrange and more hard, then are the diffinitions and principles of tieother bookes of Exclide going before’, and therefore at the firft ight or reading arenotftraight way conceiued,but ought often to be repeted, and by vie to beconfirmed.For:the propofitions following,bring vnto them much light, and facili« ticofynderitandine Firft there isa line fuppofed,and layd-before vs, which may be.a- ny:litte whatféeuer,:of what length,or fhortnes ye will : this line thus firtt fappofedis imaginéd:to hase ifuch diuifions and fo many partes asyeliftto conceiuein minde, as 3.4.5.and fo forth, which may be applied toany kinde ofmeafure,as itthall happen,as toinches.fecte,pafes.and fuch other,Vato this line faith Exclide may: be cOpared innu- merable lines, of which fome fhalbe commenfurablesandother fomeincommenfura- blesofcommenfureble lines ,fome are commenfurable bothin length andin powers other fome are cOmenfurable in pow § eronely:As ifany part of thelige pro pofed which let|be the line AB,doo -AOS O29 mefare alfo the line E F and againe ifany one fuperficies do meafure the A (aizare Of the line A Bywhich let be the {quare ABC D:and alfodao meafure the {quare of the line EF which let be the fquare EF G Hi: théisthe line EF to the fuppofed line & firit fet, name bystothieline AB,adine comnien{nra~ >: ble: bothin lehgthand' powers asye ind ydee in the firftexample here fet. Agpise oAndifitfo bethat one’& the felfe 2 oS 1 29rnl 19. 290 faine faperficies do meafure boththes: 3 09,2 Dee pm | fauares of thefe two lines A B and EE, namely thefquaresABCD, andy dya\au.s. EEGH: andnoonelinedo meafure * * the lines AB and EF: thenis the line EF of Euchdes Elementes. Fol.2306 EF ( compared,to'the fuppofed and fir{t line AB) cofimenfurablein pow eronely.As in this example ye may eafcly perceiue. For the triangle or fiipericies AC D, awife taken; ‘mea- fareth the {quareAB CD naniély; the fquare of the line ABs: and: the {elfefametriangle AC D takefi' foure times meafureth the other fquare, py dainely, thefquare of the dine EF, But tio one‘meéafure or line canbe af- figned to meature:both the lines'AB atid EF , becaufe the fides of a fquare and the diamteterare incommenfura- ble inlength the one tothe other, as afterward fhalbe fhewed . Wherefore they are in length incommenfurable, & commenfurable in power onely, that is by rea- fon of their {quares,which are commenfurable the one to the other. Agayne if it fo be that no one linemay be found to be acOdmon meafure,to meafure A both the firftline,namely,AB , andalfo the be Bobo Bue line EF: noryet any one fuperficies to mea- {ure the {quare or powers of thefe two lines, : thenis theline EF, ro the firft line fet and fuppofed , incommenfurable both in length 6 ; and in-power. As is fuppofed-to bein this example. Thus mayye fee, howto the fuppofed D Vito the few: line firft fet may be compared infinite lines, 4... ei, re fup= {ome commen{urable both in length & pow- pofe ey ides er,and fome commenfurablein poweronely, = F = _— di i and incommenfurable in length, and fome ge nis age incomnrenfirable both in power & in length. And this firftline fo fet,whereunto, and to whofe {quares the other lines and their f{quares are compared, is called a rationall line,commonly of the moft part of writers.But fome there are, which miflike that ir Why /ome fhould be called a rationallline,& that not without inft canfe . In the Greeke copyitis méfitke that called y'y7%,7ere,which fignifieth a thing that may be fpoké,& exprefled by word,athing the dine fir certaynegraunted and appoynted . Wherefore Flufates ,.amanwhich beftowed great fet fhonld be trauell and diligence in reftoring of thefe elementes of Exclideleauing this word ratio- ¢4Meda ratio- nall,calleth this line {uppofed and firft feta line certaine, becaufe the partes thereofin. %4/ “ne. to which itis deuided are certaine,and known,and may be expreffed by voyce,and alfo ¢ coumpted by number: other lines,being to this line incommenfurable,whofe parts P’/ates eal- are not diftinGly known,but are yncertayne, norcan be exprefied bynamenoraffignd “th this linea by number, whichare of other men called irrationall, he calleth yneertaine and furd /#"¢ certaine, lines. Petrus Montaureus although he doth not very wel like of the name.yet he altereth itnot,but vfeth it in al his booke.Likewife wil we doo here, for that the word hath bene and is fo vniuerfally recemmed.And therefore will we vfe the fame name, and callita ra- tidfiaitline.Por itis not fo great amatter what names we geué to thinges, fothat we fully vnderftand the thinges which the names fignifie, . This rational _ this rational! line thus here defined,is thé ground and foundation ofall the propo- /##¢ the groud fitions almoft of this whole tenth booke. And chiefly from the tenth propofition for- #4 ™aner of wardes.So that vnleffe ye firft place this rational! line,and hane a {peciall and continu- — all regard ynto it before ye begin any demonftration, ye fhall not eafely vnderftand ir, pea “4 Foritis as it were the rouch and trial! ofall other lines, by which itis known whether EN ae any of them be rationall or not. And this may be called the firftrationall line, the line 7 hae Be 3 rationall of parpofe or arationall line fet in the firtt place, and fo made diftin@ and fe- oie oc nered fromm other rationall lines,of which hall be fooken afterwarde,And this muft ye sah : OF ie ellcommit to memory,® = 6 Lines T hetenth Booke.. The fth 6 Lines which are commenfurable to this line whether in tenoth and power, defintttons orin power onely,are alo called rational],°*™ This definition needeth no declaration atall, but.iseafily perceiuedif the firt des finition be remembred,which theweth what magnitudes are commenturable, and the third which theweth what lines are commen furablein power. Here note,how aptly & naturally ,Euclide in this place vfeth thefewordescommenfurable either in leneth and power,orin power onely. Becaufethatall lines which are commenfurable in length, are alfo commenfurable in power: when he {peaketh of lines, commenfurable in lésth, he euer addeth andin power,but when he fpeaketh.of lines commenfurable in power; he addech this worde Onely,and addeth not this:worde.in length, as: he:in- the!other added this worde in power, For not ail lines which are, commenfarable,im power, are tiraight way commentfurable alfo in length, Of this definition, takethisexample, Let the firft line rationall of purpofe, nall or irrationall,So that no lines. can be called rational in length,or in power,nor like wife can they be called irrational] in leneth,orin power,Wherin vndoubtedly Campa- nus was deceiued,who yfing thofe wordes & {peaches indifferently,canfed & brought in great ob{curitie to the propofitions and demonitrations of this bok fhall | Call DE. -—-——-..nickeeeeee de Cambinus a hath caufed much ob {cure- £ie in this senth booke. of Euclides Elementef. Fol.231. eafily fee which ‘marketh with diligence the demon ftrations of Cumpanus in this booke, e 7 Lines which are incommenfurable to tke rationall line , are called ire rationall. By lines incommenfurable to the rationall line fuppofed in this place , hewnder- ftandeth {uch as be incommenfurable vnto it bothin length and in power. For there ure no lines incommen(urable in power onely: for it cannot be that any lines fhould fo be incommenturable in power onely,that they be\not alfo incommenfurablein length. What fo ever lines be incommenfurable in power , the fame be alfo incommenfurableé in length Neither can Exchide hete in this place:meane lines: incommenfurable ta length onely,for in the diffinition before , he called them rationall lines, neither may they be placed amongtt irrationall lines. Wherfore it remayneth that in this diffintion he fpeaketh onely. of thofe lines which are incommenfurableto the rational! line firkt geuen and {uppofed,both in length and in power.Which by all meanes are incommen {urableto the rationall line,& therfore moftaptly are they called irrationall lines, This diffinirion is eafy to be ynderftanded by thatwhichhath bene fayd before. Yet forthe more plainenes fee this example. Let the ftrftrationall line fuppofed, be the line. AB, whofe {quareor quadrate,iet be ABCD. And ler there be geuen an othorlincE F whichlet- beto the rationall line incommenfurablein 4 Jength and power, fo that let no one line mea- fure the length of the tewolines ;ABand EF3.: and let the {quare of the line EF be EF GH. Now ifalfo there be no one fuperficies which mieafareth the two fquaresABCD,and EF G Hiasis fuppofed to be in'this example, thé is the line E F an irrationall line, which word irrational (As beforedidthiswordrational) ¢ D mifliketh many learned in this knowledge of 4, | Sig Geometry.Flufates, as heleft the word ratio=. nall,and in fteade thereof vfed this word cery E+-—-——— F taine,fo here he leaueth the wordirrationall, — and vfeth in place thereof this word vacertaine, and euer naimeth'thefe lines yneertaine lines. Petrus Montaureusalfo mifliking the word irrationall, would rathet hage them to be called furdlines, yet becaufe this word irrationall hath ener by cuftomeand long vfeifo generally bene receiued , he vfeth continually the fame. InGreeke fuch lines are called ddcyot,alogoi,which fignifieth nameles, vnfpeakeable, vncettayne, indeterminate, and with out proportion : not that thefe irrational lines haue no proportion at alll, 'ei- ther tothe feihvationall line , or betwenethem felues:but are fonained , for thattheyr proportions té the rationallinecannot beexpreffed in number. Thatis va doubtedly very vntrue,which many write, that their proportions are vnknowne both to-vs and te nature.Is it not thinke you a thing very abfurd to fay that there is any thing in nature, and produced by naturesto be hidde from natere ;and notte. be knowne of nature ? it can not be fayd that their proportions are vttetly hidde and ynknowne to vs(mueh leffe vnto nature ) althongh we cannot gene them theitniameés ; and diftinGly expreffe them by numbers : otherwife thould Euclde haye taken all this trauelland wonderfull diligence_beftowed in this booke,in vaineand to novfe: in which hedoth nothing ells but teach the propricties.and paffions of thefe irrationall dines;:and fheweth the pro- pottions which they haue the one tothe other. : Hs 207 =: | Here is alfo to be noted , which thing alfo T artalea hath before diligently. noted, that Campanus and many other writers of Geometry ;ouer mucherred and weredeceie ued in that they wrote and taught, thar allhefe lines whofe fquareswere notfignified and-smought be exprefled by a {quare number ( although they might by anyother number: as by 11. 12, 14, and fuch others not {quare numbers ) are irrational lines. Which is manifeftly repugnant to the groundes and principles of Euchide,who wail,that alilines which are commenfurable to the rationall line, whether it be in length and power B E Fr The feuenth definition Flu [sates iw fleede of thss ‘word srratio~ nali vfeth this Word yncer= tayne, Why they are called trratige nall lines. T he tenth Bookes > powerjor in power oncly,fhould be rationall. Vndoubtedly this hath-ben¢e one of the aon of Chiefeft and greateft canfes of the wonderfullconfufion and darkenes of this booke, e 1 senife whichfo hath tofled,and tormoyled thewittes of all both writers andreaders,matters | an ome * andicholets, and {o ouerwhelmed themi,that they could not without infinite trauell bok? ce J, and {weate,attayne to the truth and perfe& ynderftanding thereof, | ik "J ~ : . - : . | ol , . e. e | . | The cighth oi] 76 Jquare which isdefcribed of the rational right-dine Juppofed , is definition, >> “atttonall. Vaull this-diffinition hath Exchde {et for:h the natureand proprictie of the firkt kinde of magnitude,namely ; ofJinesshow theyare -rationall or irrationall, now he bez ginneth to fhew how the fecond kinde of magnitudes,namely {uperficies ate one tothe otherrationall or irrational!. This diffinition is very playne. Suppofe the line A.B,to be therrationall line having his parts and:diuifions certaynely knowne-the fqnareof whichline 4 | 2- ! Lenci ig let bethe fquareA BC D.Now becaufe itis thefquareofthe baidued | rationaliline AB’, itis alfo called: rational! :andas the line A Byis thehritrationall line, vnto which otherlinesicompa- red.are icoumpt¢d rationall, or irrationall,foisthequadrat or {quare thereof, the firft rationall fuperficies vnato which all other {quares or figures compared, arexoumpted andna- ~ med rationallor irrationall. | = le SN @ ———— — oo iM } th + 4 ao i i mil} ie Eth: i a i i H ; ql | ‘ (atl aero ens’ —— 9 Such which are commenfurable vntoityare rational. sacs In this diffinition where it is favd,fuchas arecommenf{urable to the fquare of the it ratiohalldine,are not yvnderftand onely other fquares or quadrates,butail other kindes of refiline figures playne plats & fuperficieles. Whatfo ener fo thatifany fuch figure be comenfurable vnto _ errit} thatrationallfquarest x 2 Viggetecine roA B is alfo rationall.As {up : pofe that the fquare of the rationall line, whichis alfo rationall, : be ABCD: dup pole ar. fo donre-other; fquare G H asthefquare EF GH, to. be-commenfurable | to.thefame: théis the Ste yaty fquare-E F.G-H alfo ra- | ‘eterno ti9nal.So alo if the reGiline figure K L M N,whichisafigure on theone fide longer,be commenfurable vnto the fayd {quare as is fuppofed.inthis example, it is alfo a rational {uperficics and fo of all other fupcrficieles, 3 2106 Sach-which are incommenfurable¥nto it are irrational, The tenth BIT) i | defiustions oso Whereitisfaydin this diffini tionfuch which are incommenfu- rable; itis generally to be taken: aswasthis word cOmenfarablein the diffinitié before. For al fuper- ficiefes, whethet they be {quares or figures\oh:the:one fide longer, orotherwife whatmaner. of right lined figure fo euerit be,if they be incommenfurable vnto:the ratio- nall{quare fuppofed , thé are they _ os ——— —— of Euclides Elementes. Fol.23% irrationall,As letthe fquare A BC Dbethe {quare of the fyppofed rationall line which {quare therefore ts alfo rationall: fupoofe alfo alfo an other {quare, namely the {quare E,fuppofe alfo any other figure, as fer example fake a figure of one fide longer, which let be F. Now ifthe {quate Eand the igure F , be both incommenfurable to the ratio. nall {quare AB C D,then is ech of thee figures E & F irrationall. And fo of other. 11 And thefe lines whofe prveres they are, are irrationall. If they be Squares, then are their fides irrationall If they be not {quares, but ‘fome other rectiline figures then Jhall the lines whofe J quares are equall to thefe reciiline figures, beirrationell. Suppofe that the rational] {quare be ABCD. Suppofe » ___ F fe BS B alfo an other {quare,name ly the {quare E, which Jet = if c beincomé{urable to the ra ae: tionall fquare, & therefore is it irrationall: andletthe fide or line which produ- Re meee AR ceth this fquare be theline 7 G i G:then fall the line FG Sees eee by this diffinition be an ir- rationallline : becaufe itis | the fide of anirrationall (quare . Let ufo the figure H being a figureon the one fide longer ( which may be any other re@iline figure re@angled or not rectangled, triangle, pentagone,trapezite,or what fo cuer ells) be incommenfurable to the rationall {quare AB C D,then becaufe the figure His not a{quare,ithath no fide or roote to produceie yet may there bea {quare madeequallyntoit : for.thatall {ach figures may be reduced into triangles,and fo into {quares,by-the 14. of the fecond., Suppofe that the {quare Q | be equall to the irrationall figure H.The fide of which figure Q let be the line K L:then thall the line K L be alfo an irrational Lne,becanfe the power or {quare thereof,is equal to the irrationall figure H:and thus conceiue of others the like. Thefe irrationall lines and figuresare the chiefett matter and fubie&, which isen- treated ofin all this tenth bookes the nowledge , of whichis deepe, and fecret , and pertaineth to the higheftand moft worthy part of Geometrie, wherein ftandeth the pith and mary of the hole {cience: the lnowlede hereof bringeth light to all the bookes following,with out which they are hard and cannot beat all vnderitoode. And for the more plainenes,ye {hall note,that of irrationall lines there be diners fortes and kindes. But they, whofe names are fetin a table here following , and areinnumber 1 3. arethe chiefe,and in this téth boke fufficientl j for Euclides principall purpofe,difcourfed on, A mediall line, A binomiall line. A firft bimediallline , Afecond bimediall line, A greater line. Aline containing in power arational fuperficies and a medial! fuperfictes. A line containing in power two med.all {y perficieces, Arefiduall line, A firft medial! reGduall line. A fecond mediall refiduall line. A leffe line. A line making with a rational! {uperfcies the whole fuperficies mediall. A line making with a mediall { uperfices the whole fuperficies medial. Of all which kindes the diffinitions together with there declarations thalbe fet here after in their due places, DD). T he The eleuenth definstion. EO. TI. i3, it 4 , ee f tr ee th i ‘" 7 fe. - ’ ee ‘ a ig 4 ‘ . Or Um) VA “ahi << Z Pee =~ an > — = my > OS 19 = . 55 ET - a Sa = aa =—— ——————————— — ’ = > = = == = - —————— -~+ ——— a es pe =r = a. > > -- - ——— — — ~- — ——_. ow = e an = 7 _ ak 7 a GA —< = asigtly — 2 : jeg mo : te = some - = : ¢ : Fe set ea ee = —— —— “on —— . = ae Se = == = a - Sse Ke dl = a . \ = —— - anand — Construction. ae Demon ftra- Bion. A Corollarye —— ee eee et ct EOL A AE Thetenth Booke q The 1. Theoreme. The 1. Propofition. T wo vnequall magnitudes bemg geuen, if from the greater be taken away more then the halfe; and from therefidue be againe taken-away more then the halfe, and fo be done ftill continually there hall at length be left acer taine magnitude leffer then the leffe of the magnitudes firfigenen. : WEEE Fa V ppofe that there be tmo unequall magnitudes AB, anad°C, of iN NS I~ |A which let AB be the creater.:T hen 1 [ay, that if from AB, be Ps = taken away more then the halfe, and from therefidue be taken a- | gaine more then the halfe, and fo flill continually, there fhall at S | thé length be left a certaine magnitude, leffer then thé lefse mag- OM | nitude cent namely,then C. For forafmuch as€ ts the lefve mag- Se A) | zitude,therefore C may be fo multiplyed,that at the leneth it will . % ; Oe : AD fs a5 Pe ay a AS /. | be greater then the magnitude AB b 7 the §.ae fiaition of the Tift ser then AB, beDE. And deuide D E into the partes equall D untoc, which let be D F, FG, anaG E.. And from the mag- y pitudes A Btake away more then the halfe , whichtet be BH: Arie | and acaine fromA H, take away more then the halfe, which let he 1 K:.And {0 do continually untill the dinifions which are ly in the magnitude. AB, be equallin multitude vato the diui- fions whichare in-themagnitude DE.S0 that let the diuijions AK, KH,and HB, be equallin multitude vnto the dinifions H DF, FG, andG E. And forafmuch as the magnitude D-E 1 cater then the magnitude AB, and from D E is taken away leffe then the halfe, that is, E G ( whith detrattion or taking a- way is under ftand to be done by the former diuifion of the mag- nitude D Einto the partes equall unto C ; for as.a magnitude is by multiplication increaf ed , [0 1s it by dimifion diminifbed ) ana from AB 4s taken away more then the halfe, that ts, B H : therefore the refidue G D is greater then the refidue H At which thing is most true and moft-eajte ta conceaue, if we remeber this B C principle, that the rejidue of: a greater magnitude, after the ta- | 4 king away of the halfé or leffe then the halfe, as ener Crealer then the refidne of A lefse MACib~ tide, after the taking away of more then the halfe) « And fora[much as the magnitude GD is greater then the magnitude H A, and from GD 3s taken amay the halfe; that is, GF : and ‘from A His taken away more then the halfe, that is, HK: therefore the refidue D F . greater then therefidue A K (by the forefayd principle) . But the magnitude DF ts qua rt ——— unto the maenitude C (by [uppofition) . Wherefore alfo the ma cnitude Cis greater then'the o magnitude AK . Wherefore the magnitude A K isleffethen the magnitiae Cs ene of the magnitude AB is left a magnitude AK lefve then the lefve magnitude Siig on: ip then C «which was required to be prouea In like fort alfo may it be proued ifthe na Lifes 06 taken away. “eA Corollary: Of thisPropofition it followeth that any magnitude being geucn how litle’ foeuer it we , wii Aug + de leffe then it ; fo that it is impoflible that any Meo be, there may be geuena magnitud ag I a gs of Euchdes Elementes\ Fol.232. nitude thould be geuenthen which can be geuen noleffe.. «| Amother demonftration of the fames. Suppofe that the two wacquall magnitudes ceuen be AB and. Andlet C be the leffe. And forafmuch as Cis the lefie,therefare'G may [abe multipled, that it fall at the length be greater then AB. Letit befo multiplyed, and let the weitltiplexof © exceaine AB be the magnitude FM. Anddeyide¥ Minto bis partesequall unto C,.thatis,into the eaTi2h- tudes MH, HG, andGF And from AB take away more tien the halfe, which let be the magnitude BE : and lukewilefrom E-Atake amay againe more. then the halfe, name ly, the magnitude E D. And thus do continually wntill the disifious which are iw the magnitude FM , be cquall iv multitude to the dtuifions which are in the magnitiide A B: and let, thofe diuifions be the magnitudes BE;ED., and DA, And how persltiolec the magnitude F M is to the magnitude C; fo woultiplex let the maenitude-K xX. be to the magnitude DA. And denide the magnitude KX into the magnitudes equall to the magnitude D A: which Yet be 1 L,LN, and. NX. Now then the dinifions which aretn the magnitude K X, are equall unto the diuifi-- | ons which are in themaenitude MF. And foralmuch as B Eis ereater then the halfe of A B, therefore BE is greater then the rcfidie E A. Wherefore BE is much more greater then DA. But D4 is equallunte XN. A Wherefore BE 1s greater then XN. Agatne forafmuch - C 45 DE ts creater then the halfe EA, therefore aS | TS D Eis greater then the refiaue D A:but D Ais equall vnto LN : wherefore D E is greater then L NW her. fore the whole magnitude D B is greater the the whole magnitude X L. But D A is equallvntoL K . Where. fore the whole magnitude 1B js greater then the whole magnitude KX. And the whagnitude M Fis ] ereater then the magnitude BA: wherefore MF ts | much greater the RX. And forafmuch as thole mag. } mstuaesX N,N Lie» LR, are equal the one to the o- 3 | ther, cy likewifé thefe magnitudes MAG AndGF, areequal the one to the other,cy the multitude of thofe magnitudes which arein M F, is equall to the multi. tude of thofe magnitudes which aye jn KX: therfore as K ListoF G,foisL N to GH, and NX to H M . Wherefore (by the 12.0f the fifi) «sone of the antecedentes,namely, K Lis to one of the confequentes, namely, to FG, foare all the antecedentes namely, the whole K X to all the confequen tes namely, tothe whole FM. But FM is greater then RX. Wherefore FG js greater then L K. But F Gis equall untoC: and K L vnto D A (by fuppofition) Wherefore the magnitude Cis greater then the magnitude AD : which was required to be proued, =e wy See Saas ot Me = | q Ube 3 T heoreme. ; ae The 2.Pr opofition. Lwo bneguall magnitudes being geuen, ifthe lefSe be continually taken from the greater, ¢> that which remayneth meafureth at no time the ma 1g nitude going before: then are the ma enitudes geuen incommenfurable, PD§. Suppofe Couftructions- Deszom Fran F202. re a ~ A T hetenth Booke ° i; 3 , 5 > ; 1 > This propoft ¥ ppofe that there betwownequall magnitudes:A Bana . sion teacheth Al C D,and let AB be the lefse : and taking away continu- chat incont=’ |KX>) x ally by a certaine alternate detraction the le[se from the ; nuall quantt Io greater, let not the refidue meafure the magnitude going tie which the before . Then I {ay; thatthofe two matnitides A B and CD, are in- fal bee commen furable For i they be commenfurable, then (by the firft de- sn difcrete jinition of thetenth ) [ome one magnitude [hall meafure then both: quantity. Let there be [uch a magnitude, if rt be pofible, and let the fame be E. Confirattion. And let AB meafuring DF; leave a lefse then it [elfe,mamely,C F (thatis,from the greater magnitude C D; take away.a certayne part as DF, whichJet-be equallto.A B, or ifit be net.equall.vnaro it; yetletitbe fuch,thae qd thatlefle magnitude A B being more then once repeated may make the mag- nitude D F*: For this is the meaning of this ler" 4B meafuring D F, €5¢. And this decraction madejofrhé lee I fay out ofthe greater,let there be leftof the greater a certaine porton.G Fy lefe then thema snitude AB. And this isthe e -hat which in the Theoreme was faid, dad that which remaineth men meaning o ‘ fareth atno timethe magnitude going before’) . Likewife let CF meal uring BG leaue a lefse then it [elfe,namelj, AG; and do this continually as often as neede requireth, vatill there be found {uch a magnitude that B E D js lee then E.: which mus needes at the length happen (by the Propofition coing before) . Let there be found [uch a magnitude leffe then-E, which ler be AG ..And fora{much as the macnitude E meafureth the magnitude A B, but AB meafureth DF : therefore E meafu- Dewoutra- eth the magnitude D F (by this common fentence , 4 ™agnt ude menfure an other magnitude st tion leading to foal alfo meafure enery magnitude whom it meafureth ) Andait meafureth alfo the whole CD (for GH GDH’ Hy 5. Fiyy nofed to be a common meafure.to the macnitudes AB and CD). Wherefore alfozt meafureth the refidue CF (by this common fentence, if a maguitude meafarean whole and a part taken away, it foall alfo meafure the refidue ). And foralmuch as E meafureth C F, but CF msea- fareth BG, therefore E alfo meafureth B G (by the firit ‘fore{aid common fentence) : andit snealureth the whole AB .Whereforeit fall alfo meafure the refiaue AG (by the other fore- aya common fentence ) namely, the greater wa nitude [ball meafure the lefve : which is ime pofsible Wherefore no inagnitude mealureth thefe magnitudes A B ana CD. Wherefore the magnitudes AB andC D areincommenfurable . If therefore two unequall magnitudes being geuen,the leffe be continually taken from the greater ,and that which remayneth mea- fureth at no time the magnitude going before, thea are the magnitudes geuen incommen|u- rable « which was required to be demonftrated. ¢ A Corollary added by Montaureus. By this propofition itis manifeft, that if the two vnequall magnitudes geuen benot scommen{urable, but commenfurable,then the leffe being continually fu btrahed out ofthe greater,the -efidue fhall-ofneceffitie meafare thatwhich went before. aa =x Lid and ——— -- S eee + eee ee ae = RX : a = ae - = ———— — * - ’ . Son aed aes — : ; — : . 2 ; 3 : aonb pone = - = . —- —— = —< ~ -, oe = re eS =r ore = = ~ —* a = - a ; = —— — 7 : 7 : = Sa ee a le gee BO cet ed? = = “> — Sede ete = — —- - - > — E =e a ps “ = — oe ee 7 ae oS - -. ae ~ = _ - : = <2 ee =z - -— ESSERE. — —; = ~~ y - a ssa _ = ; = >= = <> tts =— = a ————- = - - == —_— on Pe a SS =. = = —<-> - — — = — —- = —- ae - a ES = ~ ee og - _ - eS ee re 3 ™ a - = ~2 a — - - < 2a = —s ~: = “ST Ste —as r= = = ~ — = = — = + q I he 1. Probleme. T he 3. Propofttion. Two magnitudes commenfurable being genen, to finde ont. their greatest common mea/ure: 1 V ppofe that thetwo commen[urable magnitudes geuen be AB & C D,of which ‘Tet AB be the le(fe.It is required to finde out the greateft common mealure of the magnitudes A Band C D.Now AB either meafureth C D,or not. If tt meajure it and feing st alfo meafureth it felfes wherfore A Bis a common meafure we ti€ Tivo cafes in ghis propa/itio. The firji cafe. of Euchides Elementets Fol.224.. the magnitudes AB andC D. And itis manifest thatit is the urea: teft common meafure to thens.F or xo nagnitude ereater then AB ‘ can vieafure AB. But now [uppofe that A Bde not meafure CDs And taking continually the lefse from the creater that which renpate 4 neth fhall at length meafure that which eoeth before/ by the corollas E ry before added ),for that AB analC D are comenfurable.Now then let AB meafuring E D whichis a part of the maenitude'C D, leaue a macnituae lee then it [elfe,namely, EC. And let BC meal uring the magnitude F B, which isa part of the mazaitude AB, leine a leffe thé tt felfe,namely,F A,and let F A precifelymeafure the mag: nitude C E (And this is the meaning of this, Thar which remayneth foal ar the length meafure that which goeth before, whenthereis nothing left after the meaturing made) Ad forafmuch as A F meafureth. the magnitude CE,butCkE meeafureth FB : wherfore AF Alfa méalureth F Ber at meafureth it (elfe : wherfore AF meafureth alfothe whole A B. But A B meafureth D E,wherfore AF allomealureth D E. And A F alfo meafureth CE, wherfore it meafureth thewhole magnitude CD.Wherforethe maguiiude AF meafureth both the magnitudes A Band C.D .Wherfore A F is a common wmieafurevnto A Band C DL fay allo that it is the kreatefi comon meafure vnte there. Ear if PB... -P G not,themts.theve {ome magnitude greater thé the magnitude AF which mealureth both the macwitudes A Band C D.Let there be {uch a one if it.bé poffible,and let the fame be the wa g nitude Gi And foralmuch as GmeafirethA Band AB meafureth ED : therefore G aifa meafureth ED, and by f appofition,it mealureth the whole C D, wherfore G meafureth alfo the refidue C E. But C E meafureth F B,wherforeG alfa meafureth F B.And by [uppofition it meafurcth the whole A B.Wherfore it mealureth the refidue AF namely the greater mag nitude the leffe,which is impoffible. Wherefore no magnitude creater then AE » mealureth thefe magnitudes A Band C D. Wherefore AF is the create common meafure unto the magnitudes A Band C D.Wherfore untotwo SA alt magnitudes ceuen, namely, A Band Dis found out their Greatelt common meafure, vamely, the magnitude AE ; which was required to be done. | Eee ages , zy. owe \ ame ors niet eeaneeeee q Corollary. Flereby it is manifeft that if a magnitude meafure tivo magnitudes, st [pall allo meafure their greate/t common mea/ure. to taken away,it fhal alfo meafure the refidues,of which one is fee by the latterende of the former demonftration, rifit meafure the wholes and the partes the greateft common meafire,as'we may eA ontaurens reduceth this Problemeinto 4 Theoreme after this maner, Two Gnequall and commenfurable magnitudes being geuen, tf the leife do meafure the greater, then is sP the greatest common meafure to them both: But tf not then the leffe being continually by amutuall detrafion (as before hath bene tangh: taken ont of the greater whenfocker the refidue precs[ely, meafureth that which went before leaning nothing, the [asd refidue foil be the greatest common encafare to bath tbe magnitudes Lewen. gf The 2.Probleme. The 4. Propofition. I bree magnitudes commenfurable beyng geuen to finde ont their greateft commonyneafure, DD iy. Suppofe F his propo» ticn teacheth, that in contt~ nial Guanti~ ty chith the 2.0f the fexeeh tencht tt erm bere, The fecend e . fe. Demonfira - ton leading to 4h abi erartics AL Corcllary. This Proe bleme reduced tod The Ga rcme, Fhis propof- tiomteacheth, that $2 contt~ nual guants- ty,whicn ibe 3.9f the fecond Sau QUEL $3 i LETSe Conftrattion. Twe cafes thes Propofi- fiott. Zhe firft cafe. Dewoufira- gion lead: ugto . an abfurditie. The fecond cafe, 22 wf Jemima ‘ o (P« saa FA SESS at » i? o€ r Fr Hroucd VE fa? g s oO se j at ? i he faliio the * : ~ ry . iE MOL VALION« 1 ROPE € oujiruciien. Yeucon fra - tiomleadin ato iy cs Git £0] RV ALT SS The renth Booke tS \9 ; Te he V’ppofe that the shree commen[urablemagnitudes geuen'be A,B,C. It ss required se oo of the{e three magnitudes to finde out the greateft common meafure.T ake (by the Aree) former propofition ) the areatef.common meafure of thetwo magnitudes A,B and let the fame be D.Now then this magnitude D either meafureth the third magnitude which is C,or not.Fir(t let it meafureC. And foralmuch as | D meafureth C, and it meafurcth alfo.the magni- >" tudes A,B + therfore D meafureth the three. magut- ' tudes A,B,C.Wherfore Dis a common meafure Un- to the magnitudes A,B,C And it is manifeft,that tt is the create/t common meajure. For no magnitude greater then D can meafurethe magnitudes.A,BsC. For if it be pofible let themagnitude E being ¢rea- ter then the magnitude D, pieafure the macnitude A,B,C. And forafmuch as E meafureth the macni- tudes A,B,C, it meafurety the twofirft macnitudes AB. Wherfore it [hall alfo (by the Corollary of the former propofition meafvre the ereatelt comon mea- fure of the magnitudes A,B whichis D vamely, the greater {hall meafure the leffe, whica 15 i772 pofsibl é. But now let’ D not mea{ure the macnitnde Cc. FirfP I fay that the magnitudes C,D, are comsmes {it~ fa 7 A? + in b-- ‘ ove nme hy eevee pene mate t-~ D 3 \ ee ee ee C\ rene. nee pm mn mree "A ts riven alto (by the corollary caine before) meafure the greate/t common mea ifure of A,B that is,Ds maenitude C.Wherfore E isa common meafure ’ wW eee eaetinaied ater camteen | QC) 6 | E W+-+— T — ter then.thre magnitude E.And let F meafure the three magnitudes A,B,C. Aud fora{much as Emseatureth the magnitudes A,B,Cyt alfo meafureth the two fir ft magnitudes A,B.Wher fore (hy the coroliary going before) it fhall alfo meafure the create/t common meafure of the “th enitudes A,B-But the ereate/t common meafure of the magnitudes A,B,is D. Wherfore F.meafureth D and it alfo meaf ureth C.Wherfore F mealureth the magnitudes CD.Wher- fare F foallal {omeafure the ereate(t common meafure of. the magnitudes C,D Bat the grea- reft common meafureof the magnitudes C,D;1s E. W. herfore F meafureth E, namely, thé ereatéy,the lelfe,which 1s impo (5 ble.Wwherfore no magnitude greater then E meafuretl the magnitudes A,B,C Wherfore E is the greatest common meajure of the magnitudes AB C. if D do not mea[ure the magnitude C-But if D do meafure C,then 15 D the greatelt comme pbb et three magnitudes commen|urable being geuen,there is found their gréae rei! common meafure «which was requixed 40 be done. Corollary: ~~ ‘y of Enclides Elemenies, Fol.235, gq Corollary, “or } Ai ereby itis manifest that ifa magnitude meafure three ma ignitudes it [hall aifo meafure their greate/t common meafure, In like fort alfo in magnitudes comme/urable hoy many foener bein g gene, may be found out their greate/t common meafure,and the corollary will ener be true. This Probleme alfo Adontaureys reduceth into'a Theoreme after this maner; Tbree magnitude! being commenfurable if rhe Lreatest Common meafure to fw e of theim,do mealure the third it foall be the greatel? common meaflure to all the three magnitudes geuen. But tf #2 do not meafure it the ereateft common mealure of the thirdand of the gréates? cowsmen wmcafure of the two firft, is ibe Ereateft commen mea fare of all the three magnitudes, 7 The 3.T heoreme. Ihe s. Propofition. Magnitudes commenfurable hae uch proportion the one to the other » as number hath to number. EAN, V ppofe that A and B be magnitudes comenlurable.T hen I fay that A hath unto B . é; » WY | tudes A,B, are commenfurable. How many vnities there are in the number D, HMA sn Fo ansrcan causal partes = Pee ee 7 Ga] inato [0 many equatt partes deuide the magnitude. ( by the 9 .of the fixt) and FA ey Pre Pee Ts ; a7 AnD kL’, f tae Fluke z ae let the MaAghibnae é be equall fd 072€ O] Lye pat tes the) of . Aid how WARY UMniULES there a¥ve in the number, E, of [0 many magnitudes equall vuto the magnitude C let the magnitude F be capofed. And let G bewnitie. Now forafmuch as how nrany vnities there are in the nuber -_ it fe Pe Sherfore what part uniti G ts of the nuber D, the fame part t the ie gnitude C of the WAG» wate Re. a Semis EE . ae yitnde A. Wherefore asthe marnituae C isto the magnitude A, fois vnitte G to the num- her-D »Butonitie G weeafureth the number Ds Whereforethe magnitude C alfo meafu- = :--- magnitude C is to the magnitude A, {0-8 V- sb ae aan I tern nl ent fateh nee Se ee | peewee dD i : nitie G tothe number D : therefore contrary- wife(by the Corollary of the fourth of the fift) : : as the magnitude A 1s to the magnitude. C, ; 0 is the number D to vnitie G.Againe foraf~ . ° much as how many vnities there ave inthe @ DG: £ number E,fo many ma enitudes alfo are there inthe magnitude F equall unto. the magut- tude C : therefore asthe magnitude C isto the magnitude F fois vnitt.G to the number E. Aad it is proued that as the magnitude A ts to the magnitude C, [07s the number D to vnt- tie G. Wherefore of equalitie (by the 22. of the fift) asthe magnitude A is to the magnitude F, [ois thé number D to the number E «But as the number: D. is to the number £; fois the macgitudeA to the ma znitude B .. Wherefore ( by the x1. of the fift ) asthe magnitude A i to the magnitude B, [01s the [ume magnitude A to the magnitude F; wherefore:A hath vn- to either of thefe magnitudes B and F one and the [ame proportion .Woercfore ( by the 9.of the fift the magnitude Bis equall unto the magnitude F . But the marnitude C meafureth the magnitude F.: wherefore st al fo. meafureth the ma gnitude Band it likewife meafureth the magnitude A. Wherefore the ma enitude G meaf ureth the magnitudes A and B.Wher- fore the magnitudes A cy Bare commen{urable. If therefore two magnitudes haue {uch pro- portion theonetothe other, as number hath to number,thofe magnitudes are comenfurable: which was required to be proued. | 3 Corollary. of Euclides Elementés. Fol.226. mat - a Corollary. Hereby it is manifeft, that if there betwo nibers,asD and _ AM Corollary. E, and aright line, as A itis poftble to gene an other line, vito which the lini A [hall haue the fame proportion, that the number D hath tothe awmber E ..F or deuide the line A into fo many equall partes as there are unities in the number D (by the .of the fixt) . And take an other line, as F, which let be compofed of {0 many partes equall to the partes of the line A,as there be unities in ihe number E . Wherefore the line A fhall be to theine F, asthe number D istathe number E. And by this meanes you may unto any line cent cene an other line com- men|urable in length . For if two lines be in proportion the one to the other,as number is to number, they [hall allo be commen- furable in length, ty this 6.7 heoreme. ne 0 ep ee Sr tng ? D £ wee I re . A x ¢ An Affumpt, Iwo numbers being geuen, and a io aright line: as the one number is to the other fo to make the {quare of § line geuen to be to the {quare of an other line. Suppofe that the numbers geuen be D and E : and let the right line cenen be A. It is required, as the number D is to the number E, foto make the fquare of the line A to be to the {quare of an other line . As the number D is tothe number E; folet the line A be tothe line F (&y the former Corollary) . And take Pen. Letiwene thofe two lines A and F the meane proportionall ( by toe 13. 0f the fixt ) which let be the line B . Now for that as the number . Demouftra- D is tothe number E, fois the line Ato the Stone line F : and as the ine A is to the line F, fo | is the fquare of the line A to the '{quare of the a F line BY by the fecond Corollary of the 20 of the fixt) . Wherefore as the number D is to Ps Al Sie Beis the number E,fo is ihe {quare of the line A to the fquare of the line B : which was required to be done. a ¢ An other demonftration of the 6. Propofition. Con ft TUCTEON « Suppofe that thife two magnitudes geuen'A and B, haue that proportion the one to the other, that the num- ber C bath to the niber D.ThEI fay; that thofe magni- tudes ave commé/urable. How many vnities there are in the number C, intofo many equall partes let the magni- tude Abe deuided, ¢y let he magnitude E be equall uns to one of thofe parte; . Wherefore as unitie is ta the num ber C, foisthe magnitude E to the magnitude A. And as the number Cis 1 the number D ,[0 is the magnitude A to themaguitudeB. Wherefore of equalitie(by the 22. of the fift)as vnitieis te the nuanber D, fo 1s the magnt- tude E to the magnitude B. But vnitie meafureth the | number na Confiructione Demonfra- tion ° . >ees¢ ° , pis + 6 ey a a eh ee Pnitie . aa = —— ss =~ — - a or LSS SO Se wt Ss iat Demonftra- tion leading to an abfurditie. This 1 the connerfeotthe former,and ts pronedby an gndirect de- wmoufiration. I. Ze T he tenth Booke sumber D . Wherefore the magritude E meafureth the magnitude B . And it alfo meafu reth the magnitude A (for that wnttie mea [ureth the number C ). Wherefore the magnitude E meafureth either of thefe magnitudes Aand B.Wherefore the magnitudes A and B are commen|furable, and the maganide Eis their common meafure. q lhe s. IT heoreme. Lhe 7. Propofition. Magnitudes incommenfurable, haue not that proportion the one to the oe ther ,that number hath to number. be demonftrated. q The 6. Theoreme. Lhe 3. Propofition. If two magnitudes hare not that proportion the one to the other that nums ~ ber hath to number, tlofe magnitudes areincommenfurable. V ppofe that thefe tu» magnitudes A and B,hane not that proportion the one to i f $s f y t fu AA 4h et y . gS 4i2 * - portion the one to the other, that a f Guare 1 umber hatoto a [quare WUBI QEK y ABE HOG ae, s 4 f the; doe CAMA Py Pe LT an Tp wee oa, | Ly I a4 Webat wa P/N t syprl ts i, DY O42 6 AIG fi “Arcos t GOT TE] UF ALLE Lid Pi Fed > Writ WAS A bat WAS veq Hei C4 MLC a wLibe Re ¢ CorroHary- Fler a. P : a 5 ol ¢}, < pe ] Fi inn 2 - BE ee fe Mytort A 2P 7% neti 2p Hereby it ismanifeft that right lines comenjurabie 1 tenet a Ale al/o.euer f 7 Si P om y ° ‘ & ap z fii py 42°n) a" ‘ conimen/urable in power. But right lines commien| 1 able m\pomers ave. uot v 5: . Cc = ee» » > “4 . a ~< commenlurablein lenoth. And right lines incomenfurable i legth J COUPLE FES Mi eeu 'd 1H eng i/e ind Fi? 7 sto e fi mit *\ . mae * j , o v (q le 7 iY }: ge . We a are not alivayes incommen|urable in power. But right untes incommen| ut Aes y -are ener alfo incommenfurable in length. For forafmuch as {quares made of right lines commen[urable in length, haue that pro- portion the one ta the other that a {auare number hath to a [quare number (by the fir{t pare ofthis propofition). ,but magnitudeswbich haue that proportion the one to the other, that number {ymply batpto number jave( by théfixt of: thetenth) commen|urable VW. herfore right lines cotamen|{urableinlength,are conmenfurable not onely. 17 length, but alfa 10 pomer Againeforafmiuch as there-axe cerbaype fquareswhich haue not that proportion the. one tothe other that m{quare nuneber hath tog (qnare numler, vut-yet haue that proportion the. oneto the other which nusiber firmply hath to nuneber: their fides in dedeare in power Cons ~ } f? *J. - ; ze rk f, f 7 mrenlavable, for that they def cribe[quares which.haue.that. proportion wi2top number {itta : - : “ IF thee > s - phbath-t0 number, which {quares ave, therfore commenfurable (Ly. the 6. of this booke) f aa I ' " M7) Rekisc ts st? but the {aid fides arbincommen(ucablein length by.the latter part of this propa Gtion. VERE. efore it is true that lines commenfurable in power, are not ftraight way comimen|urable in wAnd by the felfe fame reafon is proued alfothat third.part of the corllarys that lines incommenfurable inlength, are not abpayes incomamens, urablein power OK, they may be Ui commenf{urable in length but yet commen} urable ia power-As im thofe [quares mhiee Are te proportion the one to the other, as number isto number, bik nok as Af quare.mumvert5 60. [dnare numbers». © Het 2 | . < Banright lines incdoomenfurabletn power are almayes al{oincommen urable.in lengthy Fortf they becommen| unable in lengthsthey fhalalfo be comanen|uracle in pomer b y tiie Soft p nrt-0 f this Cor ollary Beat they AKe fuppofed tobe ICOWEIRET | UTA ble.in lengt By which is 403 - 7 : ; . . ar 0 4” 937038 2H » . is , ; furde.Wherfore right lines incommenfurable in power are Cher TAC ONNTET s[urable ie length. ‘For . = a of Euclides Elementes. _ Fol.239. _ For the better vnderftanding of this prop ofition and the other following, T' Haite here added certayne annotacions taken out of -Menraureus. And art as touching the fignification of wordes-andtermes herein vfed;which arefuch,tharvalefie they be well marked and peyled,the matter will be obf{cure and.hard,and 11 a manerinexplicable. Firit,this ye aruftnote;that lines to becommenturabletmteneth; and lines:to.be in proportion the one to the other,as number is ton omber ts allone.Sothat whatfocuet lines are commenfurableimength,are alfo mrproportion theone to the otherjas nanie ber is to number.And conuerledly what fo ener lynes are in proportién the oneto the other,as number is to number, arealfo comméfurable in length,as it is manifelt by the 5 and 6 of this booke.Likewife lines to beincommenfurableinJength,and not to bein proportion the one to the other,as number is to number is allone,asit is manifedt by the 7.and 8.of this booke. Wherforethat which is fayd in this Theoreme, ought tobe vnderitand of lines commenfurable in length,and incommenfirable inlength. This morcouer ts to be noted,thatit is not all ote,numberstaibe fguare numbers; and to be in proportié the one to the other, as a fquare number is to a fqtvare Aumber: For although fquare numbers be in proportion the one to the other, as 4 {quare num- beris toa {quare number, yet are notall thofe numbers which are in proportion. the one to the other,as a {quare number is to a {quare number, {quare numbers. For they may be like fuperficiall nunibers,and yet not {quarenumbers,which yetarein propor- tid the one to the other,asa {quare numberis-to-afquare number.Although all fquare numbers are like fuperficiall numbers.For betwene two fquare numbers there falleth one meane proportionall number (by the 11. of the eight). Butif betwene two num: bers,there fall one meane proportional! number,thofe two numbers are like fuperfici- all numbers(by the 20.0f the eight). So alfo iftwonumbers bein proportion the.one to the other,as a fquare number is to afquare number, they fhall be like fuperficiall ni- bers by the firit corollary added after the laft propofition of the eight booke. And now to know whether two fuperficiall numbers geuen,be like fuperficiallnum- bers or no,itis thus found out. Firft if betwene the two numbers geucen, there fall no meane proportionall then are not thefetwo numbers like fiuperficiall numbers(by the 18.of the eight.But if there do fall betwene them a meane proportionall.then are they like fuperficiall numbers (by the 20.0f the eight) Moreouertwo like fuperficiall nnm- bers multiplied the oneinto the other,do produce af{quare number (by the firltof.the ninth). Wherfore if they do not produce afyuare number,then are they not like fupers ficiall numbers. And if the one being multipliedinto the other, they produce a fquare num ber;then are they like fuperficiall(by the 2 .of the ninth); Moreoner if the faid two fuperficial numbers be in fuperperticular, or fuperbipartient proportion, then are they not like fuperficiall numbers. For if they fhould belike, then thould there be a meane proportionall betwene them (by the 20.0fthe eight). Butthacis contrary to the. Co- tollary of the 20.0f the eight. 3 -- And.the eafilier to conceiue the demonftrations following, take this example of that which we haue fayd. Suppofe that there be aline, namely,C, which imagine to be foure fote long: and let there be an other line Dy whichletbe three foote long. And ( by the 13. of the fixt)itake the meane propor- tionall betwene thelines A, D, which fet be the line B. Wherefore the fquare of the line B ihall be equall to the rectangle parallelogramme:contayned:vnder the line C and D ( by the 17. 0f the fixt) «Which {quare fhall contayne 12, foote, & fo much alfo fhall the parallelopramme defcribed ofthelines C & D containe. Take alfo ewo other lines E and F, of which let Ebe 3 .footelong,and let Ebeafoote long. And let the meane proportionall betwene the!ines Eand F, be the line A. Now thenthe {quare of the line A fhall containe 3.foote, as alfo doth the parallelograme defcribed of the fines E,F.Thé I fay, that the {quare of the line B, which cétaineth 12.foote, is to the {quare of the line A; which contayneth 3. foote,in that proportion that a fquare number is toafquare number . For as the number 12.18 to the number 3, fo is the {quare of the line B, which containeth 12. foote, to the {quare of the line A, which contayneth 3 foote . But the numbers 12.and 3. are like fuperficiall numbers , for the fides of 12.which are 2. and 6, are proportionall with the fides of 3. whichare: r.and 3, Wherefore the {quare of thejine B, which contayneth 12.foote, fhall be-vnto the fquare ofthe line A,which contayneth 3.foote,in that proportion thata like fuperficiall number is to alike fuperfici~ all number. But like fuperficjall numbers are in proportion the one to the other, asa fquare number is , EE A ° to a Certayne ane notations ous of Montau- ae St | Rules to know whether twa Juperficiall numbers be like OF HGe : The tenth Booke. eo a fquare nfiber, which fquare num- PI bers are 4. andi. ¢ ( by the 26. of the eight).. Wherefore B 42" * the {quare of the jine B, which con- % tayneth 12-foote, is Do ttt to the {quare of the line A,which cotai- neth 3.foote,in that proportion that a fquare number 1s to- a f{quare . number, namely , that: the number4.is to the number 1 :. which proportion !s qua- druple . For the greater {quare whi- che is 12, contay- neth the leffe {quare which is 3, pt times » Wherefore the fide of the fquare 12, Which is the line B,is double to the fide of the fauare. 3, which 1s the line A. Wher- fore the’ line Bis to the line A.,in.that proportion that nuinber is to num- Be 4% ber. Wherfore (by the-s.of this booke) thelines B & A arecommenfurable in length . Which isa fuppofition neceffary to conclude the firft partofthis Theoreme, namely, that the fquares of {uch lines are in proportion che one ro the other, thata {quare number is to.a {quare number. So alfo the naber which denominateth the greater terme of the proportion of the line B tothe line A.whichisa,ifit be mulciplyed into it felfe, it maketh a {quare number,namely, 4. Likewife the num- ber which denominateth the lefle terme,namely5t. if it be mulciplyed into it felfe, it maketh no more But z. Which vnitie is alfo in power a fquare naber.Wherfore the {quare of the line B,is to the {quare of theline A, in that proportion that a {quare number is to a {quare number,namely, that4.is to 1. By this-you fee (which thing was before noted) that itis not all one,numbers to befquare numbers, and to be in proportion the one to the other, asa {quare number isto afquarenumber, For itis manifett, thatthe numbers 12. and 3. are not fquare numbers, when yetthe {quares expreffed by thofe num- bers are in that proportion. But the fide of the {quare 12.although it can not of it felfe be expreffed by number diftinéily, to fay that the fide thereof is fo many foote long,which feete fquare taken make the whole fquare ta : yet being referred or compared to an other thyng , namely, to the fide of the {quare 3, which fide alfo of it felfe can not be exprefled by number, it is vnto the fayde fide of the{quare 3,in double proportion . For the one {quare being quadruple to the other {quare (as 1s the fquare of theline B, which contayneth 12. foote, tothe fquare of the line A, which contay- neth 3.foote) hath his fide double to the fide of the other {quare, by this generall Corollary of thezo. of the fixt; Uke reFslsne figures are sn dowble preportion the one te the other that their fides of lke proportion are. Now if aman will fay, that the fide of the f{quare 12z.'may .be meafured,forthat hys proportion which it hath to the fide of the {quare 3,is meafured by 2 (forafmuch as it is dupla propor- tionsthisis to be confidered,that in fo faying, you fay not,that that magnitude cansf itfelfe be meafu- red, but the proportion therof . For,that magnitude,namely,the fide of the fquare 12,fhould by it felfe be Se te without any re{pect of the proportion of it;to. another thing, we may fay thar the fide of the {quare, which contayneth 12.foote,is fo many foorelong,the num ber of which foore multi- plyed into it felfe thould.make that number.12. Bur this 1s not poflable,for that 12.1s not a {quare num- ber. Wherefore thus you may fay : In afmuchas that fquare 12. 1s confidered by it felf, without hauing any refpeét of the proportion ef it to any other thing,butonely as it is 1z.foote,it hath no fide which of it felfe can be expreffed by number . Bur ifitbe compared to any other thing, namely, to the {quareof 3.fo0re,then may you fay that the fide of the fquare 12.182, andthe fide of the fquare 3.is 1. But chysis the denomination of chat proportion whichis called duple, which proportion cannot be or confidered o WwW ——s ¥ of Euclides Elementes Fol.24.0. in fewer termes then.two, when as it isa relation of one thing to an other thing : wherefore 2. isnot: the number of fuch feete,of which there are 12.1n the fquare , Agayne,if the number 2. fhoulde be the ide ofthe fquare 12, fo that that fide fhould be 2, then of the multiplication of 2.into it felfe, fhoulde not be made that {quare 12,but.an other fquare which fhould be 4.foote : as ofthe number z..multiply- ed into him felfe is produced the {quare number 4. Neitheralfo ifany other number,meafure the fide of the fquare 12,and the fayd number be multiplyéd into him felfe fhall it euer make the number I2. When yer all numbers denominating the fide of any {quare number, if they be multiplyed into them felues;thev make the number-which denominateth. the (quare, whofe- fides they denominate. As 2: multiplyed into him felfe maketh 4 : 3. maketh 9 : 4.maketh46 : and folikewife ofall others. Where- fore it is notalloune,magnitudes to bein proportion the one to the other,as number is to number, and euety one of them to be mealured b him felfe without any re{peét had of the proportion . As here the fide of the fiiare 12.can of it felfe by no meanés bé meafured,b ut being com pared to any other magni- tude ,namely,to the fide of the {quare 3,the proportion thereof is exprefied by number, So alfo the fide of the fq uare 3,and of all other {quarefigures;whofe-areas yet can not be exprefied by fquare numbers. And that whith we here fay, is manifelt euen by the wordes of Ewcide inthe 5.6.7. and 8. Theoremés of this booke. Where he fayth nor, that magnitudes commen{urable and incommenfurable are of the elues or of their owne nature exprefled by numbers, but that either they haue or haue not that propor- tion which number hath to number . Which thing not being well confidered,it fhould feme hath cau- fed many to erre as hereafter fhall be made manttelt. And in deede they which haue demonftrated this Theoreme, may feme to fome rather to have demoftrated it particularly & not vniuerfally.And doubr- les I iudge there are fome which vnderftand their fayinges otherwife then they ment: when as they thinke,that they fuppofe certayne lines not onely co.unmenfurabie in length,as they are fuppofed:to be inthe Propofind, but alfo fuch, that.ech of them apart may be exprefied by fome'cer- tayne number. W herfore for wantof right | | vnderttading, this mought they fay.of their SSRs SSR Rees demonitrations: that wheras they thought | | that they had concluded that generally, SENE eR; Ck aS which isin tats theoreme of Exchde, Squares | | | | | | defcribed of lines comeénfurable in length,are in 2] j++ proportio the one to tbe other that a [quare it | | | | | ber 73 fo a fgunre 1248 298 ber: they conclude par- mera Natt Datel ict 52 ticularly, thys onely Squares defcribed of é | | lizes which may by them felues be exprefedb ry Ake eS rs ee aa : forme certaine number, ave in proportion. ee. A which yet is otherwife,and theit deméttra- tions are right & agreable with the Theoreme.Onely the picture of the figures which the Greeke boke hath, may {eeme to bring fome doubr.-For the {quares are fo.defcribed with certayn litle areas,that the number of them may be denominated by a {quare number : whereby it mought feeme that the lines A-& B which defctibe the {quares,ofight to ; efuch that they.may be exprefled by fome : : certaine ntiber,Ass the line A to be s.foote, and the ine B 3. foote.As the two former fi eures here fet declare. Which thyng yet Ez- cide fuppofeth not, but only requireth that 2,5 sees they be commenfurable in length, asin the fornier. example of the two fquares, the whole area-of-one.of which is12; and the ? whole area of the otheris 3. For although their fides ca not by them felues be expref- L4 5 fed by fome cértaynenumber , yet are they commentfurable in length . Moreouer thys defcribing of the {quares of the liues A and B,denided by certaine litle areas,may caufe this error, thae aman ihould thinke thatitis all one two numbers to be {quare numbers , and.to be in proportion the one to the other as a {quare number is toa fqnare number. For the number of the areas in the {quare of the line A is a {quare number , namely) 2: produced of the roore 5, which is the length of thelline A. Likewife the number of the areas of thefquare of the line B; is a {quareniiber, namely, 9, which is pro- duced of the roote 3, which is the length of the line B. Bue we haue before declared that it is not alf oue,numbers to be called {quare numbers,and to bein proportion the one to the other,asa fquare nii- ber is toafquare number , Wherefore as touching thofe areas contayned in the greater fquare, which is of the line A, and which are in number 25 ,they do exprefie that fquare number 25, which is produced of the numbers multiplyed into him felfe , which number s.is the greater extreme of the proportion betwenes. and 3, which is the proportion of the lines Aand B . And this proportion, namely, of 5.to 3.cauleth that the limes A and B are commenf{urable in length (by the 6.of this booke) . The fame may be {aya alo of the areas ofthe leffe {quare «Neither is it of neceflitie that you ynderitand thofe areas to be fquares,as either feete fquare or pates {quare which make the whole {quare, although in deede they | EE.iij. may Note; The tenth Booke may be fuch,fo that the fides of thofe fquares be ft many foote long, as 5. foote or 3.foote . Howbeit thys is of neceffitie that the numbers which exprele the number oF the feete {quare or pafes {quare,cé- tayned in the {quares,be either both of thé {quare numbers, as in thefe {quare figures ofthelines A,B, or that both of them be like fuperficiall numbers, as in the former {fquares which were 1z.and 3: of which nii- bers itis manifeft by that which hath be- 2$ tibiae hae fore bene faid,that they are like fuperficiall numbers, and therefore haue that propor- e tion the one to the other, thata fquareni- A bér hath toa fquare number. And therfore you may defcribe the fquares of the lines oe | A,B, Without any diftinétion of fuch litle Cae ig areas, fo that the {quares may be voyde and emptié,and contayned onely of foure right lines,a in the figure here put. @ An Affumpt. Forafmuch as in the eight Looke in the .6. propofition it was pronea, that luke playne numbers haue that proportion the one to theither, thatafquare number hath toa {quare ee number: and likewife in the 24.. of the fame sooke it was proued, that if two numbers bane Ehes affumpt that proportion the one tothe other, thatafiuare number hati toa (quare number, thofe a8 D ho " OV wer] 2 ana j : , Me Le ig - ca as ‘ ‘i 1 yo s " 7 ' 7 nas eee oF Pet 3 PECOnHCII® — aunebers are like plaine numbers.Hereb y tt ti MARTY [t,tiat vulike Pate numbers, thatis, he as fd whof ¢ fides are i ot proporttonall raibe not tint proportion the one to the other that a {quare z number hath to afauare number. For if the hane,then fhould they be like plaine numbers, which 1$ Co atrary to the [uppofition .W herfor unlike plaine numbers haue not that propor- tion the one tothe other that a fquare number hath toa {quare naber. And therfore [quares whith have that proportion the one to the other ,that unlike plaine numbers haue, [hall haue their fides incoramenfurable in length (by the laft part of the former propofition) for that thofe fauares haue not that proportion the on to the other that a{quare number hath to a - I {quare number. q Thes. Iheoreme. The 10. Propofttion. If foure magnitudes be proportiozall, and if the firft be commenfurable ‘nto the fecond,the third alfo [hd be commenfurable ynto the fourth. And if the fir{t be incommenfurable vito the fecond, the third fhall alfo be ine commenfurable nto the fourth. rs V ppofe that thefe foure macnitides A,B,C,D be propor tional. As Ais to B, | fo let C be to D,and let A be conmen{urable vate B. T hen I fay that Cis alfo Demonfiratio x om >; commen{urable unto D. For prafiuch AS (Als commen|urable Unto B, it ihe Bak CIPS SE dk at hasten Ole maabor Lath tows of the fir Z4A| hath ( by the fift of the tenth) thit proportion that number hath to number Bus parte as Ais to B,fois Cto DWherfore C alfo hats unto D that pro artion that number hath to number. Wherpre C is commen- | furdble unto D (by the 6.of the tenth). But now {uppofe that | Demonfra- themagnitude A be incommenfurable vito the magmituae | tion of tise Bi Then I fay that the magnitude C alfo is incommen|urable fecondpart* — unto the magnitude D.F or fora{much as Ais incommen|ura- : ble unto Bs therfore (by the 7. of this booke).A hath not-unte | B (uch proportion as number hath to number. But as At to B, ee fois Cto D. Wherefore C hath not vnto D jich proportion as | number bath to number. Wherfore (by the $.0f- the tenth) C is : incommen{urable unto D. If therefore ther be fouremagni- = tudes proportional and if the firft be commet{urable unto the : | | [econd, yw Geren eee ere ite nee, DB of Euctides Elementes. Fol.z4t. fecond,the third alfo hall be commenfurable unto the fourth. And if the fir t be incommen: forable unto the fecond, the third fhall alfo be incomsmenfurable unto the fourth : which | was required to be proned. @7A Corollary added by Montaureus; If there be foure lines proportional and if the two fir 5¢,0r the two last be commenfarable im power onely the other two alfo {hall be commenfurable in power onely. This is proued bythe 22. of the 44 orollary. fixt,and by this tenth propofition.And this Corollary Euclide vfeth in the 27.and 28. propolitions of this booke,and in other propofitions alfo. q Lhe 3. Probleme. he i1.Propofttion. Vnto aright line firft fet and geuen (which is called a rationall line) to m Re ae et ‘ id n I ‘ finde out two right lines incommenjurable,the one in length onely, and the other wi length and alfo in power. fa V ppofe that the right line fir et and geuen,which is called a rationall line of NZ _| perpofe be A.It is required vito the [aid line A,to findg out two right lines in- Vy | comzmenfurable,the onein length onely, the other bath in length and in power. eee! Take (by that which was aaded after the 9 .oropofition of this booke) txo nuni- bers B and C,not having that proportion the one tothe other,thata {quare number hath t0. 5 i 10 in lon oth afquare number,that is,let them not be like plane msinbers (for like plaine numbers by the onely tothe 26. of the eight haue that proportion the one tothe other that a {quare number hath to a line geuen, {quare number ).And as the number B-isto the number Cfo let the {quare of the line A be unto the [quare of an other line, samely of D (baw todo this was tanght in the afumpt put before the 6. propofition of this booke.) Wherfore the {quare of the line A,is unto the fquare of the line D commenfurable (by the fixt of the tenth.) | And fora{much as the number B hath not vntethenum ber C, that proportion that a {quare number hath to a Square nuber, therfore the (quare of the line Ahath nat vuto the {quare of y line D, that proportid thatafyuare to finde ont the frF line wncommenfu- number hath to a nuber Wherfore by the 9.of the tenth, z the line Ais unto thelive D incovmenlurablein length oncly. And {ois found out the fir[t line,namely,D incor- ot eh ow menfurableinleneth onelyzotheline cenen A, Acaine ~-\ %\ % sir ef * co o iy ¢ &é 4 + Lay i x. er : To finde ous take (Gy the 13.0f the fixt) the meane proportional be- . she fecond line twene the lines dand D,and let the fame bed Wherfore { ; incommenf{u- as the line A istothe line D [oisthe {quare of the line A : ; rableLothin tothe {quare of the line E (by the Corollary of the 20. of | © +. length andin the fixt). But the line Ais vate the line D iniommen- B power sates ee ASF C line geuttte furable in length Wherfore alfo the fquare of the line A 4s unto the {quare of the line E incommenfurable by the fecond part of the former propofition. Now ford{muchas the [quare of the line A is incdme- furable to the {quare of the line E,it followeth (by the definition of incommen{urable lynes) that the line _Ais incommen|urablein power tothe line E Wherfore vnto-the right line ge- ten,and fir {tet Awhich is a rationall line,and whichis Juppofed to hane {uch dinifions and {0 many partes asye Lift to conceyue in minde,asin this exam ple 11, whereunto, as was de- clared in the §.definttion of this booke,may be compared infinite other lines, either commen- EE My. furable | 1 | | : : i] hit : 7 / ' ; { bia : , ; 1)? \ t : ! - - ‘ aa itd ul Th J { : 1} : he - } } | ig i \'! i ; : Hi ' y ih ‘ at Mh ' ; M shit 4 \ ; zz | : 4 i] \ } { i ni itt } ¢ i t \ : ‘ 7 ' | HH 1} , i} t : ie uy i | : i i " ' | | pl ; : iil ii tik ; ' : 7 - by ‘ } ; He 5 : i : + ih } t ’ i Bt | { ! \ : At ’ Bit ‘ i y ; i Ht i i { t a Pie Ea ; ha] | i hl } mi | Lh | : : p iv ' Te ! itl f 4 ly i : : Bis A ' \ ; a Pi h i ‘ - if \ - t] ' | t | : \! } H wit ne : : i} ni f Halt } | i) a} ; ' : J ea hi ; ’ " i : ; j r ' ‘ i 4 ; § ‘a H a eh | t Hea) el " 4 - 7 ath rr 3 { , t Ti a a If aE ey ‘ Wo Pe i sh ' ' fi iv | P| ae wee 1 i { AWD By aie weet ya * 1% : ie the : 4 i ia it} ies Thi P| ‘ H ; ' im H ; j aie 14 fy ‘ .) : int ‘ ey, ‘ " : a Ee ia } Hi : : t - HME Ba at } i : Ver H yy ' iia ea fe i ae H 72 4 ' ut ee i : th { j ; Hi ee Saal Tae) j 1 “ \ : ip b ‘ ! " t ! ) i } f, + an : ! nap 1 I | hain wht en ae - ui ihe ‘ ‘ : a , ! : ’ } ial) aa a ; in i; / ) } 1 : han f : ih “4 ; ; im bi i | | | . i ; 4) i] ' | it 1 i Confiruciion. Demonfilra- tidus Thee73 with Zameert an A[surapt, but Grrerly smpro- perly: Fluffates maketh st 4Co~ rollary,but the G7 eeke and Montanrens enate tt a pro~ po/stion : bute wery Way an snfallible truth Sf bbe —- Lhertenth Booke {erable orincommenfurable,is found out the line D.incommenfurable in length onely Wher forthe line Dis rationall (by the fixt definitio of this booke). for that it is incommen{ura- bleinlength onely to the line A,which is the firft line fet,and is by {uppofitio rational.T here is al{o found out the line E,which is unto the fame line A incommen|urable, not onely in length but alfo in power which line E compared tothe rationall line_A, is by the definition irrationall.F or-Euclide alwayescalleth thofe lines irrationall, which are incommen|{urable both in length and in power to the line firft fet,and by [uppofition rationall. q. Lhe 9. I'heoreme. he 12. Propofition. Magnitudes commenfurable to one and the felfe ame magnitude : are alfo oa ¢ comimenfurable the one tothe other. I { a ld ‘ z 4 £ > + PNM. Vio le that either of thele macnitudes A and B,be commen furable unto the Wag- Sy LL Ee tgs Hea, Be , a7 PPT. 4 ‘ ee id - I ; . 4: BFS q P eG nitude C.T hen I fay that the magnitude A is commen{urable unto the magnitude HONG B. For foraf wuuch as the magnitude Ais commen furable vato the magnitude Cy therefore (by thes of the tenth) A hath unto C {uch : proportion as pumber bath to number. Let A haue vnte C that proportion that the number D hath to | T 1S RECEP AS Se eee od tn aes c {yn ch RQ; Ke ; tC WUHBOE Ls .\_A CALE [OVALE aS Db t5§ COHWWEH “a = ber. Let Chane nto B that proportion that the number F hath vate thenumber G.Now then take the leaft numbers in continual proportion and in thefe proportions.geuen., namely , that the number D hath tothenumber E , and that the number F hath tothe number G(by the 4.0f the eight): which furable unto C,therefore (by the felfe fame)C hath unto B that proportion that number hath to num- ] : AP Gas let be the numbers i1, K, L. Sothatasthenumber D> ...... beg ae D is to the number E,falet the number Hbetothe EB .... store e. number K and as the niber F is to the nuber G, fo : let the nuber K be to tne nuber L. Now for that as K.. Ais to C,fois D toE, but as Disto EfoisHtoK, : SS therfore as Ais to C,foisH to K. Againefor that | asCistoB, fois F toG, butas FistoG fois KtoL: therefore asC istoB, foisKtoL. But it is now proued that as Ats toC; foisH to K . Wherefore of equalitie (by the 22.0f the fift)as Aisto Bfois the number H to the number L.Wherefore A hath unto B {uch propor= tion as number hath to number. Wherefore(by the fixt of the tenth the magnitude Ais com men{urable unto the magnitude B. Magnitudes therefore commen{urable to one and the felfefame magnitude,are alfo commenfurable the one to the other : which was required to be proued. ¢ An Affumpt. If there be to magnitudes compared to one and the felfe fame magnitude, and if the one of them be commenfurable’pnto it,and the other incommens furable:thofe magnitudes are imcommen[wrable the one to the other, Suppofe — of Euchides Elemente’. Fol.242. We, V ppofe that there be twomagnitudes, namely, A andB and let C be a certayne othey Demonfira- Fmarantuae And let Awe commenfurable 3 | tion leading to KONE vate Cand let B be comméfurablevnte the A a abfrditicn felfe {amec.-Fhend fay that the magnitude Ais in- commenfurable unto B. Fer tf A be. commenfurable vito Bs forafmsichas A is alfa comme fir able Uinta CB vm rsiinienncn i g therefore by the 12.0f the tenth) B is allo commefura. . ble- —p_—-_— furable , of which the magnitude A i$ incommenfurable vnto B, wherefore by this 13 -propofition the magnitudes Cand-B are al(o 5 ¢-——-emmnpnine incommen{urable: but the magnitudes B and Dare cOmenf{urable wherefore by the fame,or by the former affumpt , the magnitudes ST SC RARER oy, CandD are incommenfurable the one tothe other. This corolla- ty, Theen victh often times asin the2a.26:and.36 propofitions of | ~--*-—— this booke,and in other propofitions alfo..» q An Affumpt. Two bnequall right lines being genen,to finde out how much the greater is in power more then the leffe. Suppofe that the tmounequall right lines gewen,be A Band C, of which ler AB be the | greater. it ; th eh i Me Tk { 4 ‘thy f bi t mi (imal? ; y 4 i J i » ae ii \! i i me |)! ih i | | ie J | a ae ‘ | ‘ 7 } ny " | 1, \ “Conftruction. Demonflra- tion, wf Corollary. Dewmonfira- tiGM> T he tenth Booke greater.It is required to finde out how much more rb theline:A Bis then the line C. Defcribe upon the line A Ba femicircle. AD B.Anavnto i from the point A apply (by the firft of the fourth) aright line A D, equalluntothe tine «: Cand draw a right line from D. to B.. Now itis mani=- fest that the angle A D Bisa right angle (by the 31.-0f° the third). and that the line.A.B is in power more then the line A D,thatis,then the line C by theline.D B, by | the 4.7. of the firft. | for And like in forte, two right lines being geuen , by this \- meanes may be founde out a right Lyne whsch contayneth them 4 & both im power. Sippofe that the tivorright lines eeuen be ADand D B.Itis required tofinde out aright lne | that contayneth them both in pomer.Let the lines AB und DB be fo put, that they compre- hend aright angie A D B,and draw aright line from A to B.Now agayne it ts mantfe/t (by the 47. of the firft) that the line A B contayneth in power the lines A D and D B. D ne eemeeann qT be. 11. Eheoreme, T he. 14.Propofition. If there be forwer right lines proportional, and if the firft bein power more then the fecond by the fquare of a right line commenfurable in length vnto the firft,the third alfo fhalbe in power more then the fourth, by the {quare ofa right line commenfurableynto the third. Andif the firft be in pos sver more then the fecond by the fquare of 4 right line incommenfus rable in length vuto the firft,the third alfo {hall be in power more then the fourth by the fquave of a right line incommenfurable in length to the third. KY ppofe that thefe foure right lines A,B,C,D ,be proportional. As Ais to B,foler CRS C be to D. And let A be in power more then B by the [quare of the line E.And » ‘A likewife let C be in power more then D by the [quare of the line F. Then I fag 624 that if A be commenfurable in length unto the line E,C alfo [hall be commene [euenth of the fift). Wherfore (by the 17.0f the fift)as the ‘furablein length unto the line F And if A be incommen- furablein length to the line E, C alfo fhall be incommen- F fauare of the line Eis 0 the [quare of the line B, fo is the ree | [quare of the line F to the {quare of the line D.W. herfore alfoas the line E isto the line B; ois the line F to the line D (by the fecond parte of the 22. of the fixt) wherefore contrari- urable in length.to the line F For. for that.as A ss to B; fo is Cto.D, therefore as the fquare of the line listo the t {quare of the line B, fois the {quare of the line C to the | [quare of the line D (by the 22.0f the fixt). But by {uppo- | fition vato the {quare of the line A are equall the {quares of the lines E and B,and unto the {quare of: théline C are’ | equall the (quares of the of thé lines D and F : Where-. | fore as the (quares of the lines E.and B.(which areequall | to the fquare of the line A) are to the {quare of the line B, foare the {quares of the lines D and F (which are equall to | | A the [quare of the line C) to the {quare of the line D( by the cae aoe ee eee i. aoenene steed Seed 0 -+— pife (by the Corollary of the fourth of thefift) as B isto fois D to F.But( by [uppofition) as er fee ee eo ime of Euchdesdlementess F01.23% hihis to B fois€.toeD,Whirfareofeqmallitie (bythe waa the pyr) WAH to E, fois C7 F.df therfore A be commen|ur able in length vnto E,C allo fpaltbecommen[irablein leek unto F : and if it ba incommen{urable in length unto E,C alfo fhalbe tncommenfurablel in kincth unto F, by the roxofthisbobke. If therfore therObefoure Hecht tlines proportional, and if the fir{t bein power more then the fecondby the [quare of a right line commenfurable teleneth somre tbe pep) thevhird alfo [haltbe th power more then tbe FoOurtD; by the fouare aright line comm enfurablem length outo the third: and if the fir ft bein power morethe the fecond , by the [quare of aright line incomen|urable.tn length veto the prt, the third. aifo:fralléeimipemer more thé the.fourth sby the f Giare of a ticht labeincommenfurable it dencth to theibivd * whith’ was required to be prowed. ) | iD as a | Note that the line Amay be proued to. be.in proportion to the line E,as the line.G is to the line F by an other way, naniély,by conuerfiohiof proportion (of foniejas we have ‘before nered, called inuerfe proportion) by the 19.0f the fift. For,forafmuch as the foure lines A,B,C,D, are proportionall:ther- fore(by the 22.0f the fixt) their f{quares alfo are proportionall. And forafinych as the antecedeng name- lythe fQuare Of che hne A excedethche'confequent, namely; the fquare ofthe line BY, by the {quare of the line E.s.and-theerherantecedent,namely,the fquare of che line. Cyexeedeth the otherconfequent, namely, the fquare of the line Dy by the {quare of the line F, therefore 2s the{quare of the line “A isto the exceffesiamelpite the (qaareor theline E,fois the fquare of the line'C to the exceffe,aaticly,to the {clare of the line F. Wherefore (by the fecond part.oithé 22.of thie Gxt) asthe line A isto the line E, {ois the line C to the line F. Si . i _-g Lhe 12. iT beoreme. he 15. Propofition. | on Uf twoynagnitudes commenfuyable.be conpofed,the whole magnitnde coma pafed-alfoshall be commenfurable to either of the.two. partes. And tf the wi Whelemaonitude compofed be commenfarable to any one of the two partes, thofe two partes [hall alfo be cominjen/yrable. - | * aatne, Et thefe two commenfurable magnitudes.d Rand BC.beconspofed or added t0gt- epy te 4B and BC. Far forafmndl ad Band BC age commenfurablestieloe pervs (by the firft defmition af the tenth) [ome one magnitude meafurcth there both, Let there be a magnitude that mealureth tOep ed ever vor vos. & Yds re letsthe {acme te DB -Naw forafenueleas DMR SA. moe ah BAS Ress ABandBC, tt frallallomealure the whale magete \——— BR tude commpaled. A. C,.by.thiscomumen fentence, Wat Po ws = fresexrmagnitnde racafureth tro okhar magnitudess...« EY ye [hail alfa meafune the magnitude iat afthemp. But the [ame D mealtreth AB and BC Gy ppalition... I heref ore Dimes ‘ureth A BB Cand A ae W. herefore cA Gis commen| i rableto citherof thefe magnitudes ALB And BG.» 0 he } aad won fappofe that the whale campo ed magnithde A C be commen|urable to Any one of thefetmomaace tudes AB orBC, let it be commen{urable I fay upto-A B. T. hen LJay, that tetwospagnitudes A Band BC.are commenfurable ~F or foralmuch 4s AB-and-A Care commenfurable, {ome one magnitude meafureth them (by the frst definition of the tenth). Ket fame-maguitude meafure them,and lerthelamebe D Now forafmuch as D meafureth AB and AC, italfomeafureth therefidue BC, tythiscommen [entences what [oeuer mea fereth the whole and the part takeh andy, hall alfo mealure the refidue.B ut the fame-D mea- urcth the magnitude A B (by [uppofition ) Wherefore D. megfureth either of thefe magni- tudes AB and BC. Wherefore the magnitudes AB ana B C are cominen{urable . If ther- poretwoonarnituilescommen|urablebe conpofeds the mhole-maguitude .compofed. alfo fhall he commer urable to either ofthetwo partes. And 1f the wholémagnitude compofed ve com oe | en{urable feck ther..T hen Lf 4 that wiembhole macmitude AC iscomenfurable to either of thee An other way to brone that the lines A, iB (, Fare b*vO~ a a porlschahe Demonflra- tton of the bh AS s Jerse part, wiat® . s 4 Demon Tratts on of the fea cond pate which is the conuerfe of the firdt. A Corollary , Demonfira- sion of the fersé pare by an argument beadindg to an abfurdities Demon fratie on of the fe- cond patt leading a!fo to an impoff:bi- btite. And this fecond part is the conuerfe of the first. % Thetenth Booke menfurable toany one of the two partes shofe two partes fhall alfa be commenfurable : which was tequired to be demoniirated. | q AC orollary added by Aontaureus. Uf au Wwholewmagnitude be corsmenfurable to one of the twe magnitudes Which make the whole magnitude, st {hall alo be commenfurable to the other of the two magnitudes. For ifthe whole mag- nitude A C be commenfurable vnto the magnitude B C, then by the 2.part of thys 15 Propofition: the magnitudes A B and'B C are commenfurable . Wherefore (by the firft part of the fame) the magnitude AC thall be commenturable to either of thefe magnitudes ABandBC. This Corollary 7'heen vfeth in the demonttration of the 17.Propofition and alfo of other Propofitions . Howbeit Enclide leftit out for that it {cemed eafie as in a maner do all other Corollaryes: : q Lhe 13. I heoreme. Lhe 16, Propofition. If two magnitudes incommenfurable be compofed,the whole magnitude ale A c ~ ° fo [hall be incommenfurable vato either of the two partes coponentes. And if the whole be incommenfurable to one of the partes componentes , thofe jrft mAvnitudes alfo hall be incommen|lurable. aw pW ; = : i et , N f Kao rdatecether Then I {ay, that the whole magnitude AC, is incommenfurable to fist ZA Ff age | : es 7 ay ‘ ; ie CAND) dither of thefe magnitudes A Band BE. For if AC and AP be not incom- - gj wenfurable, then fome one magnitude meafureth them ( by the firft definition (BATES of the tenth )..Let there be [uch a magnitude, if tt be pofible, and let the fame be D . Naw forafwiuchas D meafureth CA and AB, it al{o meafureth the refidue BC, Cr it lthkemife meafu- reth AB Wherefore D meaf{ureth AB and B C.Wher- fore (by the firft definitio of the tenth) the magnitudes AB and RC are Aibaerabl . Bat it is fuppofed that they are. incommenf{irable : which is impofsible . Wherefore no magnitude doth meafure the magnitudes AB and AC. Wheres fore the magnitudes CA and AB are seg Oa . In like [ort alfo may we prowe,thas the magnitudes AC andC B are incommenfurable . But now [uppofe that the magnithde AC beinctommenfurable to one of thefe magnitudes A B or BC, and firft let it be ixcommenfurable unto AB .ThenT fay, that the magnitudes A Band BC aretncommenfurable. For if they be commenfurable fome one magnitude mea~ fureth ther ..Let fome one magnitude meafure them, & let the famebe D . Now forafmuch 4s.D mitafureth A B and BC, it alfo meafureth the whole magnitude AC. And itmeafi- reth AB: Wherefore D-meafureth thefemagnitudes C A and AB .Wherefor? CAG AB are commenfurable., And they are (uppofed to be incimen{urable : which is impofable Wher- fore no macnitude meafureth AB and BC «Wherefore the macnitudes ABand BE are tn commenturable . Andin like fort may they be proned to be incommen{urable, if the magni~ tudé AC be [uppo{ed to be incommen|urable vnto BC . If therefore there be two maenitudes incommenfurablecompofed,the whole alfo fhall be incommenfurable onto either of the two partes component, and if the whole be incommenfurableto one of the partes component ,thofe Fift magnitudes [hall be incommen{urable < which was required to be proxed. eS Ngo Et thefetwo incommenfurable magnitudes A B.c> BC, be conapofed, or added t= h, sta . is mw B | G io fds @ A Corollary added by ¢ontaureus. If an whole magnitude beeincommenfirable to one of the nvo magnitudes Which make the whole magnitude, itfhall alforbe sncommenfarable to the other of the two magnitudes » FOF if the “ ee whole st i) forafmuch as GB is afquare,therefore the line of Euchdes Elementess. Fol.24.44 whole magnitude A C be incéménfiirable vnto the magnitude B C,then by the a.part of this 16.Thed- reme,the magnitudes A Band B-C thall be incommenfurable. Wherefore by thesfirft part of the fame Theoreme, the magnitude A Cifhiall be incommenfurable to either of thefe magnitudes A Band BC, This Gorollary Theow vfech in the demoaltration of the 73. Theoreme,& alfowf other Propofitions. @An Aflumpt. If ypon a right line be applied'a parallelogramme wanting in fioure by a |quare: the paralleligramme fo applied, 13 equal to that ‘parallelogramme whichis contayned ynder the fegmentes of the right line, which fezmentes are made by reafon of that application. Suppofethat upon aright lined B be applied a parallelograme 4 G,wan ting informe by the {quareG B. Then I fay, that AG is equall unto that whichis contayned under AD and D B,which thing is of it [elf manifeft.F or GC D G is equali unite the line DB: and the pa- rallelogramme AG is that which is contayned . under the lines AD andDG;that is, vn- der the lines ABD and DB. If therefore vp- on aright line be applied a parallelogramme A wanting in figure by a{quare.: the parallelo- qrame applied is equallto the parallelograme | which is contayned under the fegmentes of the right line, which are made by veafon of that application .. which was required to be aenzon trated. : D B This fig I betore added as a Corollary ot of Fluffates after the 28. Pros polition of the fixt booke. Satay “wt q Lhe 14. Theoreme....... Fhe 17. Propojfition. If there be two right lines ‘ynequall, and if opon the greater be applied a parallelogramme equall nto the fourth part of the fauare of the leffe line, and wanting in figure by a fquare, if alfo the parallelocramme thus applie ed denide the line-where pon it is applied into partes commenfurable in . length :then fhall the greater line beinpower more then the lefse, by the Square of a line commenfurable in length vnto the greater. Andif the greas ter bein power more then the le(Se bythe fauare of a right line commenfue rable in length ‘ynto the greater, and if alfovpon the greater be apphed a parallelograme equall bnto the fourth part of the ‘fquare of the lefse line, and Wanting in figure by a [quare: then {hall it deuide the greater line ine to partes commenfurable, i” pole that thefe two right lines A and BC, be unequall: of which le Ze BC be the greater. And vpon the line BC let there be applied (by the 28. of the fixt ) a parallelogramme equal unto the fourth part of the {quare of the ENG line A being the lefve (that is, equall unto the {quare defcribed vpon halfe a ae FF 4. the it But | | ’ I ir i | | ma ia | ) ; mi maa ‘ Ht Conftrutlicn. Demonftratio of the first pers. Demonftra- tion of the fecond part which ts the conuerfe of the firft, The tenth Booke the tine A’) and wanting tw foure by 4 {quare . And let the fame parallelporamme be that which iscontained under the lines B Dana DC. And (by fuppofition) let the lines BD and-D€ be commenfurablein length. Then I faysthat theline BC, isin power more then the line A, by the {quare of aline commen{urablein leneth vnto the fayd line. BC. Denil, by the 10.0f the firft )the line BC intotwo equall partes in the point | OOS: AO eae an Evtval( by the third of the firft) vatothe line D E put A aa equall line EF. Wherefore the vefidue DC is equal , unto the refdueB F.And forafmuch as the right line . | B C15 denided into two equall partes inthe point E,and into two unequal partes in the point D,therefore(by the 5.of the [econd ) the rectangle figure tom prehended under the lines B D and D C together with the {quareof the line E D,1s equall to the {quare of the line E C.And in the fame proportion are they eche being taken fower times by thers of the fifth . Wherfore ubwhichis contained under thelinesB Dand D C taken fawer times together with the {quare of the line E D taken alfo fower times,is equall to the (quare of the line EC takes fower times. But vitto that which ts cotained under the lines BD ch DC foure times ts equal ‘the [askare of the line A by [uppofition: for the parallelograme contained under the lines B D and DC once is fuppofed to be equal to the fourth part of the (quare of the line A. And wnto the fauare of the line D E taken fower times ts equall the {quare of the line D F , for the line D F is double to the line D E. And unto the {quare of the line E C fower times ta- ken,is equall the {quare of the line B C,for the line B Cis alfo double to the line C E. Where- forethe [quares of the lines A and D F are equal vnto the {quare of the line B C.Wherefore the [quare.of theline BC, is greater then.the [quare of the line A , by the {quare of the line DT Wherefore the creater line BC 1s in power more then the lefve line A , by the {quare of the line D F.Now refteth to prone that the line B Cis commenfurable in length vato the line D F.F oralianch as by fupofition the line B Dis commenfurable in length unto D C., there- fore (by the 15.0f the tenth ) the whole line B Cis commenf{urablein lencth unto the line D C:but the line D C equall to the line B F. Wherefore the whole line BC is commenfurablein length vate the lines BF cy C D.Let the two lines B F and C D beimagined to be {o compo- fed that they makeonelline.Now fora{much as the whole line B-€ is commenfurable in length to the two lines B F and C D taken as one line: therefore the lives B F and C D taken as one line aré commenfirablein length to the line F D(by the'z.part of. the 15 of thetenth).Wher fore alfarshevhole line BC is commenfurablein length to theline F D by the firft part of the (ame. this may al{o beproued by the corollary put after the 16. propofition.of this booke. Whereforethe line BC is in power more then the line A by the {quare of a line commmen{u- rable in length unto theline BC. | | “But now fuppofe that the line B C be in power more then the line A , by the {quare of a line commenfurable in length unto the tine B C. And vpon the line BC let there be applied avectancleparallelocrame eqnall vnta the fourth part of the {quare of the line A,ana wan- hing.in fieureby afquare, and let the fayd parallelograme be that which is contained under the linesB D and DC.T hen muft weproue that the line B D is unto the lineD C commen- furablein length. The fame confiructsons and {uppofitions, that were before,remayning, we may in like fort prone that the line B C 4s sn power more then the line A, by the [quare of the line F D. But by fuppofitio the line B C is in power more the theline A by the [quare of 4 line comenfurable unto it in length Wherfore the line BC is unto the line F D comen{urable 18 leneth Wherefore the line compofed of the two lines BP and D Cis comen{urable in length auto the line F D.(by the [econd part.of the.x5..of thetenth ) . Wherefore (-by the 12. of the tenth or by the first part of the 15 of the tenth) the line BC is commen{urable in length to the hne compofed of B F and D.C. But the whole line conpofed of BF and D C is commenfura- ble it lencth vnto D. .For BF (as before hath bene proued)ss equall toDC. Wien fe ine er ee oad of Euclides Elementes: Fol.245. line BC is comsimen{urable in length unto the line D C(by the 12. of thetenth) . Wherefore alfo the line B.D 1s comsmen{urablein length untothe line D C(by thefecond part of the 13. of the textin) If therfore there be tworight lines vnequall,and if upon the greater be applied a parallelograme equall unto the fourth part of the {quare of the leffeand wanting in figure. by afquare, if alfa the parallelograme thus applied deuide the line whereupon itis applied into partes commenfurable in length: then {hall the greater line be in power more then the leffe by the {quare of a tine commenfurable in length unto the greater. And if the greater be. in power more then the lefve by the [quare of a linecomme{urablein length unto the greater; and if allouponthe greater be applied a pvrallelograme eqnall unto the fourth part of the Square made of the lefeand wanting in figure by.afquare: then {hall it deuide the greater line inte partes conmenfurable in lencth :which was required to be proued Campane after this propofition teacheth how:we may redily apply vponthe line, B Ca parallelogrameequall to the fourth part of the {quare of halfe of the line A, and wanting in figure bya {quare,after this maner. Deuide the line B C into twolines in fuch fort that halfe of the line A fhalbe the meane propor- tionall betwene thofe two lines, which is poffible,when as the line B C is fuppofed ro be greater then the line A,and may thus be done.Deuide the line B C into two equal partes in the point E and defcribe ypon the line B € a femicircle B H C.And ynto the line B C,and from the point C erecta perpédicular line C K and put the line C K equail to halfe of the line A. And by the oaths K draw he the lineECa parallelline a «5 K H cutting the femicircle in the point H, (which it muft needes cut; foraimuch asthe line BC is ereater then the line A) And f6 the point H draw ynto the line BC aper pendicular line H D: which line H D, forafmuch as by the 34.0f the firflitis equall vnto the line K C,fhall alfo be e- guall to halfe of the line A : draw thelines BH andH C. Now then by the 1.of the third the angle B H C isaright B E angle. Wherefore by the corollary of the eight of the fixe A booke the line HD isthe meane proportionall betwene (9 thelines BD andD CG. Wherefore thehalfe of the line A which is equall vnto the line H Dis the meane proportio- nall betwene the lines B Dand DC. Wherefore that which is contained vnder thelines B DandD é isequall to the fourth pare of the fquare of the line A .And {0 if Vpon the lineB D be defcribed 4 rect. angle parallelograme hating his other fide equall to the line DC. there fhalbe applied vpon the line B Ca rectangle parallelograme equall vnto the {quare of halfeof theline A sand wanting in figure bya iquare: which Was required to be done. | q The 1s.I heoreme. Ihe 18. Propofition. If there be two right lines vnequall., and if pon-the greater be applieda parallelograme equall ynto the fourth part.of the Square of leffe, and wane ting in figure by a fquare-; if alfo the parallelogrameithus applied deuide the line ‘whereupon it 1s applied into’ partes incommenfurable in length: the greater line fhalbé in power more then the leffe line ‘by the fquare of _. a line incommenfurablein length nto the greater line. And if the oreas ter-tine be in power more then the le/Se line by the [quare of a line incommés »sfurablein length vnto the greater, and if alfo.vponthe greater be applied “a parallelograme equall ynto the fourth part of the quare of the leffe and “i wanting tn figure by a quire: then ‘praltse denide the yreater line into partes incommenfurable in length, | Sees th eae How to de- urde the line B C redely i” K > fuch fort as is required 2 she propofitive - - . - 9s ¢ BAe 4 oe oe . x = Senate Sete -- - — - 4 — = a = - - ——— — s} — > —— ——— oe SS sen = = - ” ain Se tien af SeOrS ne — ee hie vue fi wey! : , aah j ri { 4) ay wei Ly Deimonstra- tid of the firft part. Demon Fr tt- on of the je- coud part which 2s the canuerfe of the former. T hetenth Booke ane Vppofe thar thefe tworight lines Aand BC bé wnequall the oneto the other , of. S whichlet BC be the greater. And vpon the fame BC apply.a parallelograme equall Ne wnto the fourth part of the {qnare of the line Asand wanting in fieure by a {quare: which hove to doo was before taught inthe end of the former propofition . And let the fayd parallelograme BF EB tis ck he that which is contained under the lines B.D and DC: And let BD beincommenfarable in length vn 10D €.T hen I fay that the line BC 181m power more then the line A,by the [quare of aright line incommen|urable in length vnto the line BC. First let the [ame order of. conftruction,and demonflration be obferned in this which was in the former propofition.. And we may in lake fort proue that the line BC 1s 0 power more thé the line A by the [quare of the line D F.Now then muft we prove that the lines BC and DF dreintommen{urablein length. Forafmuch as by Juppofition the line BD 1s incommen| 1. rable in leeth unto the line D C: therefore( by the 16 .of the tenth )the line B C is incommen- furable in leneth vato the lineC D. But DC is commen{urable to thefe two lines B F and D C added together.F or B F is equall unto D C. Wherefore ( by the 13. of the tenth ) BC is incominen{urableuuto thefe two lines b F and D.C compofed. Wherefore by the fecond part of the t6.0f the tenth the line compofed of the lines BF and D C taken as one line is incem- moen{urable tn levieth unto the line F D Wherefore by the firf! part of the (ame 16. propofitt- on the line B Cis incommen{urable in length onto the line F D .. Wherefore the line BC is f ; wir ; . ¢ ~ : : Les sae Pak i > Se eee , in tower moore then the line A by the {quare of a line incommenfui able iz length unto the i ey fine BC. But new [uppoj e thatthe line BC bein power more then the line A by the fynare of a line incommen| urable in length unto BC.And upon the line BC let there be apt lied a parallelo- gramme equal unto the fourth part of the {quare a the line A, and wanting in figure by 4 [quare,and let the {aid parallelogramme be that which ts contained under the lines B D ¢ D C.T hen-muft we prone that the line B.D 1s unto the line D C incommen{urable in legth. The (ame order of conftruction and demonftration being kept,we may in like fort prowe that the jine B.C is in power more then the line A by the [quare of the line F D.But now(by fi ‘5 pofition) the line B & is in power more then the line.A by the {quare of a line. incommen{u- sablein length vito BC. Whereforethe line BCis unto the line F Dincommenf{urable in beneth.Wherfore the line compofed of BF and D C taken as one line, [hall be incommen{u- rable in length to the line F D (by the fecond part of the 16 .of the tenth). wherefore alfo by the fir|t part of the fame,theline B C fall be incommenfurable in length to the line compo- fed of the lines B F and D C.But thé line compofed of the lines B F and D Cis commen{u- rable in length to the line D C (for that B F (as before hath bene proued) is equall to D C). PWherfore the line BC ts incomenfurable iw length to the line D C (by the 13. of the tenth). Pwherefore by the fecond part of the 26-6 the tenth, the line B D is incomme{urable in legth ainto the line D Colf therfore there be two right lines vnequall,and if upon the greater be applied a parallelagramme equall vato.the fourth partof the {quare of the leffe liner wAn- ting in figure by a {quare, if alfo the parallelogramme thus applied deuidethe line wherupon itis applied into partes incommenfurablein length: the greater line phall bein power more then the lelfe line by the [quare of a line incommenfurable in length unto the greater. C And if the ereater line be in power more then the leffe,by pee of a line enEOmIMED IZ wibletn length ware the ereater and if alfo-vpon the wreater be applied a parallelogramme equallintothe fourth part of the quare of. the leffehimesand wanting in fi cure by a {quare, shex-fhallit-dewidethe greatér line iotopartes incommenfurable in length.: which was re- aed quired to be demonftrated. a5, This Propofition may alfo be demonftrated by the former propofition,namely, the Grft part of og . * ——<=«—_ on oo of Euclides Elenientess Fol.24.6. by the fecond part of the forrmer,and the fecond part of this by. the firftspart of the former,by an argue pient leading to an abfurditie. Fonas touching the fink part of this propofition,the line B C contayning | in power niore'then the line A by the fquare of the line F Dyifthe line B C be not iucommenturable yn co'the line F Di theiwisdt coimmenfurabled ynto it. Wherfore (by the fecond part ofthe 17 propofition) the lines BD and. D € al are'conimenfurable,which is impoflible,for they are fuppofed to be incom- iwenfurable. So likewile as touching the fecond parte of the fame, the line B C contayning in power more then thelinéA by the fquare of the line F D,if the line D B be notincommenfurable to the lyne D C,then is it commenfurable vnto it: wherfore(by the firft part ofthe 17.propofition) the lines B C and F Dare alfo commenfurable, which were abfurde.For the lines B Cand F Dare fuppofedto be in- con:menfurable : which was required to be proued. @q An affumpr. se ° ‘ x ts ‘ " - 7 4 os a2 AwNs LK Foralmuch as it path bene proued that lines commenfurablein length, are almayes alfa bonimen| uvavle in power ,but lines commen af urableix power are wot abvayes Conmmen|{ ura ble in length but may be in length both consnien ifuvable and alfo tncommmen|trable:it is mant- fefl that if unto the line propounded, which ts called vationall of purpofe, a certayne line be comenfurable in length,it ought to be called rationall and comenfurableunto tt, not only in length but alfo in power : for lines commenfurable in length are alfo alwayes para ble ix power But if unto the lize propoundea which is called rationall of purpofe, a certayne line be commenfuratle in power, tien if it be alfo commen{urable unto it in length, itis cal- fed rationall and commenf{urable untoxt bethin length and in power But againe if unto the faid line cenen which is called rationall,a certayne line be commenlurable in power and in- - commenfurable in length that alfo is called rationall,commen|urablein power one. An annotacion of Proclus. He calleth thefe lines rationall which are vnto the rationall line fir(t fet comen{urablein length & in power,or in power only. And there are al[o other right lines, which ave vate the rationall line fir[t fet, incommen[urable in leneth, and ave unto it commen urable in power only.and therfore they are called rationall, c comme[urable the one to the other: for whith canfe they are rationall.But enen thefe lines may be commenfurable the one to the other, either in lexeth, and therefore in power, or els in power onely . Now if they be commen{i- rable in length,then are thofe lines called rattonall,commenf urablein length, but yet [othat they be underfland to bein power commenfurable-but if they be commenfurable the one to the other in porver onely,they alfo are called rational commen{urablein power onely. «A Corollary. And that two lines or more being rationall and commenfurable in length to, the rational line firft fet,are alfocommenfurable the one to the other in length hereby itis manifeit;for forafmuch as they are rationall and commenfurable.in length to the rationall line firft fet, but thofe magnitudes whiche are commenfurabletoone and the felfe fase magnitude,are alfo commenfurable the one to the other (by the 12.0fthe tenth) whérfore the rationall lines,;commentfurable in length to the rational lyne firtt fet, ate alio commenfurable\in length the one tothe other.And as touching thofe which are rationall com menfurablein power onely to the rationall line firft a eee fetjthey-alfo muft needes be at the leaft. commenfura- ble 2g tet the one to the other. For forafmuch as... gp their {quares are rationall they fhall bee commenfus rable to the {quare of the rational] line firft fet. Wher- fore by the 12, of this booke,they are alfecommenfu- G ———____, table the one tothe other. Wherefore their lines are at the Jeait commenfurable in power the one to the other.And itis polible alfo that they may be comméfurable in Jégth the one to the orher: For fuppofe shat Abe a rationall line firft fet,and let the line B be vnto the fame rationall line A commenfurablein power onely, that is, incommenf{urable in length vnto it. Let there be alfo an other line C commen- . EF 3, {ura- Se SS ee ee Se ee An other de- monSiration Ly an argumece leading to an abfurditiee An Afsuntpte 4 Corchary added by Montaurense hae ay ee ANU | ite i 1) Deg aes | i mt) \e 1 if t te} \ Rie Re Canfe ¢ Caufe of ine creajing the difficulty of thts bookee Note, Conft rutiion. Demosffra- £10. Diwers cafes in $his Dr9p0- fitzon. | The fecond cafe. T he tenth Booke furablein Ierigth to the lyneB (which is poffible by the principlesof this booke. JNow by the 13.0f the tenth, it is manifeit chat the line C is incommenturable in length vnto the line A. But the {quare of che line Ais céitéfurable to the fquare of the line B by fuppofition,and the {quare of the line C is alfo commenfurable to the fquare of the line B by fuppofition. Wherefore by the 12. of this booke, the {quare of the line C is commenfurable to the {quate of the line A. Wherfere by the definition, the line C thall be rational! commenfurable in power onely to the line A,as alfo is the line B. Wherefore there aré geuen tworationall lines comm enf{urablein power onely to the rationall line firit fer,and commé- furable in length the one to the other. Here is to be noted which thing alfo we before noted in the definitions, that Cam- pane and others which followed him, brought in thefe phrafes of {peaches, to callfome lynes rationall in power onely,and other fome rationall in length and in power, which we cannot finde that Euclide ever vfed.Por thefe wordes in length and in power are ne- nerreferted to rationalitie orirrationalitie,but alwayes to the commenturabilitie or incommienfurablitie of lines. Which peruerting of wordes(as was there declared) hath much increafed the difficulty and obfcurenes of this booke.And now I thinkeit good agayne to put youin minde,thatin thefe propofitions which follow,we muft euer have before our eyes the rational] line firft {er, vnto which other lines compared are either rationall or irrational! according to their commenfurability or incommenturabilite. q The 16. Theoreme. The 19. Propofition. 4 A reélangle figure comprehended ‘ynder right lines commenfurable in lengthe, being rational according to one of the forefaide “wayes : is rae tionall. ee Sexig Vppofe that. this rectangle figure AC be comprehended under thefe right lines A Bikte. (67:5 : ; ats 5 B and B.C being commenf{urable in length,and rationall according to one of the le fore/aid mayes.T hen I fay that the [uperfictes A Cis rationall,de{cribe (by the 46. So 7 ofthe firft) upon theline A Ba fquare AD. Wherfore that [quare A D is rationallby the ¢ = ‘~ definition. And forafmuch as the line A B is Ee ere itis eae ee commenfurable in len ath unto the line BC, | and the line A Bis equall unto the lyne BD; therefore the lyne B D1s commen urable in lenethunto the line BC. And as the line B D is tothe line BC, [ois the [quare D Ato the fuperficies AC ( by the firft of the fixt):but it is proued that the line B Dis commenfurable “nto the line B C,wherfore (by the o.of the tenth) the {quare D A is commenfurable unto the rettanele [uperfictes A C.But the [quare D Aisrationall, wherfore the rettanele fu- perficies AC fe is rationall by the definition. A rectangle figure therfore com rehended under right lines commen{urable in length beyng rationall accordyng to one of the forefaya wares is rationall : which wasrequired to be proued. . Whereas in the former demonftration the {quate was defcribed vpon the leffe line, we may alfo demonttrate tie Propofiton, if we defcribe the {quare vpon the greater line,and that after thys maner. Suppofe that the rectangle fuperficies BC be contayned of thefe vnequall lines A B and A C, which let be rationall commenfurable the one to the other in length.And|et the line A C be the ereater.And | | vpon Lal of Euchdes Elementess vpon che line A C defcribe the fquare DC. Thea I fay, that the parallelogramme BC is rationall. D For the line A C iscommenfurable in length vnto the line A B by fuppofition , and theliné'D A is e~ quallto theline AC . Wherefotethe line D\A is commenfurable in length to the line AB . But what proportion the line DA hath to the line AB, the fanic hath the {quare D C to the para le- lograinme C B (by the firit of the fixt).. Wherefore (by the 10,of this booke) the {quare DC is com- menfurable to the parallelogramme CB . Buritis manifelt, thatthe fquare D Cis rationall, for that itis the {quare of a rarionall line, namely, AC, C Wherefore (by the definition) the parallelograme alfo.C Bisrationall. | Moreouer,forafmuch as thofe two former demonftrations feeme to fpeake of thar parallelograme which is made of two lines, of which any one may be the fine firit fet,whichis‘called the firft racionall line, from which(we fayd)oughtto be taken the meafures of the other lines compared ynto it, and. the others comméfurable in length to the fame firlt rationall line, which is the firit kinde of rationall lines comenf{urablein length : I thinke it good here to fet an other cafe of the otherkinde of rational! lines; of lines I {ay rationall comenfurable in length compared to another rationall line firtt fet,to declare the generali truth of this Theoreme,and that we might {ee that this particle according to any of the forefayd ayes Was hot here ia vaine put . Now then fuppofe firita rationall line AB . Let there be alfoa paral- lelograme C D contaynéd vnder the lines C Eand E D;which lines let berationall; C that is commenfurable in length to. the firft rationall line propounded A B. How- beit, lecthofe two lines CE and ED be diuers and vnequall lines vnto the firit ra- uonallline AB.Then I fay, thatthe pa- rallelogramme C D isrationall . Defcribe the {quare of the line DE, whieh let be DF . Firititis manifeit ( by che 12.0f this D booke)that the lines C E & ED,are com- menfurable in légth the one to the other, For either of them is fuppofed to be com- ad meniurable in length vnto-the line AB. But the line ED is equallto the line EF. Wherefore the line Fol.24.7. L dg a E rr Sr neenedl meinem nomena nen eee oe es ‘CE 1s commentfurable in length to the line E FY But as the hae C Eisto thé linc E F, fo is the paralle- dogramme C D to thefquare D F (by the firftofthe fixt).. Wherefore (by the 10.0f this booke) the pa- rallelogramme C D fhall be commenfurable to the {quare D EF. Burthe {quare D F is commenfurable to the fquare of the line AB which is the firft rational line propounded. Wherfore (by the 12.0f this booke) the parailelogramme C Dis tommenfuratileto the fquare of the line AB. But the fquare of theline A B is rational! (by the définition). Wherfore by the definition alfo of rationall figures, the pa- tallelogramme C€ D thal be rationall, Now ‘refteth.an other cafe of the thirde kindeof rationall lines commenfurable in length the one to the other,which are to the rationallline AB firft fet commenfurable in power onely, and yerare therfore rationall lines. And let the lines C E and E D becomenfurable in length the one to the other. Now then let the felfe fame conttruétion remaine that was in the fornter.: fo that ler the lines C E and E D be rationall commenfurable in power onely vnto théline AB . Burlérthem becommenturable in Jength the one to the other. Then I fay, that in this cafe alfo the parallelogramme C D is rational! .Firft it may be proued as before, thatthe parallelograthnie C.D is commenfurable to the {quare D F. Wher- fore (by the'tz.of this booke) the parallclogramme CD fhall be commenfurable to the (quare of the dine A B . But the {quare of thelline A:B is racionall : Wherefore ( by the definition) the parallelograme C D ihall be alfo rational . This cafe is well to benoted, For it ferueth to the demonfiration and yn- derftanding of the 25 .Propofition of thisbooke. - wes @ The 17: T heoveme. The 20. Propofition. Ifypon arationall line be applied a rational retZancle parallelogramme: the other fide that maketh the breadth thereof [hall be a rational line and ~commenfurable in length nto that line-whernpon the rationall parallelos gramme ts applied, —~ : FF iii. Suppofe The firft hing of rational lines com. menfarable in fen F443 b, This particle in the propofie t1ou (accor. ding to any of the [orefayde wayes) was uot in Vayne put. The fecond kinde of rati-’ onall (ines co« menfurable in leng.b. ; The third cafe, The third kinde of ratio= nak lines come menfurable i length, The fourth ¢ af be ThetenthBooke SM ppofe that this rational: rectanele parallelocramme AC, be applied vpon | theline A.B,which let berationall according to any one of the forefaiad wayes This propof- od | (hether it be the first raiionall line fet, or- any other line com gion ts the co Si ( fi fet, 'y r line commenfurable to Bre of tbe . SS) the rational line firft fet,and thatin length and in power, or in power onely:for former propo~ One of thefe three Ayes As WAS aeclared i” the Afsuimpt put before the 19. Propofition of this Sition. booke, is atine called rationall) and muking in breaath the line B.C.T ben I fay that the line ConSruttion. B Cis rationall and.commen{urable inlencth vnto the line BA. Defcvibe (by the 20 .of the firft) vpor the line B Aalquare AD.Wher- 7 ge : 3 yaa / bon +4 > Fe: a: z° 2 , vie sath Cc fore (by the 9 .acfinitio oj the tenth thi {quare : | AD is vationall . But the parallelogramme : Se AC alfois rationall (by [uppofition).. Where- | Demon{ira- fore (ly the conuerfion of the definition. of ra- | re te rionall figures, or by the 12. 0f thes beike) the | fquare D A is commenfurable vnto the pa- ‘a Leh oe ay sg vallelogramme AC.Butas the {quariD Aw to the parallelogramme AC; {ois the kne D B to the line BC{ by the firft of the fixt).Wher- fore (by the 10. of the tenth) the line D B is commen[urable vuto the lineBC. But the line DB is equallvntothe lineB A. Wherefore the line A Bis comenf{urable unto thelineB C. Butthe line AB isvationall. Wherefwe the lineBC alfoisrationalland commenfurable in length unto the line B A. If thereforevpon a rationall line be applied avationall rectangle parallelogramme, the ther fide that waketh the breadth therof {hall be a rational line com menf{urable in len eth unto that line wheren pon the rationall parallelogramsme is arplica: ay hich WAS reg wired ko be dena onftrated. q An Affumpt. A line contayning in power an ivrationall [uperficies is irrational, vin Affampt. * vig; | Rg) aa ppt , | BOT" ~~ Suppofethat the line A B cotaine in power an trrationall fuperficies,thatis,let the {quare defcribed upon the line A B, be equal vnto an irrationall fuperficies . T hex I fay, that the live AB is irrationall.F or if the line A B berationall, thé {hall | the [quare of the line AB be alforatimall. Forfowasitputin A B the definitions . But (by {uppofition ) itis not . Wherefore the line AB isixrationall . Aline therefwe contayning in power an irrational [uperficses, $5 ir ratiovall. q Uhe 13. Theorene. T he 21. Propofition. A vettanele ficure comprehended ‘ynder two rationall right lines come menfurable in power onely, ts irrationall . And the line which in power contayneth that reétangle figure's irrationall, eis called a meatall line, =V ppofethat this rectangle ficure AC be. comprehended wider. thefe rational right lines AB and B Ccommenfurable in power onely. TZ ren I fay, that the [u- lperficies AC is irrational « and the line which combayneth it in power ts1rration Conflraction + A A fquare is fayd to be applied vpon aline,whenitjor a parallelograme equall ynto it,is applied vpomthe fayd line; If vpon arationallline genen we will apply a reGanslé parallelograme equall to the fquare of a mediall line gétteti;and fo of any line ceven,we muft,by the 11.of the fixt, finde out the third line proportional! with thé tational! Hie and the mediall line genen + fo-yet that the rationall line be the firtt,and the medial) ine geuen,(which containeth in power the fqnaréto be applied )'be the fecond . Fortheni the fuperficies contained vnderthe firft and the third , fhalbe equal! to the {quare of the midle line; by the 17. of the fixr. q Ihe 20. T heoreme. The 23. Propofition: A right line commenfurable to a mediallline,is alfo a mediall line. J, medial line.Let there be put arationall liwe C.D. And upon the linéC D, apply MNO rectangle parallelograme CE , equall unto the {quare of the line A ,and mar king in breadib theline E D.Wherefore (by the propofition going before)the line E D ts ra sionall and incommenlurable in length unto thelineC-D. And againe vpon the line€ D $9 a rectangle parallelograme C F equall unto the {quare of the line B , and making ip readth the line D F . And fora{much as 7 the line Ais commenf{urablevnto the line ai D "odes B therefore the [quare of the line A is commenfurable tothe {quare of the line B But theparallelograme E Cis equall to the {quare of the line A,and the parallelo- game CF isequall tothe [quare of the ine B-wherefore the parallelograme EC eS 3 ye is comenfurable vuto.the parallelagrame . CF. But as the parallelograme E C., is to BRI 2 she parallelograme CF, fois thelineE D if tothe line D F (by the firft of ee ). Wherefore { by the to. of the tenth) the line E Dis commenfurable in leneth unto the line D F. But the tine E Dis ¢ationall and incommen{ura- blein length unto the line D C , wherefore the line D F is rationall and incommen|urable in length unto the line D C ( by the 13.0f the tenth ):Wherefore the linesCD and D F are rationall commenfurable in power onely . But arectanele figure comprehended under ratio- wall right. lines commen|urable in power onely,s (by the 21.0f the tenth irrational and the dine that containeth it in power is irrationall,and is called a mediall line. W, herefore the line that containeth in power that which is comprehended under thelinés C D and D Fis a me- aiall line.But the line B containeth in power the parallelograme which is comprehended un- der so Flow afquare Ty fayde tobe applied vppon a line, Demon Fra- £70 i» T he tenth Booke der the lines C D-and D F Wherefore the line B.is.a.mediallline..A right line therfore conse menfurabletoamediall line,is alfoa mediallline: whichwas requirid to be prowed. ri Corollary. Hereby it ts manifeft that.a ‘[uperficies commenfurable’vnio a mediall fitz perficies js alfo a medial fuperficies. | For the lines which contatae-in power thofe [uperficieces are commenturable in power , of which the one isa medzall line( by the definitio of 4 mediall linein the 21 .of this tenth): Wherefore the other allo sa medial line by this 23. propofitio.. And as it Was fayd of rational lines fo alfo is it tobe faya of mediall lines, namely ,-that alinecommenfurableto a mediall line;is alfo a mediall line, aline lfay which +s commenfurable unto a mediallline,whether it be sommenfnrabl in length , and alfoin 204 wer or-ells sn power onely. For vninerfally stestrue,that lines commenfuravlesn length , are alfo coms menfuravle in power. Now sf unto amedtall line there be alinecommenfursble in power, if tt be cons emenfurable in length,thé are thofe lines called mediall lines commenfurablein length e ia power.\Bxt of they be commenfurablean power onely they ave calied mediall lines commen furable i power onely»< | There are.alfo other right lines incommentfurable in length.to the medial] eenmdchee Ge ble in power onely to the fame:and thefe lines are alfo called mediall, for that they are commen{ura~ ble in power to the mediall line. And in as much as they are mediall lines , they are commenturable in power the one to the other.But being compared the one to the other, they may be commenfurable ei- ther in length,and therefore in power;or ell$ in power onely.. And then if.they be commenfurable in Jength ,they are called alfo mediall lines commenturable in length,and fo confequently they are ynder- ftanded to be commenfurable in power . Butif they be commenturable in power onely , yet notwith- fianding they alfo are.called mediail lines commenfurable in power onely, Flufates.after this. propofition teacheth how te come to the vnderitanding of mee diall fuperficieces and lines, by furd numbers, after this maner. Namely to expreffe the mediall {uperficicces by the rootes of numbers which are not {qaare numbers : and the lines cotainingin power fuch medial fuperficieces, by the rootes of rootes of numbets not fquare. Mediall linés'alfo commenfurable, are expreffed by she rootes of rootes of likefuperficial numbers,but yet not {quare, but fuch as haue that proportion that the dquares of {quare numbers have. For the rootes of thofe numbers and the rootes of rootes are in proportion as numbers are, namely, ifthe {quares be proportional! the fides alfo fhalbe proportional! (by the 22.0f the fixt) . But medialllines incommenfu- rable in power,are the rootes of rootes of numbers, which haue not that proportion, thatf{quarenumbers-haue. For theirrootes are the powers of medialllines , which are incommenfurable (by the 9.of the tenth) . But mediall lines commenfurable in power onely , are the rootes of rootes ofnumbers, which haue that proportion that fimple {quare numbers haue,and not which the fquares of f{quareé haue. For the rootes(which are the powers of the medialllines are comméfurable,but the rootes of rootes (which exprefle the fayd mediall lines)are incommenfurable. Wherefore there-may be found ont infinite mediall. lines incommenfurable in power, by comparing infinite vnlike playne numbers the one to the other. For vnlike playne numbers, which;haue not the proportion of fquare numbers , doo make the rootes which exprefle the {uperficieces of mediall lines incOmen‘urable(by the 9-of the tenth).And therefore the mediall lines containing in power thofe fuperficieces are incé menfurabl¢in length. For lines incommenfurable’in power, arealwayes incommenfu- sable in length(by the corrollary of the.g.of the tenth), { ) ait ! i H H i 4 N ‘ i My ) t ! b Hf ; f { j it; i te at Tit { 4 uy : a) 1 y hs i ‘ oh # ni it, f ‘ a Ania i} ay Ve Hn i , ly ao a} 7 At ou WAS jibe i q f° ie : |. ire i mt q Lhe 21.I beoreme. T he 24. Propofition. . J rectangle parallelogramme comprehended ynder mediall lines comene furdble in length is a mediall reffangle parallelogramme. | Suppofe Lg of Euclides Elementess Fol.2506 BUN Yio ; | | (ye wchtallricht lines AB and BC,which let becommenfurable in length . Then OS SOKE thefivlt) anon the ine ABu fanare AD .Wherefore the [quare A D is 2 me- Jur eu = ; ‘wafmushas theline A Biscom- | mepfurabietn length onto the line BC, and the lige» BERLE & Demon fira- S thot, j » . oe arts SPANK os SS 4 Bis equallanto the line B Dy therefore the line BD | | ' as commmrennfurn ble iL. length U73t0 the line B Cw Butas Rie, Rage | the Bue. DB iwstothetiae BG, {ods the [quare D Ato | ; Jaorammed Cl by the fx of the fixct) WV her- ae ee the parallelogramme y tbe fix ofthe fixt) WV F reins fore (by the 10:0f the tenth thefquare DA is commen- furableonto the paralldogramme A Cs But the {quare Dod ismediall; for tharit is deferibed upan'a medial line. Wherefore AC alfois a mediall J parallelograme( by the farmer Gordilary).. reclanglescxt: which was required tobe proued. gf The 22. Theoremes 0 Ehe'2s, Propofition. A retiangle parallelogramme comprehended pnder medial right lines conmenfurable in power onely, is either rationall,or medtall, REE .. Medill line being genen, there may be found another line commenfura- OS UANS) blewntoit, im power ancl). ( by the 14.0f this eooke ) a5 was taught there tox- Bh La \iag rauonall upes, Now then fuppofe that. the rectangle parallelogramme Vs PA \ AC be cumprehtnded under thee mediall right lines AB BC.T hen I fay, LES) Noid S shat the parallelogramme A C is either rationall,or megiall Deferibe (by the Con[truttion 46 .of the firft) upon the lines A Band BC, their [quares A D and BE .Wherefore either of thefe(quares A D.gnaB E is. mediall ( by the 21.0f the tenth) . Let there be put arationall like FG “And vpon toe line F G, let there be applieda rectangle parallelogramme G H €- qual to the fquare A L, and making the breadth the line F H .( Howto do this was taught inthe 22.0f this booke ) - And upon the line H M, apply a rectangle parallelogramme M K, equallto the parallelogiamme AC, and making in breadth the line H K: (todo this ye muft take a fourth line propationall with the lines HM, A B cy BC (by the 12.0f the fixt which ‘fourth line let be HK: wherefore (by the 26.of the fit) that which is.contayned under the extremes H-Mand HX, is equal to the parallelogramme contayned vader the meanes AB and BC). And more- oxer-uponthe line K Ix apply arectangle paral. | lelogramme NM Lequal tathe [quare B Es.ane raking in breadth tt line K im Webexfore the lines B Hall Kerk L, arein one.and the feift Lage right line CF 9 ihofe parallelogrammes foapplied upo the lines | | F G,H M,and KN, are rectangle , and theangles FHM and KH M are equall totwe xight. angles, for they treright angles : wherefore the lines F H.and HK are in one right bac, by the 14.0f the fff So alfa may be fayd of the angles HK Nand LKN). Ane ilsi GG. 4. forafmush Demonftra- $20Me _ ™ a ~ : - — - = —™, = —————— —. — - a | = SS = = = = ~er ~ \f hh ‘ 4 We » TWh ' 4 >= - Peet - teece-r — ~es eu par a 5 eS es = a — Inthefe ,, fewe words ,, confelteth the full de- ,, monflratrd ,, of the corol- ,, tary folow- 23 stig. 33 Note, The tenth Bovke | forafmmich as-either of thefe{quares AD-andB E is mediall; and the fquare AD is equall to the parallelogramme G H, and the{quare BE tothe parallelogramme NL : wherefore eitherof thefe parallelogrammes GH andN Lis mediall...And they are applyed upon a ra- tionallline,namely,F G . Wherefore (bythe 22.0f the tenth.) either of thefericht lines F if and K Lis avationall line,and incomen{urablein length vntothe line F.G..And foralmmuch as the {quare A D1s commen{urableto the {quare BE (for the lines A Band BC are [uppe- fed to be commen{urable in power ) therefore the parallelogrammeG H is commenfurable to the parallelogramme Nek ( for they aveequall vnto the fayd{quares) « But as the paralle- logramme G H is to the parallelogramme NL; {o-(by the first of the fixt isthe line FH to the line K L .Wherefore (by the 10.0f the'tenth) the line F His commenf{urable in length te the line K L . Wherefore thefe right lines:P H and K-Lyare rationall commenfurable in length the one to the other ( commenfurable in length, I fay;the-one tothe other,for unto thé, line F G, by reafon of which they arerationall, they. are incomenf{urable in length asit hath, beneproned ) .Whereforethe paralleloeramme contayned under the lines F Hand K Lyis. vationall ( by the 1.9.0f the tenth ) . | “¢ | And fora{muchastheline D:Bis.equall to the line BA, and thelineX B to the line BC: therefore as the line D B is to the line BC, fois the line AB tothe line BX . But astheline D Bas totheline BC, [o( by the firft of the fixt)is the [quare DA to the parallelograme AC. And as the'line AB is tothe line BX, foisthe parallelooramme AC to the {quare CX. Wherefore as thé {quare D A is to the parallelo¢ramme'A C; fois the parallelocramme AC tothe {quare CX .. But the (quare A D is equall to the parallelogramme G H, and to the pa- rallelocramme AC isthe parallelogramme M K ‘alfo equall, and to the parallelocramme NL is equallthe{quare BE > Wherefore as the parallelocramme GH isto the parallelo- cramme M K, [04s the parallelogramme MK to the parallelooramme NL. Wherefore ( by the fixft of the fixt) a> pels | die. ) the tine F W145 to the “ES © line H K, fo is theline | HK ‘to the line KT. * Wherefore ( by the 17. of the fixt ) the paralle- lograme contayned Un- der the.lines FH and KL, is equall to the {i quare ofthe line HK. cae But the parallelocrame pisses eon wes F i K contayned under the : lines F H and K L, is rationall, as hath before bene proued.Wherefore the [quare of the line H K is alforationall. Wherefore alfo the line H K is rationall, bai And now if theline H K be commenfurable in leneth unto the line H M,that is, ontothe line F G, whichis equall to the line H M, then (by the 19 .of the tenth) the parallelogramme be NH is rationall . But if it be incommenfurable in length unto the line F G;then the lines HK and M arerationall commenfurablein power onely. And (0 fhall the parallelocrame H N_be mediall. Wherefore the parallelocramme H N is either rationall, or mediall . But the parallelogramme H N is equall to the parallelugramme AC. Wherefore the parallela- gramme AC 1s either rationall, or mediall. A rectangle parallelogramme therefore com- prehendea under mediall right lines commenf{urable in power onely, 1s either rationall, or mediall : which was required to be demon|ftrated. ) How to finde mediall lines commenfurable in power onely contayning a rationall parallclogramme;and alfo other tediall lines commenfurable in power ‘contayning'a | mediall tee aw bt of Euchides Elementes. Fol.251. mediall paraliclogtamme, thallafterward betaughtin the.27 .and.28,Propofitions of. Gres bGOREA Sn weeny lala (Maan shrio. es ake ed) rao CM GA Corollary Hereby it is manifest that a rettanple parallelogramme contaynea under 69 right lines is the eseane proportionall betwene the [quares of the fayd lines. As it was manifeft (by the firit ofthe fixt) that that which is:contayned vnder the lifes‘ AB and B'C, isthe meane proportionall betwene the {quates AD and CX . This Corollary is put afcérche\s3.Propofition of this booke asan Affumpt,and chere déition trated, which there in his place you fhall finde, But becaufe it followeth of this Propoft- tion fo euidently and briefy without farther demonftration jl thought it not amiffe here by the way to- note it. Le | q Uhe 23. T heoreme. The 26, Propofition. A mediall fuperficies excedeth not a mediall fuperficies bya rational [ue perficies. SS 4 AL Or if it be pofiible,let A B being a mediall [uperficiesexceede AC being alfoa X(N, medial [uperficies,by D B being arationall fuperficies . And let there be put a RM } rational right line EF. And-vupon the line E F apply 4 rectangle parallelo- gramme F H, equall vuto the medial fuperficies A B, whofe other fide let be line G His rationall and commenfurableinlengthuntothe line EF , But the line E G is ra- tenth) the [quare of the line E Gis incommenfurable unto the parallelogramme contayned Wherefore the [quares of the lines EG and GH are incommen{urable unto the parallelo- A Corollary. Conkructions Demonflra- tion leading te an abfwrditi¢e Contruftion. Demon [tra- 10Me Square of the line EH ds incomenfurable to the [quares TherenthBooke= namely, as Wyitiers tothe number 2; o¥ as-2543 40-48 and therefore (by the b: of this bookey shey are commenfurable ). Wherefore (by the 13. of the tenth) the Squares of thetines BG and G H are incommen{urable vnto that whichis contayned under the lines EG and GH twife .( This is more briefly concluded by the corollary 3 ofthe 13.0f the tenth) . But the {quares of the lines E.G and. GH together. with that which 1s contayued under the lines EG and G H.twife.are equalbta the {quareof theline EH (-b5 the g. of thefecona Y" Wherefore the: of the lines EG and G H (by the 16.0f the tenth) . But the {quares of the lines E G ¢ GH. are rationall Wher- fore the [quare of thelimeE H isirnatianall . Wherefore the line alfo E H is irrationall . But it hath before bene prodedtoberationall : which is impuible . Wherefore. 4 mediall fuperficiés exceedeth not a mediall [uperfictes Ly a rational! [uperficies : which was required to be proud. q.Uhe 4.Probleme. The 27.Propofition. L'ofiideout mediall lines commen/urable.in power onelys contayning 4 rational! parallelogramme: gn GBPS. Et therebe put two rationall limes commenfirablein power onely, namely; A and A © eS & B. And (by the 13.0f the fix) take the meine proportional betwene the lines A and ) EES? Band let the [ame line bECX ADA as the line Ai 10'the Line B [0 (by the 12. of the fixt) let the line C be tothe line D. And foralmuch as A and Bare vationall lines commen{irable in power onely therfore (by the 21-0f the tenth) that which is contay ned-vnaer-the lines A-and B,that ts;the fquare of theline-C: For the [quare’ of the line C is equall to the parallelogramme contayned onder the lines A anh B (by the 7. of the fixth)is mediallsther fore C alfois a mediall ine. And for that as the line A is to the line B,fo is thelineC to the line D, therfore as the {quare of the line-Ais'to thefquare of the lyne B, fo is thé {quare of the line’ to the {quare of the line D (by the 22. of thefixth): But the “i Na oa {quares of the ines A and B are commenfurable, for the lines A ana B are {uppofed to be rational comme. Jurable ta power onely. Wherefore alfo the fauares of rs the lines C and D are commenfurable*(by the'ro.’ of thetenth) wherforethe lines C and 'D are commen|u- Bye Sag | rable in power unely.And Cis a mediall line. Wherfore (by the 23. of thetenth) D alfois a medial line WherforeC and D are mediall lynes commen|urable in power onely. Now alfo I [ay that they contayné'arationall parallelogramme.F or for that as the line A is to the line B. fois the line C to the line D : therfore alternately alfo (by the 16.0f the fift) as the line # 45 to the line C.fois the tyne B to the lyne D But as the Lyne A is to the lyne C, fo is the line C to the lyne B : wherfore as the line Cis to the line B,foisthe line B to the lyne D. Wherfore the parallelogramecotayned under the lines C and D is equal to the |quare of the line B.But the [quare of the lyne B is rational. Wherfore the parallelocrame which is contayned under the lynes C and D is alfo rational. Wherfore there are found out mediall lines commen{ura- | ble BO Be of Euchdes Elementess Fol.252. ble in power onelyscontayning a.rationall parallelogramme : whichwas requited to.be doney. The 5: Probleme. I'he 2a. Propofition. Lo finde ont mediall right lynes commenfurable in power onely,contayning a mediall parallelacramme. Mg Et there be put three rationall right lines commen{urablein power only,name- e,/),A,B,and C,and(Ly the 13. of the fixt)take the meane proportional betwene \ the lines A and B,c> let the fame be D.And asthe line B is to the lineC fo(by ithe 12.0f the fixt) let the line D beto the line E. And forafmuch as the lines PPADS A and B are rationall commenfurable in power onely therefore (by the 21. of the tenth) that which is contained under the lines A and B, that is the {quare of the line D, is medtall, 4 RLS Wherfore D is a mediall line. And foralmuch as the lines B and C are commen{urablein fener onely,and p RR 128 as the line Bis totheline C,fo isthe line D to the line E : wherfore the lines Dand E are commenfurable 3 Br 8 in power onely (by the corollary of the tenth of this booke) but D is amediall line.WhereforeE alfoisa ¢ BS mediallline (by the 23 of this booke.) Wherfore D gp E are medialllines commenfurable in power onely. 1 BR 7 % fay alfa that they containe a meatal parallelograme. For for that as the line B is to the line C,fo is the line D to the line E : therfore alternately (by the 16 of the fift)as the line Bis to the line D,fois the line C toy line E. But as the lyne Bis tothe line D, {ois the tine D to the line A, by conuer{e proportion ( which is proued by the corollary of the fourth of the fifth) Wherforeastheline D is to the line A, {ois the tine C tothe line EWherfore that whichis contained under the lines Ac C,is (by the 16.0f. the fist) equall to that whichis contayned under the lines D & E, But. that which is contained onder the lines:A and Cis medial(by the 21.0f the tenth.) Wherfore that which is cotained onder-thelines:D and E is mediall,Wherfore there are found out mediall lines commen{u- rable in power onely,containing a mediall [uperficies : which was required to be done. : An Affumpt. To finde ont tvo fanare numbers, which added together make a _{quare number. ; Let there be put two like [uperficiall numbers CAB and BC (which how to finde out, hath bene taught after the 9: propofition of this booke) And let them both be either euen gumbers or odde. And let the greater number be AB. And fora{much asif from any ever pumber be taken away an enen wumber,or fro an odde number be taken away an bdde num- Arie De wie Gas ees B ber ,the refidue hall be exen( oy the 24:and 20 | | of the ninth): If therfore from AB being an'euen number be taken away BC aneuen. num- ber, or from A B being an odde number be taken away B C being alfo odde : the refidue AC. foall be euen. Deuide the number AC into two equall partes in D: wherefore the number mich is produced of A Binto BC together with the {quare number of C D; is (bythe fixt of the fecond;as Barlaam demanftrateth it in numbers) séquall to the{quare number of BD.- GG.ty.* but Conruttion, >. De monStran FON, A Laroliary, To finde out two fquare numbers ex ceeding the one the other by a fquare muNIDER, An Afumpte T he renth Booke Burthatwhich is produced of A BintoB Cis afquare nuber.. For it was prowed (by the firft of the ninth) that if two like plaine numbers multiplieng the one the other, produce any nu ber,the number produced fhalbe a {quare number. Wherfore there are found out two [quare numbers,the one being the {quare number which is produced of A B into BC, and the other the {quare number produced of C D which added together makéa{quare number, namely, the [quare number produced of BD multiplied into himfelfe, fora{muc h as they were demo- flrated equall to it. x. A Corollary. And hereby itis manifest,that there arefound out two Square numbers namely,the one the [quare aumber of B.D, and the other the {quare number. of C D, fo that that number wherin thone excedeth the other, the number (1 fay) which is produced of A B intoB C, ss alfoa {quare number : namel,when AB BC are like playne numbers. But when they > A vvverD pocorn © ieeeens B are not like playne numbers, then are there | found out two {quare numbers,the [quare number of B D,and the {quare number of DC, whofe excefse,that ts,the number wherby.the greater excedeth the le(e,namely, that whichis produced of A Binto BC,ts nit a [quare number. « An Affumpt. To finde out two fqnare numbers which added together make not a {quare Snwaber. LeeA Band BC be likeplayne numbers fo that ( by the firft of the ninth) that which is roduced of A B into BC 63 {quare number,and let AC be anenen number.And deuide C ‘A into two equal partes in D: Now by that which hath before bene {ayd inthe former af- fumpt, itis mamfef? that the{quare number produced of ABinto BC , together with the {quare number of OD ,isequall to the fquare number of BD: Take away from CD vmitte DE. Wherfore that which is produced of A Binto BC together with the {quarejof CE is leffe then'the [quare number of | BD. Now then fay thatthe A..GseH DE. F oo) © vives B {quare number produced of AB into B C added to the fauare number of C E,make not a [quare number. For if they do make a[quare numbert hen that [quare number which they makests either greater thé the ‘{quare number of B E,or equall unt it,or le(’e then it. Firft, greater it cannot be, for it is already. proued that the (quare numbir produced of A B into BC together with the [quare number of © Bits leffe then the [quarenumber of BD. But betwene the (quare number of BD, and the fa nare number of B-E,thire is no meane fquare number. F or the number BD excedeth phe number E onely by. vnitie : which vnitie can by no meanes be deuided into. numbers. Or if the number produced of A Binto BC together with the {quare of the nuber C E,fhould be greater then the fquare of the number B E, then fhould the felfe fame number produced of A Binto BC together with the {quare of thenumber CE be equall to the {quarc of the number BD ;the contrary wher of is already proued.Wherfore if # be poffible,let that which is prodnced of A B into. B C together with egeee number of the number CE be equall ta the (qntre number of B Ew _And let G A he double to vnitie D.E,that is, let tt be the num- ber twos Now foralmuich as the whole number A Cts by fuppofition double to the whole number CD of which the number 2G isdouble to vnitie D Et herfore (by the 7 af tielt if | : meres. of Euchides Elementeses Fol.253. wenth) the refidue;namely the number G.C is double tothe refidueswameely,to the number B C.Wherforethe number G Cts denided into twoeguall partes inE. Wherefore that which is. produced of GBinto.B C together with the [quare number of C E is equall to the (quare nit be? of BE But that which 1s produced of A B into B C,togetber with the (quare number of CE,ts {uppofed ta be equal ta the {quare number of BE: wherfore thatwhich is produced of G Binte b C together with the {quare number of C-E-is equall ts that which is produced of AB into BC, together with the [quare number of CE. Where ore taking away the {quare number of C E,which ts common to them both,the number A Bfhall be equal to the mum- ber G B,namely the greater to the leffe, which is ampof jible Wherfore thatwhich is produced of AB into BC together with the [quare number of C Ess not equall torhe f quare number of B El [ay alfo that that which is produced of ABinto BC tozether with the (quare mans bet of CE 1s not lefse then the (quare number of BE.For sf it bi pofSsble, thé fall it be equal to fome {quare number leffe then the {quare number of B E.Wherfore let the number prodit. ced of A B into B Ctogether with the {quare of the number C Eve equal to the fanare nui- ber of BF And let the number H A be double tothe number D FT hé alfo it followeth that the number i C is double to the number CF, fo that HC alfe is denided into twocequall partes in F and therfore alfothe number whichzs produced of H B into BC, together with the {auare number of F C,ts.equall tothe [qnare number of thenmmber BF. But by {i Dpofi- sion,the number which is produced of AB into B€ together with the {quare nimber of C E is equall to the {qnare number. of B F.Wherfore it follaveth thet the number produced of A B into BC together withthe {quare number of C E,is equall tothat whichis produced of H Binto BC together with the {quare number C F which isimpo fible. Foxif tt fhould be e- qual, then forafmuch asthe fauare of C F 1s leffe then the {quare of C E,the number prodt- ced of H B into BC fhould be greater then the number produced of A BintoBC. And fouls fe fhould the number H B be zreater then the number AB when yet tt i: liffe then it. Wher- fore the number produced of ABinto BC together with the {qsare number of CE, is not lelfe thenthe fquare nitber of BE. And itis alfo proued that it cansot be equall to the fauare number of B E neither greater then it. Wherfore that which ts produced af AB into BC added to the {quare number of C E,maketh not a {quare number. And although it be poffible to demonftrate thes nanny other. mayes yet this femeth taws ‘fuffcientsleaft the matter beyng ewer long ,fhould feeme to much tedious. SAW INST OD he 6: Probleme. The zy. Propofition. Lo finde out two fuch rationall right lynes commenfurable in power on- Ly,thatthe greater fhall be in power more then the lefseby the [quare of 4 right line commenfurablein length vuto the greater. Ss Et there be put a rational line AB, andtakeal. eee (0 tivo [uch (quare numbers. D.and DE,that % their exce(fe CE be not a [quare number (by the corolary of the firft-al[umpt of the 28.0f the tenth) And vpon the line AB defcribe a femicircled F B. And by the corollary of the 6 of thetenth, as. the. number D Cis tothenumber GE,fo let the{quare of the lyne BA be tothe {quare of theline AF .Anddraw a line frou F ta By Now for that as the (qnare-of the line BA ista thé... Crore Bs... D fquare of the line AF fois the number CD to the num- | GG sig. | ber Conftruttion. Demon ftra~ t2@n, TZ he tenth Booke ber CE, therfore the {quare of the line B A hath to the{quare of the line:A Pf, that propors tion that the niber CG D hath to the number C E.Wherfore the fquare of the line BA is tos we [urable to the{quare of the ine A F (by. the 6.0f the téth).But the {quare of the line AB és rational. Wherfore.al{o the {quare of the line AF is rational Wherfore alfo the line AFts rationall.And fora{much as the number D hath notwuto the number © E that proporti-s on that a{quare number hath to.a{quare number,therfore neither alfohath the {quare of the line AB to the{quareof theline AF that proportion < that a{quare number hath to a {qnare number. W herfore (by the 9. of the teth)the line A Bis vintothe line.A F in- commen{urablein length. Wherforethe lines .A F and.A B are rationall commen{urablein power onely.. And.for that asthe number. DC is to the number CE, [01s the {quare of the line A B.tothe {quare of theline.A F ther: fore by conuer{ion-or ener{e proportiowhich is demonfira- ted ( by.the corollary of the 1.9. of the fifth).as. the: number C Dis tothe number DE, fo1s thefquare of the live AB to the {quare of the line BF, woichis the excelje ofthe fquare of the line AB aboue the {quave of the line LAF (bythe afvumpt put before the.14. of this booke). But the number C D hath to the number D E.that proportion that a {quare number hath toa fquare number: wherfore the [guare of the line A B hath tothe {quare of the line BF, that proportion that a fquare number bath toa lquare number. Wherefore (by the 9. of the tenth) the line AB is commen{urable in leagth unto the line BE: And (bythe47. of the firft) the fquare of the line AB is equallto the {quares of the lines AF and F B. Wherforethe line A Bisin power more then the line.A F by the {quare of the-line BE whichis commen|urable in length vn» tothe line AB. Wherefore there are foundout twofichrationall lines commen{urable in power.onely namely,.A Band A F [othat the greater line A B is in power. more then the leffe line AF ,by the {quare of the line F B,which is commen{urableinlength unto theline A. B.: which was required to be done. 7 i ans ¢ Lhe.2.. I heoremes:: Ihe 30. Propofition. Montaurens LF finde out two [uch rationall lines commenfurable in power onely , that wee 1 sd the greater fhalbe in power more then the lefse by the {quare of a right line an SHUMIPT. . Da i - sgt he incommen|urablein length to the greater. text feemeth eek | Ras Et there be put a rationall line AB , and take alfo (by the 2..affumpt of the 28. of but Without 4 AWC? the tenth) two fquare numbers C E and ED , which being added together make Conf aifien 3) WES? not a [quare number and let the numbers CE and E D added together make the a number CD. And upon the line A Badefcribe a fencircle AF B. And ( by the corollary of the 6. of the tenth ) asthe number D C is tothe : : | number C E,folet the {quare of the line A B be to the fquare of the line AF and draw a line from¥ to B. And we may in Demonftra~ like fort,as we didin the former propofition , proue that the $i0fte lines BA and AF arerationall commenfurable in power one- ly. And for thatas the number DC isto the number CE, \) | foisthe {quare of the line AB to the fquare of the ine AF: 47 , B. therefore by conuerfion( by the corollary of the 19-0f the fifte) ) | 7 as the number C Dis to the number DE; foisthe fquareof C......... Eee. D the line AB to the {quare to the line EB: But the number C - | ‘ ” ; ot IAG At todo tkewife =. of Eucliles Elemente Fl.254. ~ cA D hath not to the number DE that proportion bhava /quare wunaber hath toa fyivares number . Wherefore neither a the fguareofehe tine A Br bach tothe fouae ofthe q line B F that proportign that.a ware number bath rouyipnare wat ber Wherefore the line A Bis(bythey of thetenth )incommenfurable imenathtorhe tine BP wuhd the bine AB és in power nore then.thé line A F by the [quareof the rieht bhoBP yw whith 4s incomntns furable in length vntothe line AB. Wherforethe tints AB und AP urevasionall commen. furable in power onely. And the lint A B i m power nore then the time Al Puy she(qaurcof the line Bwhichis commenfurable in length vutd helime AB wwhich was required to If there be'two right lines hinnig betmene them feluesany proportion + ay" the one right line ts to.the other , [01s the parallelograiné contained Vader both thevight lines ta the [quare of the lefse of thofe two lines. | Sappofe that thee tworight A.B avd BG be Fu. « D iw ome certaine proportion. I hen L fay that.as the.| | line A. Bis tothe live B.C,fois the parallelograme.|... .. contained vider A.B and B.C tothe:[guare.of Bi. C. Deferibe the {quare of the line B.C and let the}... fame be CC Dyand make perfeth the parallelogranz \... | ae AD wowitis manif csi thatas.the lneAB ts to. eS Rees Sa A the line BC; f07s the parallelograme AD to the | et parallelograme or fqnare BE (by the fir of theyixt’) But the parallelocrame A. Dis that which is boxtained under the lines A B and BC. for the line BC is equall to the ke BD and the paraltelagrame B E.isthe fauare of the line B&).Wherefore asthe line A Bis tothe line BC {07s the paralleloerame coutained under thelinesA Band BC to the {quare of the line B C,which was required to be proued. q The 8. Probleme. T he 31. Propofition. sac, d.0fihde out two medial lines commen|urable in power onely ,comprehene din g arationall /uperficies fo that the greater [hall be in pawer more then the lefSe by the {quare of a line commenfurable in length dnto the greater. Et there be taken (by the 29.0f the tenth) twerationall lines commen{urable in pow- % s| cr onely A and B,fo that let the line A betie the greater be in power more then the Poe |'lineB, being the re bythe fquare of a line commen{urablein leneth unto the line A,.And let the fquare of the line C be equallto the parallelograme contained under the lines AvandB which ts done by finding out the meane proportional line , namely the lineC be tmene the lines A and B(by the 13.0f the fixt).Now the parallelograme contained under the lines Rand Bis mediall( by the 21. ofthis booke ) . Wherefore (by the corollary of the 23. of the tenth )the fquare al[o of the line C is medtall. Wher fore the line C alfois mediall:V nto the [quane of theline B let the perallelograme contained under the lines C and D be equall (Ly findingout a third line proportional ) namely the line D to the twalines C and B (by the 11.0f the fixt) But the aware of the line Bis rational. Wherfore the parallelograme ton- tained vnaer the line C and Dus rational «And for that as theline A is tothe be B; . 8 a , +. ~ Leake Oe ¢ . a} “ . : ‘és . + te 4&4 5 x . , + ++. “¢ ’ : : > ‘ d va BQ >) | a t a ; : : é . ay avs — a : a a An Affurn tv ; UT UR ¢ " ; ; @re¢ ack VS ; ; , 4 ’ a ° : Mi 2 This Afvumpe . fetteth forth | nothing els but that » which the frft . ofthe fixt fet- ~ teth forth,and therefore in ” fome exam- pla 5 it $5 108 founde. Construction s Demon ftra- ston. Couftractiote Demonfirae tt0M. BULA ThetenthBookew> fosssbiparallelagtante contained under thedines Arse wy AG anaB to thefquareof the ineB ( by. the afvumpt.go- Si stn ec ae | | . ) | ‘ wi ing hefore).But unto the parallelograme. ontaineds Bie 68.5 the {quare of the line D,anavnto that which iS contayned vnder the lines Bey C; is equall C eee Phnt which 23 comtayned Under the lines D and E .Whereforeas the line A is to the line C, fo is theifauaveof the line D;t0 thativhich is contayned Under the lines D and E . But as the [quare of theline Dis tothat which is contayned vider the lines D and E, fois the line D to the line E (by the A iumpe put before the 22.0f thetenth:) . Wherefore as the line A is to the line C; [08 theline D totheline E . But theline Ais unto the lineC commenf{urable in power onely . Wherefore the line Dis unto theline E commenfurablein power onely . But D 432 mediall line : Wherefore by the 23 .of the tenth) Ealfoisa mediallline. And for that as the line Ais to the line Cfots the line D tothe line E,and the line A is in power more then the line C, by the [quare of a line commenfurablein length vnto the line A. Wherefore (Ly the 14.0f the tevith ) D is in power miore then Es by the {quare of a line commenfurable in leneth unio the line D . I {ay moreouer that that whichis-contayned under the lines D and FE i$ medial SF or forafmuch as that whichis contayned under the lines B Gy C, is equall to that which is tomtayned vader the lines D and Es but that which is contayned under the lines B and Cis medial . Wherefore that which is contayned under the lines .D and E is al- fo mediall: diet ety there are found ont two medialllines D.and E,comen{urable in power onely, comprehen ing a mediall [uperficies, fo that the greater 1s in bower more then the lefe, by the [quare of aline commenfirable in leneth tothe greater: which was — to be ace dope — wv ———— ee . Re sf Coufirictione Demonstra- L0H. eae NANA aes ior T he tenth Booke 1 iy | i | a done. Aad thus-dss edidest > bow sm like forte. gay be founde out tuo medial ines commenfurable in power Ai | Bi ef Corgliay ys onely comtaysing « medial {uperficies,fo thatthe greater fhall bein power mote then the Jeffe, by. the [qnare of @ | if ii] fe encemminen{urble sx length Unto the greater. Whey the ling Ais in power more then the line Cs ae ai by the [quare of a.line incomenfurable intength vato the liseA:as the thirtethteacheth vt. ATE ii Bed sa: aia to. AnAtumpte. Ce Ge a Suppofethat there be axcctangle triangle A B C,haning the angle BAC aright angle. Ha And (by the 12.0f the firft ) from the poynt A to the right line. BC ,.4 perpendicular line HR icing drawen AD: then I fay firft, that the parallelogramme contayned under the lines vii ie r. CB and BD, isequall to.the{quareofthe line BA... Secondly I fay;that the parallelo- Wi gramme contayned vader the lines BC and C D,isequell to the {quareof the line C A. Wa i ThikdlyLfay,that the parallelogrammecontayned under the lines B.D and D.C,1s equallto Ni Val 3. the {quate of the line A D.«. And fourthly fay that the parallelogramme contayned under ee 4. the lines BC cy A Dis equall tothe paraelocramme cotayred vader the lines BA cy AC. A ne | As tonching the firft,that the parallelocramme contaynednder the lines C Band B D, a at envalltothe{quareof the line A.B, isthis proved. | i i Kor forafwinch as intherechangle triangle BAC, Vana) from the right angleunto the bafe is draven.aper- ae peadicubar ting A Dythexfore( by the-8..of the foxt) nT ie tit tbe triangles CLBD. and. AD Csare diketo the a whole trinazle ABC, andvare alfo likethe oneto ie We the other. Aud for that the triangle ABC -isilike ts a to the tridnols.A D-Bs therefore both thetriangles: ue ih areeguiancle. by the defines io of like figures.W her- Wi ei | fore (ly the 4.of the fixt) asthe line C B is to the HE HE line B.A, {0 is the line CAB to the-line BD. Hil Hiliii * A Corollary added by I.Dee.~ * Wherefore ( by the | Ae i * Therefore if you deaide the fquare of the og ‘§ 6 gad ae il ie 4 fice AB by the fideB C #the quotient will be parallelogramme cath : _ ea at' BD. Which maketh D C alfo knowen: ‘by ei- tained under the lines BC cre Dts equall to the, fqnare. of | i uh ther of which (by the 47.0f the firft) the ai the line AB. ih ai stented ro ie —e ce oo aeaden As touching the fecond, that the parellelogramme con- Ht aii three hes AB, A Cyan BC; are kiiowen or tained vader thetines BC ind CD, ts equall to the fquare TA geuen. of the linesG-is by the fel fame veafon proued . For the triangle A BC is.tike to thetiangl AD C.W. herefore as the I.Dee *ThefecoxdCorollzry, line BC ista the line AC, [os the line; AC tothe line DC. i . K Whi | : or, contained under thelines * Therefore if yon denide the fquate ofthe sa herefore the pagrree Pere he Lieesd Cn fide A C,by the fide B.C, the:portion DC; will BC and.G Ds - 5% the {quare of t CURED Sos Kee be the produdt. &¢. asin the former Corol- ching the third, thatthe parallelogramme contained under lary . the lines BD. ahd DC, isequall to the fc quare.of: the line D A, as this préwea For; foralmudh as if in a rectangle triangle be drawn fre the right angle tothe bale a perpendicular line, the perpendicular fodrawen is the © woeane proportional betwene the feemets of the bafe( by the corollary of the 8.0f' the fixt):ther fore as inedine B.D istothe line D-A,; fo is the line A Dtdiheline DC - Wher e for €( by the F 7.0f the fisct.).the parallelogramme contayned under thelines BD and DC, is equal to the {quareof the line D-A.s~ As touching the fourth, that the par allelograt sre COME ained wiser the line\ BG aad 4D, is equall.to.the parallelogranme contained under the lines B Aand AC, isthus proued.. For forafmuch as (as we have already declared) the triangle ABC.is like, and therefore equianglestothe triangle ABD, therefore as the hue B oe te | | ye ; eget) a Gg ; ‘ep A i fi ae tH 4 : ' ] te HH b ie ai Weg .7 Aili 1 ' ‘ye fT q ¢ ws 4 i ie eel ith bi ! iit th ; he AE oa ee +t ) Dae | ih aan Hi 4 } i : of. Euchaes Elementes. Fol.256 the line AC, forts the line B At thé line AD (4 the 4. of TDee *T he thirdeCorollary, the fixt) .* But if there be foweright lines proportionall, Pe that which is contained under the firft and the last,is equall ANA be denied 'e : ——— of es to that which is contained underthe two meanes ( by the.16. geyethe perpendicular A. Thefe three Co- bf che fixt) Wherefore that which is contained under the xollaryes mpragtife Logifticall and Geometri- lines B Cand AD, is equal! to that which is contayned under call are profitable . the ines B Aand AC. : 1 fay woreouer, thatsf therebe made a parallelogramme complete, contained under the An other de- lines BC and A D, which let bea C : andiflikewife be made complete the (mabey monStration contained under the linesB A and AC, which letbe AF ; it may by an other way be promed of this fou rth that the parallelogramme E C is quall tothe parallelogramme AF . For, foralmuch as e- L re of the ther of them is double to the triangle A CB (by the.gr. of the firft ) sand thinges whith are Clermiatro. double to one and the felfe [ame thing, are equall the:owe to the other. Wherefore that which is contained under the lines BC and A Dyis equall to that which is contained under the linesB Aand AC, ? 2. @AnAffumpt. If avight line be deusded into two bnequall partes: as the greater part is to the lefse , fo is the parallelocramme contayned bnder the whole line and the greater part, tatheparallelogramme contayned ‘bnder-the whole line and the lefSe part. 7 An Affumpt. Denide the right line.A Bintotwo-unequall partes in the point E : And let AE. be the greater part . Then I fay, that asthe line _A E.ssto the line E B, fois the parallelogramme contained under the lines B A asd AE tothe parallelocramme contained under the lines BAC BE. Defcribe the (quareof the line A Byand tet the fame be AGD B. And from thepoint E draw into either B E A. of thefe lines AC and D Bea piralleltline EF Now itis | manifeft that as theline. AE sstothe line EB; fo is the pa- rallelogramme A F tothe paralidogramme BF (bythe firft | of the fixt) . But the parallelogranme A F is contayned un- der the lines B.A and AE (for the line AC is equalltothe line AB) and the parallelogranme BF is contained-vn- der the lines A Band B E ( forte line DB is equall-to the line AB). Wherefore as thelim A Eis to the line EB; fo bs is the parallelogramme containeaunder the lines B A and AE, to the parallelogramme costained under the lines A Band BE: which was regai- = > =, Co xed to be dentunfirated, ‘ses, This Affumprdiffereth life from the firft Propofition of the fixt booke. ae. 3. a An Affiampe: * pare If therebetwovnequall right lines and if the lefse be denided into two ee quall partes : the paralelogramime contained bnder the two ‘dnequall lines, is double to the paralleb eramme contained Ynder the greater line ex halfe of the leffe line. ae weal a : . ABS. Suppofe Con Fraltion. Demanftra- $5G%« fore by compofition of proportion, as the. whole line ee ene T betentb Booke Suppofe that there be two unequall right lines A B and BC, of which let AB. bethe greater,and deuide the line B Cento two.equall partes in the point D. Then 1 fay, that the pa- rallelogramme contained under the lines A B cy B.C, is double to the parallelogramme con- tained under the lines AB and BD. From the poiut Braifevp vpon the right line BCs perpendicular line B E,and let B E beequalltothe line B.A. And drawing from the point Cand D,the lines C G and D F parallels and equall | toBE : and then drawing therightlineGFE,theg 7 figure is complete... Now for that as the line D B is tothe line D C,fo is the parallelogrammme BF to the parallelogramme DG by the 1. of the fixt) : thers BG is tothe line D C,fois the parallelogramme BG $0 the parallelogramme DG ( by the 18. of the fift). Bat the line BC1s double tothe line DC. Where. ' fore the parallelogramme BG is double to the pa- rallelogramme D G . But the parallelogramme BG 's contained under the lines A B and BC.) for the Line A Bis equall to the line b E ( and the parallelo- giamme DG. is coutayned under the lines AR.and BD) for the line BD is equall tothe dine D C,and the line AB tothe line DF : which wasrequired to be demonftrated. q Lhe 10. Probleme. T he 33. Propofition. | Lo finde out two right lines incommenfurable in power whofe [quares ade ded together make arationall fuperfictesjand the parallelogramme contate ned bnder them make a mediall fuperficies.’ Ake by the 30. of the tenth, twa rational richt lines commenfurable in power onely, namely, AB and BC; fothat let-theline.AB, being the greater, be it power more then the line BC being the leffesby the {quare of a line incommen« \& furable in length vate theline AB. And bythe 10. of the firft, deuide the 3! line BC into two equal partes in the point D. And vpon the line AB apply s paratlelogramme equall to the fauare yaa either of the line B D or of the ineD C, and wanting in ficure by a [quarts by the 28 .of the fixth, and let that pas» vallelocramme be that which #§ coms ~ tained under the lines AE and BB. Audupomthe line AB deferibe a fe- « micircle A FB. And bythe 11. of the fir ft. from the point E,raife vp unto theline AB, a A of Euchdes Elementes. Fol.257, shelineB Aund AE isequall tothe [quare of the hve AF jby thé fecond part of the firft af je? 4 fumpt before put..And that whichas contained under the lines.ABand BE is by the firft part of the famealfumpt, equalltotbe/quare ofthe line BF Wherforethe{quare of the line A F ts incommenfurable to the {quare of the line B FiWherforethe lines AF and. BE are incommen{urablein power. And forafmuch as .A Bsa rationall line (by fuppofition ther- fore (by the 7 definition of the tenth) the {quare of thelineA Bas rational. Wherefore. alfa the fquaresof the lines AF and F B added together make a rationall{uperficies.Fan(ly the. 47 of the firft they are equal to the fquare of the linexA B. Again forafimach as( by thethird part of the first afiumpt going before that whith is contained under the lines AE and. BB is equall to the {quare of the line E F. But by {uppofition that whith is contained vnder.the lines A E and E Bis equall to the (quare of the line BD. Wherfore the line F E is equall to the line BD. Wherforethe line BC1s double to the ine F E.VW herfare(by the third afsurmpt going before) that which 1s contained under the lines.A Band BC; is double to that which 4s contained under the lines: A Band E F But that which ts. contained vuder the lines A Band B C,is by fuppofition medial Wherfore( by the corollary of the23 sof the tenth) that which is contained under the lines ABand EF is. allo medially but that whiche is -con- tayned under the lines A Band EF, is (by thelaf parte of the firft afvumpt eoyne be- fore) equall to.that which is contained under the lines AF and F B. Wherefore that which is contained under the lines. AF CF Bis.aimediall fuperficies. And it is protucd,that that whichis compofed of the{quares of the lines A F and F Badded together is rationall. Wher fore thexeare. found out two right lines AF and FB incommen{urable in power, whale fauares added together,make a rationall [uperficies,and the parallelogramme contained vit der thems, is a mediall{uperfictes : which was required to be done. | ) q The 11. Probleme. ~The 34. Propofition. I 0 finde out tworight linesincommenfurable in powers, whofe fquares ade ded together make a mediall fuperficies, and the parallelogramme contay- ned ynder them,makea rationall fuperficies. | Ake (by the 31 .0f the tenth) two medial lines AB and BC, commenfurable in ipower onely comprehending a rational {uperficies, fo that let the line AB be in power more then the line BC. Ly the fauare of a line juciminen{urable 72 length ZV upto the line A B.« And defcribe vponthe line A Bua femicircle A D B.And by the ro. of the firft, deuide the line BC ‘vito two-equall partes in the point E. And (by the28.of the fixt upon the line As a. AB apply a parallelocramme equall to | i | | soNS the {quare of the line B E, and wantyng | ly in figure by a fquare, and let that paral. © lelogramme be that which is contayned under the lines A F and F B.Wherfore Laat the line A F is incommenfurablein length unto the lineF B (by the 2.part of the 18. of the tenth). And from the point F unto the right line A B,raife vp (by the rr.of the firft) a per- pendiculer line F D,and draw lines from Ato D,and from D to B. And feralmuch as the line A F is incommen|urable unto the line F B : but (by the fecond afsumpt coing beforethe 23 0f thetenth) asthe line AF istothe line FB fo'ts the parallelocramme contayned under thé lines BA and AF ,to the parallelogramme contained under the lines-B A and BFE wher fore (ly the tenth of thetenth) that which is contained vader the lines ‘B.A and AF isin- commen|urable to that which ts contayned under the lints.A B and BE + but that which is es HH.4. - 6673- ry . Psy & & A ise firft part ro : of the deter: M2 dI20? yi sabi Gites 1¢cn cluded, the fecond part cocludths Confirn ig » Demonstra~ 70%» The firft part of the deter mination COn- ¢f & a ed. The fecond part cocluded, The total coucln fon. Confirntlion. Demonitra- $107 The firft part concluded, Thetenth Booke contained under the lines B Aand A F is equall tothe fauare of the line AD, and that which is contained under the lines A Band B F is alfo equallto the {quare.of the line DB (by the fecond part of the firft afumpt going before the 33. of the teth) wherfore the [quare of the line A D isincomenfurable tothe.» fquare of the line DB. Wherefore the vs renitiieke lines AD and D Bare incommen{ura- | ble in power. And forafmuch as the athe fquare of the line A B is mediall, there- fore alfo the {uperficies made of the “ : 3 fquares of the lines A Dand D Baa- ded together is mediall.For the {quares of the ines AD and DB are ( by the 47. of the firft ) equall to the fquare of the line A B.. And forafmuch as theline BC is double to the line F D (asit was proued in the pro- ofition going before) therefore the parallelogramme contained under the lines AB and and B Cis double tothe parallelogramme contained under the lines AB and F D (by the shird alfumpt coing before the 33. propofition) wherefore tt ts allo commen urable unto st (by the fixt of the tenth) Bat that whichis contained Under the lines A Band BC is fu po rd to berationall Wherfore that which iscontained under the lines AB and F Dis alfora- tional. But that whichis contained under the lines A Band F D,is equall to that which is contained under the lines A D and DB (by the laft part of the firft afumpt going before the 33 of thetenth) Wherfore that which ts contayned under the lines AD and D Bis alfo ra- tionall. Whereforethere are found out tworight lines AD and DB incommenfurable in power, whofe [quares added together make a mediall fuperficies,and the parallelogramme Ca- sayned vader them make a rationall fuperficies : which was required to be done. E A qf Tbe rz. Probleme. The 35. Propofition. T 0 finde out two right lines incommenfurable in power whofe [quares ads ded together make a medtall [uperficies and the parallelogramme contate ned pnder them,make alfo a mediall fuperficies, which parallelogramme moreouer [ball be incommenfurable to the fuperficies made of the [quares of thofe lines added together, B. And upon the line AB defcribe afemicircle A DB, and let the reft of the conftruction be as tt was in the two former propofitions. And forafmuchas (bythe 2 part of the 8.ofthetenth) the line AF ts the oo commenfurableinlength vnto the gE B line FB, therforethe line.AD is | incommen{urablein power unto the line D B (by that whith was demonftrated in the pra- pofitio going before)..And forafmuch as the {quare of the line A B is mediall, therefore that alfo which is compofed of the [quares of the lines A D and D B (which [quares areequall to the (quare of the line A B by the 4. of the first )is mediall.And foralmuch as that which $s AA 60%- of Euclides Elementes. Fol.258s contained under the lines AF and F Byis equal to either of thefquares of the lines E Baud F D, for by {uppofition the parallelogramwce contained under the lines A F and F Bis ¢ quall tothe fquare of the line E B,and the fame parallelogramme ts eqtall tothe {quare of the line D F (by the third part of the firfi affumpt going before the 73.0f theteth).Wherfore the line BE is cawallto the tine D F Wherfore the lineB Cis doubletothetineF D. Where- ‘fore that which is contained under thelines AB and BC 1s double tothat which iscontai- ned wnderthelines A Band F D.Wherfore they are commen{urable by the fixt of this boke: but that which is contained under the lines A Band WC is medial by [uppofition Wherfore alfo that which is Contained wnder the lines A Bana F D 1s redial (by the corollaryof the 23,0f the tenth) but thar which is contained under the lines AB and F D,is (bythe fourth part of the ‘firft apmmpt going before the 33.0f the tenth) equallto that whichis contained a ts wnilerthe lines D and D B : wherfore that which is contained under the lines A D ana, #6E}s 0 nd D Bis alfo medial And forafmush as the line A Bis incommen nrablein length unto the part cicludede line BC.ButthelineB C is commenfurable in length unto the line BEW berfore( by thers of the teith) the line AB isincommenfurable in length unto the line BE. Wherefore the ware of the line AB ts incommenfurable to that which is contained under the lines AB and B E (by the firff of the fixt andr0.of this booke).Batunto the fquare of the line AB are equallthe fonares of the lines A D and DB added together ( by the 47 .of the firft): ava. unto that which is contayned under thelines AB and B E, is equall. that which 1s contai- ved under thé lines AB and F D,that is,which is contained under AD and DB. For the parallelogramise contained under the lines AB and F Disequall to the parallelogramme contained under the lines A D.and D B (by the lift part of the firft afiumpt going before the : 33.0f this tenth booke).Wherfore that which is conspofed of the {quares of the lines AD and The — = D Bis incommenfurable to that which is contained unaer she lines AD and D B..Where. PT C0Hses fore there are found out two right lines AB and D.B-incommenfurable in power, whofe __ fguares added together make a medial [uperficies,and the parallelogramme contayned: UR- der them:make alfoa medtall uperficies which parallelogramime moveouer is incommen|u- yableto the [uperficies compofed of the [quares of thofe lines added together, which was re- The total cov clufion, * ’ : quired to be done, $a The beginning ofthe Senaries by Compofition. shale a beg Dhe 2s. Liheoreme. The 36. Propofition: If two rationall lines commenfurable in power onely be added together:the - dps ; = ‘whole lineis irrational and is called abinominm or a binomiall line. pofition, } } V ppofe ‘that thefetwo rationall right lines AB and BC being commen[urable \\ie power ontly be added together ( the t1.of the tenth teacheth to finde out two | (uch lines). Then fay thatthe whole line A C 1s irrationall . For forafmuch As | Merete the line A B is incominenfurableinlengthwnto the line BC, (for they are fup- Demonstra=- pofe ed-tobecommen|urable in power onely) Butas the ; a line A Bis tothe lineB C , [0 ( by the afcumpt put be A Be & jsmeee fore 4 he 22.0f the tenth) is the parallélograme contat. wea under theliner AB and BC tothe fauare ofthe . ; Bre BC whertfore (by the r0.0f this booke ) the parallelograme contayned uuyder the lines A 08 . HH. iy. Band ve — ' 4 1 HE HY | ' i: 4 (6 T aa lait iH HB ‘Abn i ‘ \ ae " in} ! } { yi i ' " ‘ 7 wa of i ah { uy on i 1 H Pe | by! a " 1. Tam) y fae . ‘ae vie & aie 4 Wan. id he: a \ hh he wie {a WY . WM | Hy | We au) i Rally Sie 4M saa / i HS |i fi Ate ew ie, ay Tat ie : BR } aN) aii ' ; HI i te | f Hy i } ‘4 fy a Diffinition of a binomial bite Sexe kindes of bruomiall biMCS Dewion{tra- £802» TZ be tenth Booke B and BG is incommenfirable tothe fquare of the | line BC. But unto the parallelograme contained A B Cc > 48 commen{urable the under the lines AB and BG parallelograme contained vader ABand BC twif e . (bythe 6.of the tenth ) : wherefore that whichis contained under AB and B Ctwifeis in. commenf{urable tothe [quare of the line BC ( by the 13 of the tenth) . But unto the (quare of the line B C is commen|urable that which is compofed of the [quares of the lines.A Band BC (by the 15.0f the tenth) ,for by fuppofition the lines AB and BC are commen{urablein power onely . Wherefore(by the 13.0f the tenth ) that which is compofed of the fquares of the lines -A B and BC added together is sncommen{urable.to that which ts contained under the lines AB and BC twife, Wherefore’ ( by the 16.0f the tenth ) that whichis contained vn- der AB and BC twife together with the {quares of the lines A Band BC , which ( by the 4. of the fecord )is equall tothe [quare of the whole line A C,is incommen{urableto that which is cormpofed of the{quaresof AB and BC added together . But that which is compofed of the fquares of A B and B C added together is rationall, for itis commen{urable to either of the {guares of the lines A.B and BG of which either of them 4s rationall by [uppofition:wherfore the {quare of the line.A C is( by the 10.definition of thetenth )irrationall Wherefore the line AC alfois irrational and is called a binomiall line. This propofition fheweth the generation and produGion of the fecond kinde of ir- rational lines which is. called a. binemium,or a binomial line. The definition whereof is fully gathered ont of this propofition,and that thus. A binemium or a binomial line,is an irrationall line compofed of two rational lines commenfi- rable the one to the other in power onely. And itis calieda binomium , thatis ,; having twe naines,becant itis made oftwo fuchlines-as of his partes which aré onely commen{u- rable'in powerand notin length: and therefore ech part or line, orat the leafttheone of themi,as touching iength,1s vncertaine and vnknowne.Wherefore beingioyned.to- gether their quantitie cannot be expreffed by any one number or name _, but.cch.part remayneth ‘to be feuerally named in{uch fort asit may . Andof thefe binomuall lines there are fixe feuerall kindes , the firft binomiall , the fecond, the third, the fourth, the fifth and the fixt, of what nature and condition ech of thefe is fhalbe knowne by the definitious which are afterward fet in their due place. q I he'zs. I heoreme. The 37-Propofition. If two mediall lines commenfurable in power onely containing arationalt fuperficies’; be added together:the ‘wholestineis.irrationall and ts called a firft bimediall line. Sr raet thefe twa medial lines A BandB C being commen{urablein power onely , and P KS! contayning a rationall [uperficies(the 27. of the tenth teacheth to finde out two fuch @ , a} | , j ; ee : ; ye 4 lines be com poled » Then 1 fay that the wholeline AG 1s irrationally For ast was | fayd in the propofition vext going before that whichis composed of the fquares of the lines A Band B C is incomme{urableto that which is contat-\ \ . Widens wea under the lines ABand BC twifeswherefore( bythe... su : ane A B cS 16.0f the tenth that which is compofed- of the Quares Of << Ra—a the lines ABandBC together with that. whichis cone tained under the lines AB and BC twifesthat is, the WBE ? {quare of. the line A Cus incommen|urableto that whichis contayned under.the lines AB and B Ctwife.But that whichis contayned under the lines AB and BC tile is commen- [rable to that which #8 contayned under the lines AB and B C once(y the 6.of the tenth) wherfore of Euchiles Elementes, Fol.259. wherefore the {quare of the whole line A C , is (by the 13.0f the tenth). incommenfurabléto that which ts contained underthe lines A Band BC once. Butbyfuppofition the lines AB and B C comprehend a rationall [uperficies. Wherefore the (quare of the whole line A C is irvationall: wherefore alfo the lime AC 4s irrationall.And it is called a firftbimediall line. The third irrational line which is called a fir bimediall line,is fhewed by this pro- polition,and the definition thereofis by it made manifeft,which is this, firft bimedsall iine,ts an irrational line,which is compofed of two medial lines commenfurable in power onely con- tayning arationall parallelograme Jt is called a firft bimediall line, bycaufethe two mediall lines or partes whercof itis compofed contayne a rationall fuperficies which is prefer- red before an irrationall, q Ibe 26. Fheoreme. Ihe 38.Propofition. If two medial lines commenfurable in power onely contayning a mediall fuperficies be added together :. the whole line isirrationall, and is called a fecond bimediall line. | Et thefe two medial lines A Band BC being commenfurable in power one- | ly,and contayning a mediall fuperfictes( the 28 of the tenth teacheth to ‘finde | put two [uch lines be added together.T hen I fay that thewhole line A C is irvationall.T ake a rationall line DE. And (by the 24.of the firft) upon the ee whofe other fide let be the live DG. And forafmuch as the {quare of the line A Cis (by the 4iof the eveaal equallto that which at ek is compofed of the fquares of the linésA BandBC, to- A B C cether with that which is contained vader the lines A B | and BC twife-but the {quare of theline AC isequall eas Go tothe parallelograme DY Wherefore the parallelograme D Fis equall to that which is compofed of the {quares of the lines A BandBC together withthat which is con- tayned under the lines A Band BC twife. Now then a- gajyne(by the.44,0f the frit) wpon the line DE py the parallelograme EH eqnall to the [quares of thelines A 5 andB ©. Wherefore the parallelocrame remayning, wamely APs eqnallto that which 1s contained under K F the lines ABand BC twife. And fora{much as either of thefe lines A Band BC is mediall,therefore the {quares of the lines A B andB C are al- fo mediall. And that which is contained under. thelines AB and B.Ctwife is(by the corolla x), of the 24.0f the tenth mediall.F or by the 6.of this booke it is commé{urable to that which is contained vader the lines A Band B.C once,whith is by {uppofition medial. But vato the fgeares of the lines ABvandB C13 equallthe-parallelograme EH , andvuntothat whichis contayned under thelines AB. and BC tee equall the parallelograme H F : wherefore either of thefe parallelogrames i Kand AF is mediall:and they are applyed upon the ratio- wallline B DWherefore (by the 22.0f the tenth) either of thefe lines DH and HG isa ra- tionall lingandincommen| table in length vutothe line D E.And foralmuch as(by fuppo- jition)theline A Bis incommen{urable in length unto the line B C...But as the line A-Bis tothe line BC foisthe fauare of the line A B to the parallelograme which is contayned UN- der the ines ABandBEe (by the first of the fixt)...Wherefore(by the.10. of this booke ) the {quare of theline A Bis incommenfurable to the parallel ograme contayned under the lines HH sig. line DE apply the parallelograme D F equal to the {quare of the lineAC, Diffinition of afirft bimedi- all line, ConStruction. Demon Sra 30. i Ds Fenition of a fecond bime dick line. Demon fra- tidite T hetenth Booke A BuadB C.But tothe (quare of the line A Bis com- wienfrable that which is compofed of the fquares of the. ™ a digest B and BC ( bythe 1s.of thetenth). For the Squares of the lines AB and BC ure commenfurable ++, ie (when asthe lines AB and BC are put to be commen- SoG. ease 3 furable in power onely) «And to that whichis contayned | 3 under the lines A Band BC 4s tommenfurable that ' which is contained under the lines AB and BC twife | | (by the 6 of the tenth) wherefore that which is compofed | of the{quares of the lines A B and BC is incommen|u- | rable to that which is contayned under the lines AB and BC twife. Buttothe fqaaresof the lines A Band ? B:C és equall the parallelugrame RH. And to that K | which is contayned vader the lines A Band BC twife is equall the parallelograme FH. Whierfore the parallelograme F His iacommenfurable to the parallelograme HE, Wherfore thelizeD His incommen|nrable werlencth to the lineHeG (by the 1 of the fixt and 10 of A. ISA i f F f re. ae Y E 7 : SRE ETS ae as | CF, — ry tis booxe). And itis proued that they are rationall lines .W herefore the ines DH GHG az PS - > SL OS 2 . j g aA Fal ee Be 44 f, - 144 2 [ 4A 2 V5 y . i + LAs d are rationall commenfuradle in power onely .Wherefore ( by the 36 . of the tenth )the whole ‘ 5 ae ‘ 4 f a pes : ee we 4 i 7 > iti, feae IDs 18 irrationails And the lise Dis rationall.But-a vettanele fuperficies. compre- . : 7 ; i Pate ; ° f mi 2 r - + a bended under a rationall line and an irrationallline is (by the corollary added after the 21 of / i J j I \- a! fap DDD hin ies Ck Fgh! tenets : '‘ ay the tint irrational. Wherefore the faperficies DF 4s trrationall.And the line alfo which y se ah LS i 2 fog apts } ah re) [33 2 fj Sh - A s {i ng éontatnets it in pomerts rational .Buttheline DC containeth in power the fuperficies D wT. +re>f r ; _ / ee a, 7 GU . i OR, = " ’ ‘ EP. Wherefore thetine A Cis irrational. And itiscalled a fecond bimediall line. A : This Propofition theweth the generation of the fourth irrationaltline, called a fecond bimediall line . The definition wherof is euident by this Propofition; which is. thus. ed fecond vimediall line isan irrationall line, which 1s made of two mediall lines commenfurable in power onely ioyned together, Which comprehend a mediall (aperficies. And itis.called a fecond bi- mediall, becaufe thetwo mediall lines of which itis compofed,cotaine a mediall fuper- ficies,and not a tationall. Now a medialis by nature & in knowledge after a rational. q Lhe 27. I heoreme. T he 309. Propo/ition. If two right lines incomenfurable in power be added together, haning that which ts compofed of the /quares of themrationall, andthe parallelocrame contayned Ynder them mediall : the whole right line is irrationall, andis 7 “q > J Aa A 4A : . called a greater line. h _— : Fallelocranme Contained twife onder the lines AB and BC ismeaiall. (For that which ws and BC once (by the 6.of the tenth). Wherefore (by the Corollary of tie 23. of the tenth) that which is contained under AB PP BC a is mediall) But by [vsppofition that which is compofed of the [auares of the lines AB and BC, isrationall . Wherefore that which ts contained under thé lines AB and BC twife ts incommenfurable to that which $s compofed ‘of the of Euclides Elementes. Fol.260. ofthe quares of the lines A Band BC. Wher-~ daddies. a caooe | fore (by the 16 .of the tenth\that which is com. A hikes B a poled of the qnares of the lines AB and BC to- | eether with that which is contayned under the’ wx ee: lines AB EBC twife, which is (by the 4.0f the fecond) equall to the {quare of theline AC, is intommen|urable to that which ts com ve of the [qnares of the lines AB and BC : Bs that which ts compofed of the {quares of the lines AB and BC, is rationall .Wherefore ‘he fquare of the whole line AC; is trrationall. Wherefore the line AC alfo is irrational . Anil is called a greater line. And itis called a greater line for that that which is compofed of the {quares of the lines AB BC which are rationall,i; ereater then that which is contayned onder the lines A Bana B Gtwife,which ave mediall . Now it is meete that the name jhoula be geuen according to the propertie of the rational. | | An Affumpt. And that the [uperficies compofed of the [quares of the ines AB and BC, 15 ereater then that which ts contained under the lines AB and BC twife, maythus be proued . Firft it manifell, that thelines AB ana BC ave vnequall: for if they were equall , then the '{quares of the lines AB and BC fhoula be equall to that which is contained under the lines AB and BC twife,fo that that whichis cotained under AB and BC, fhould alfo berationall: whichis contrary A D _ ave, to the (uppofition . Wherefore the lines AB and SE eae a pe eS B Care vnequall . Suppofe then that the line_A B = be the greater, ¢ let the line B D be equall to the line B C. Wherfore(by the 7.0f the [econd,) the (quares of the lines AB and BD, are equall to that which is contained under the lines AB and BC twife and to the {quare of the line AD . But the line. D.B.is equall to the line BC. Wherefore the [quaresof thelines A Band BC,are equal! to that which is contained under the lines.A Band BC twife and to thef{quare of theline. A D. Wherefore the fquares of the lines A B and B € are greater then that which is contayned under the lints A B-and BC twife, by the {quare of the line AD : which was required to be proned. This Propofition teacheth the prodution of the fiftirrationall line,which is called agreater line: which is by the feafe of this Propofition thus.defined. ed greater liness an tratt onall line, Which ss compofed of two right lines Which are incommenfu- vablesn power,thefquares of Which added togethér, make a rationall Juperficies, and the parallelo- gramme Which they containe, is medial, Ieis therefore called a greater line, as T beon fayth, becaufe the {quares of the twolines of which itiscompofed,added together. being ra- tionall,are greater then the mediall {uperficies contained vnder, them twife.. And it Is conuenient thatthe denomination be taken of the proprietie of the rationall part, ra- ther then of the mediall part. | q The 29. T heoreme.- ewer), 40. Propofition. If two right lines incomenfurable in power be added together, hauing that which 1s made of the fquares of themadded together mediall ,and the pac yallelogramme contayned ynder them ratsonall : the whole right line is ire rationall, and is called a line contayning im power avationall and 4 mediall ‘nperficies. Let An Afsumpte Diffinition of 4 greater lines Thetenth Booke Re Et thefe two richt lines A BandBC being incommen{urableyn pomer, Cr making se that whith 1s required in the Propofition be added together. (The 3 4.0f the tenth j iso teacheth to finde out two [uch lines ).. Then L fay, that the wholeline AC isirra- Denson firde tionall . For fora{much as that whichis. compofed of the {quares, of the lines AB s50Ne end BC is medial, but that whichsscontained under thelines A Band BC twifeis ratio- nall, therefore that which 15 compofed of the {quares of the lines A B and B.C, is incommen: furable io that which 1s contained under the lines AB and BC twife. Wherefore (by the 16, of the tenth) that which is compofed of the {quares of the lines A B and BC together with that which is contayned under the lines AB and BC | awife,which (by the g.of the fecond ) is the {quare of she mbole line A Cts incommen[urableto that which A B CG 45 contained under the lines A B and BCitwife. But that which is contained under the lines AB andBC twife,is rationall , for that which is contained under the lines AB and BC once is put to be rationall .Wherefore the {quare of the whole line A Cis irrationall .Wherefore the line AC alfa isirrationall.. Aud itss therefore called a line containing in power a rationalland a me- diall{uperfi cies becaufe the power thereof. contayneth two [uperficieces, whereof the one is ra tional namely that which 1s contained under the two paries,and the other mecdiall,namely, that which is made of the [quares of the partes added together ._And by reafon that thera- tionall isin order of nature & of knowledge before the medtall; therfore the firft part of the denomination is taken of it , and the fecond part is taken of the mediall: which was required tobe prettt. eo Inthis Rropofition is taught the generationof the fixt irrationall line, which is cal- jéda linc whofe powerisrationalland mediall. The definition of which is gathered of thys Propofition afterthis maner. re | “UL linteSwhofe powers rationall and mediall, is an irrationall line Which is made of two right lines Diffinition of \incomuenfiurable ipower addedtogether, whofefynares added together make a mediall fuperficies, al:newhofe butsbarfuperfities Which they containe is rational. The reafon of the name is before fet forth power ts rati-: in the Propofition. onall ana Z wicdiall q Lhe 29. I’heoreme. The 41. Propofition. If two right lines incommenfurablein power be added together, banyng Yo uote | that which ts compofed of the [quares of them added together mediall, and awywienis o> theparallelogramme'contayned onder them mediall, and alfo incommene ‘furable to'that which is compofed of the /quares of them added together: the whole right linets irrationall sand is called a line contayning in power two medials. SaEt thefetworightlines A Band B C being incommen|urablein power,and hauing Gi that whichis compofed of the {quares of the lines A B and BC medtall,and the pa- (eas lr allelogramme which is contayned under the lines AB and BC meaiall, and alfo tacommen|urableto that which is compofed of the [quares of the lines AB and BC added Contraltion .tdgetherjbeadded together (the 25.of chetenth teacheth topinde out two fuch lines) Then I fay that thé whole line A.C 1s irrationall. Take arationallline DE, and (by -he 44. of the ficht) upon the line DE apply the parallelogramme DF equall to the [quares of thelines A Band B C.And vpon the line G F whith is equall to the line D E, apply the parallelograme G H equall to that which ts contained vader the lines AB and BC twife. Wherefore Hi . WHE of Euchdes Elementess: Fol.261. whole parallelogramme D Hisequall to the {quare of the line AC (by theg. of the fecond.) DemonStra- And forafmuchias that whichis compofed of thefquares of the lines A Band BC is mediall tion, | and is equall.to the parrllelogrammeD F : therfore D F alfois medial (by that which was {aid iwthe 3 8.propofition of thisbooke). And at is applied upon therationall line D E.. Wherefore the line D G 1s rationall and incommen|{urable inlégth A | B G. vaio y line D E( by the 22.0f the teth). And by the famercafon the lyneG Kts rational cy incomefura~ ble in length unto the line G F ,that is,vntothe line D E.And forafmuachas that whichis compofed of 5 fqnares of the lines AB cy BC. added togethersis by {uppofition incomenfurabletothat whichis cotained vndler the lines A Band BC tmife; therfore alfo the parallelogramme DF is incommzenfurable unto the parallelogramme G H.Wherforealfo the line DGits- incommenf{uranle unto the line G K (by the first of | the fixt) and by the tenth of thetenth. But itis mom en proued that they are rationall: Wherfore the lines D GandG K are rationall commenf{urable in power onely. Wherfore (by the 36.of the tenth) the whole line D K is rationall,and is called binomiall line»batthe line D E is irrationall.Wherfore the parallelograrmme D H is irrationall( by the corollary added after the 22 propofitio of the tenth). Wherfore alfa the line whichtcontaineth itin power is irrationall: but the ine AC containcth it in power. Wherfore the line A Cisirrationall, and is called a line contayning in power tro medials.It is called a line containing in power two medials, for that it contay- neth in powertwo medial fuperficieces;one of whichis compofed of the fynares of the bynes ABand BC added together,and the other is that which is contained vnder the lines ABC? BC twife : which was required to be demonfirated. i je In this propofition is taught the nature of the 7.kinde of ifrationall lines which is ‘called a line whofe poweras two medials, The definition whereof is taken of this pro» pofition after this maner. ; . | . A line whofe power is two medials,ts. an srrationall line whichis compofed of two right lines in- Diffinition 0 eommenfurable in power,the {quares of Which added together, make amediall fuperficies, and that f oe] : Sago TU aline contai- which ts contained under them ts alfo mediall, and morecouer it is incommenfurable to that which és ning in p St as compofed ofthe two fquares added together. ATT) | two medtals | e The reafon why this lineis called a line whofe power is two medials, was before int the ende of the demonftration declared, And that the faid irrationall ines aré denided one way onely that is, in one point onely, into the right lines of which-they are compofed, and which make every one Ve the kindes of thafe irrational! lines,fhall firaight way be demonftrated: but firft will we demonftrate two alfumptes here following. a WT Nannsise st } | ¢ An Affumpe. Take aright line and let the fame be A B, and deuide itd#t6 two oneqrall partes itt the 3 i? th ea * MES on Sas ; 6 . An Afumpt. port C and againe deutde the [ame line A B into two other ma partes,inan other point namely, in D, and let the line A C (by [uppofition) be greater then theline D B. Then 1 fay ca ag that + Thetenth Booke shat the (quares of the lines AC and BC added together,are greater then the f{quares of the lines AD and DB added together: Denide the line AB (by the to: of thefirft) into two e- quall partes in the point E.And fora{much asthe line A-Gss greater then the line D B, take away the line D C which ts common to them both : wherfore the refidue.A D is greater then the refidue C B,but the line A E is equallto the line E B.Wherfore the line D Eis lefve ther the lime E C. Wherfore the pointes Cand D are not equally diftant from the point E,whichis the point. pte . ‘" of the fection into two equall partes. And foraf- Pr 58 much as (by the 5. of the fecond) that which is con- taynea under the lines A C and C B together with ow of the line EC is equal! tothe fquare of the line E B. And by the fame reafon that whichis coatayned under the lines A D and D B together with the {quare of the line D E3is alfo tquallto the felf fame {quare of the lipeE B : wherfore that which is contained Onder the lines AC and CB together with the {quare of the line E Cis equall to that which is contained under the lines AD and DB to- gether with the {quare of the line DE : of which the {quareof the line D Bis le(Se then the Square of the line EC ( for it was proued that the line D Eis leffe then the line EC). Wher= fore te parallelograsmme remayning contayned under the lines AC and C Bis lefSe the the parallelogramme vemayning contayned under the lines AD and D B. Wherfore alfo that which is contayned under the lines A Cand CB twifeis lee then that which is contayned under the lines A D and D B twife.But(by the fourth of the fecond)the {quare of the whole line AB isequaltothat which is compofed of the {qwares of the lines A C and CB toge. ther with that which is contained under. the lynes AC and CB twife, and by the fone réafon the {Guare-of the. whole line AB is equall to that which is compofed of the {quares of the lines A-D ghd DB together with that which is contayned under the lynes A D and D B twife.: wherfore that which is = of the [quares of the lynes AC and CB together with that which is comtayned under the lynes.A C andC B twife, is equall to that which is compofed of the {quares of the lynes A D and D.B,together with that which és con- tayned wnder the lynes A D and D B tmife. But it 1s alread) proned that that which is con- tayned under the lynes AC andC B twife, is lee then that which is contayned under the Gnts AD oF D Btwife:Wherfore the refidue,namely sthat which ts compofed of the fquares of the lines AC and C Bis oreater then the refidue, namely, then that whith is c ompofed of she {awares of the lines A D and DB: which was required to be demonftrated. q An Affumpr. _ A rational fuperficies exceedeth a rational fuperficies, by a rational fue perpicies. : Let AD bea rational Superficies, and let it exceede AF being alfo a rationall Super ficies by the fuperficies EDA hen 1 fay that the [uperfictes ED is.alfo rational. or the paral- Kclogramme A D ts commenfurable tothe parallelogramme A F for that either of them is rationall. W. herefore (by the Second part of the 15.0f the tenth) the parallelogramme _A F 13 commenfurable to the parallelogrammeé E D. But the the parallelogramme AF is rational] Wherfore alfo the pa- rallelogramme ED isrationall, ° q Tbe Mie Lat? *-# of Euchdes Elementes: \ Fol.262, ¢ Ihe 30.T beoveme, °°? The 42 Propofition. Ai binomiall line isin one point onely denided into his names, Yee V ppofe that A Bota binomial line, and in she point C letit be dewided into | x ws | his names, that is into the lines wherof the whole line AB 4s compofed. Where- We. fore thefe lines AC andC B ave rationall commenfurable in power onely.Now I SSS [ay that the line AB cannot in any other point befides C be denided into two ra- tonal lines commen|urablein power onely.For Nw ; if it be poffible,let it be deuided in the prep we PE Sicheacan Se fothat let the lines. AD and DB be rationall comuscnfurable in power onely Fir ft itis manifest that neither of thele pointes C and-D ae uideth the right line.A Bintotwe equall partes, Otherwife the lines A Gand CB frond be rationall comimen{urable in length and fo likewife fhould the lines :A D snd DB be. For €- uery line meafureth it felfe, and any other line equall to it ‘felfe. Mareasser the line DB is either one and the famewith the lize AC,that 1S,48 equall to the line AC, or els it is ercater then the line A C,either els it is leffe then we DB be equall ta the live A-C,then putting the line D B vpen the line AC eche endes of the one, fhall agree with eche endes af the other. Wiherforeputting the potnt Bvpon the point .A,the port D alfo fhall fail upon the point C, and the line AD whichis the refi of theline AC, fall alfo be equall to the line C B,which 4s the veft of the line DB.Wherfore the line AB is dénided into his namesin the point C. And fo XN alfo [hal the line AB being deutded in the point D be deuided in the felf. ‘fame point that the. Self fame line AB was before deuidedin the point C,which is cotrary to the Suppofitia.For by fuppofito it was denided in fundry pointes,namely, in C & D.But if the line D B be greater the the line AC,let the line_AB be dewided into tivo equal partes in the point E.Wherfore the points CR D fhal nat equally be diftant fro the point E.Now (by the fir ft afsiipt going before this propofitio)that which is copofed of the (qnares of § lines ‘AD er-D Bis ereater the that which is compofed of the fauares of the lines. AC cy CB. Bat that which is compofed of the Squares of the lines AD ¢ DB together with that which is cotained under the lines A D oO D B twife,is equall to that which is compofed of the [quares of the lines AC & CB togethe with that whichis contained under the lines AC and C B twife,for either of them is equall to the {quare of the whole lineA B ( by the 4.of the fecond) wherefore how much that which is copofed of the fauares of the lines AD and D B added together is greater then that which 4s compofed of the (quares of the lines .A Cand CB added together, (0 much is that which is contained vpder the lines AC and CB twwife greater then that which is contained under the lines A D and D Btwife. But that whichis compofed of the [quares of the lines A D and D Bexcedeth that which is compofed of the [quares of the lines A Cand CB Ly 4 rationall [u- perficies(by the 2. afvumpt going before this propofition.) For that which is compofed of the [quares of the lines A D and D Bis rational, and [0 alfo is that which is compofed of the {quares of the lines AC and@ B:for the lines A D and D B are put to be rationall commen- furablein power orely,and fo likewi[e are the lines_AC andC B. Wherfore alfo that whichis contained under the lines AC and CR twife, exceedeth that whichis contained vnder the lines AD Ce DB twife by a rational [uperficies whe yet notwithflading they are both medial Juperficteces (by the 21. of the tenth) swhich (by the.26.of the fame) is impofable. Andif the line D B belelfe thenthe line A C,we may by the like demon|ftration proue the felfe fame im- pofscbilitie: Wherfore a binomiall line is in one point onely denided into his names : Which was required to be demonfirated. IY. j. Corrollary The feronad Se” nary by com- pofition, Demon{lratio on feadine to pice Sega an impofabi~ hitée. — i. Pat 118 i | 1} | i ) it Hi ia Pik . ; Bi } hi | at WHE bet | ’ } : TTS A, Corollarye Demo fire- tion leading to an unpofibt- detiee Demonfrario heading 10 an gin pof]ibiltites T hetenth‘Booke Corollary added by Fluffates. T wo vationall lines commenfurable in power onely being added together cannot be equall to twe other rationall lines commenfurable in power onely added together - For either of them fhould make a binomiall line,and fo fhould a binomiall line be deunided into hisnames in moe poyntes then one:which by this propofition is proued to be impoffible, The like fhall follow in the fiue{nextirrationall lities as touching their two names. q i be 31. Probleme. EF he43. Propofition. A firft bimediall line ts in one poynt onelydenided into bis names. SS Tl Y ppofe that AB bea firft bimediall line, and let it be dewided into his partes in /. e a |the point ©, that let the lines AC and © B be mediall comen[urable in power CMY |X} only and containing a rational fuperficies.T hen I {ay that the line A B can not KS = ZSA5 bedenided into his names in any other poynt then in C. For af it be poffible let xt he deuided into his names in the poynt D’, fothat let ADe> DB be mediall ines commen wrablein power, Pp ¢ B onely,comprehending a rational f uperficies. Now for-- alinucl-as bow much that which 4s contayned Under the lines AD and D B twife differrethfrom that which is contayned under the lines A-C and C Btwife;fo much differret h that which is compofed of the {quares of the lines AD and © B from that whichis compofed of the fquares of the'lines A C and CB : but that which is contaxned under the lines A Dand DB twife differreth from that which is contay ved wonder the lines A Cand C B twife;by a rationall fuperficies ( by the fecond affumpt go- ine before the 41 of thetenth ).For either of thofe fi uperficieces is rationall . Wherefore that which 15 compofed of the fauares of the lines A C and C Bdiffereth from that which is com pofed of the {quares of the lines AD and D B by arationall [uperficies , when yet they are both mediall fuperficieces: which is impoffible. Wherefore a fir{t bimedtall line is in one poynt onely denided into his names: which was required to be proued. q The 32. I heoreme. The 44. Propofition. A fecond bimediall line is in one poynt onely denided into bis names. a V ppofe that the line AB being a fecond bimediall line, be deuided into hys Eke » names in the poynt C : [0 that let the lines AC and CB be mediall lines com- DRO men furable in power onely comprehending a mediall [uperficies. It is manifeft YL that the poynt C deuideth not the whole line AB into two equall partes . For . ~. \ ~ a the lines A C and C Bare not commenf{urable in length the one to the other. Now I [ay that theline NB cannot be deuided into his names in any other poynt but onelyim C . For if it be paffible,let it he deuided into his names in the poyat D,fo that let not the line AC be one and the [amé,that is,let it not be equall,with the line D B.But lett be greater then it. Now itis manifest (by the firft affumpt going before the 42. propofition of this booke ) that the {quares of the lines A C and C B are greater then the {quares 0 the lines AD and DB. Andalfo that the lines A) and) B are mediall lines commenfurablein power onely comprehending a mediall fuperficies T ake a rationall line E EF. And(by the 44-0f t he firft) vpon the ine EE apply a rectangle parallelagrame EK equall to the [quare of the line AB. From which pa- rallelograme take away the parallelograme E G equall tothe {quares of the lines AC and CB Wherefore the refidue,namely,the parallelograme 11K zs equall to that which is con re ne of Euchdes Elementess Fol.2.63 ned under the ints A C and CB twife.Avayne fromthe the parallelograme EK take away. the parallelocrame BL equalt to the quares of the lines AD: andD B.which are lefe then the (quares of thelines A.C and C Bi Wherefore there- fidue,namely, the parallelocrame MK is equallto that A fo weDeae B which is contayned under the lines A Dand DB tnife. | And foraf{muchas the {quares of the lines AC andCB a are mediall, therefore the parallelograme EG alfoisme- E ©. SRE = N k oo os diall. And it is apply ed upon the rationall ine EEF -where- | | cigs ree fore the line Hs rational and incommenf{urable in an impeforbse leneth tothe line EY. And by the fame reafon , the | litses parallelograme TAK 1s mediall (for that whichis e- | quall unto it namely that which ts contayned under the kines A C and C B twife ts mediall) therefore the line S | : F J Le “£ HN is alfovaiionall and incoremenfurablein length un- to the line BE. And forafinuch as thelines AC and GB are medtatl lines commrenfurable ist power onely, therefore © the line C is incommenfurable tn leneth unto the line CB. But as thé lize AC 1s tothe line C B,foisthe (quare of the line A C to that which4s contayned vader the lines A.C and CB(by the 1.0f the fixt). Wherefore the {quareof. the line AC is incommenfurable so shat which is contayned under the lines A C and C B.But (by the 16.0f the tenth unto the {quare of the line A C are commenfurable the {quares of the lines AC andC B added to- gether, for the lines A C and Barecomment urable in power onely. And unto that which #5 contayned vnder the lines AC and CB is commenfurable that which is contayned un- der the lines A C and C Bawife. Wherefore that which is compofed of the {quares of the lines A Cand €.B, is incommenfuvable to that whichis contained under the lines AC and C Btwife. But to the {quares of the lines A Cand C Bis equall the parallelagrame E Gand to that which is contained under the lines A C and CB twife is equall the para- lelograme HK.Wherfore the parallelograme E G is incommen{urable to the parallelograme H K wherefore alfo the line 8H is incommenfurable in length to thelineliN . And the lines EH and HN are rationalt. Wherefore theyare rationall commen{urable in power one ly but if two ratiqnall lines commen{urable in power onely be added tovether the whole lineis Prationall,andis called a binomiall line ( by the 36. of the tenth) Wherefore the binomta it lineE N is in the poynt 4 denrded into his names, “And by the fame rca{on alfo mayit be proued thatthe lines EM andM N arerationall lines commen|urablei# power onely. Wher fore EN being a binomiall line is denided into his names in [undry poyntes namely ; in H and M, neither isthe lineEH one and the ‘[ame;,thatis,equal with MN.For the fqaares of the lines 8 Cand C Bare greater then the [quares of the lines B Dand AD (by thet.af= fumpt put after the g1.of the tenth ) . But the ajnares of the lines A D and D Bare greater then that which ss tontayned under the lines R Diund DB twife (by the affumpt put after the 3.9 .0f the tenth). Wherefore the (quares of the lines AC and C B3that is,the parallelos grame EG 15 much greater then that which is contained under the lines AD anadDB twife,that is then'the parallelograme M K:Wherfore (by the fir ft of the fix?) the line H is greater then theline MN. Wherefore®: His not one and the fame with MN .Wherefore a binomial line is in two fundry poyntes denided into bis names Which is impoffible. T he felfe [ame abjarditie alfo will follow if theline AC be [uppofed to be lefse then the line DBA, fecond binomial line therefore is not denided into bis names in funary poyntes Wherefore i 45 denided in one tnely:which was required tobe demonfirated. °° q I he 33. L heoreme. The 45. Propofition. A creater line is in one poynt onely deuided into bis names. WH, Le rT T betenth Booke- te, Et AB being a greater line be deuided into his namesinthe poynt, C fothat let the BsBilines AC anid C B be rationall incommen{urablein power, hauing that which is NN comm pof ed of the {quares of the lines A.C and C Berationall,and that which is con- Demonftrati- taimed under the lines A Cand C B.mediall:T hen 1 fay that the line AB- can notin any 0- on leading to an im pofSibte Sib. Demonftrati- on leading to an impofibi- biti. ther poynt then in C be deuided into his names. For if it be poffible,let it be deuided into his wames in the poynt D,fo that let AD and DB belines incomme- “ fiurablein power ,haning that whichistompofed of | the (quares of the lines A.D and D Brationall; and that whichis contayned under the lines A D aud D B mediall._Now forafmuch as how much the {quares of the lines A Cand CB differ froms the {quares of the lines A DuandDB,fo much differeth that whichgs contained under the lines A. 1) and D B twifefrom thatwhichis cowtained under the lines AC and C Biwife,by thofe thinges which haue.bene fayd inthe demonstration of the 42. propofiti- ee + lene. C and C Bitwifeby a rationall fuperficies:when as either of them is a mediall fuper- fictes Which ts impof [ible (by the 26.0f the tenth) . Wherefore a greater line is in one poynt oncly deuided into his names :which was required to be proued. q Ehe.34. I beoreme. The 46. Propofition. A line contayning in power arationall and a mediall, is in one point ones ly deuided into his names. Et.A B being 2 line.containing in power a rationall anda mediall, be deui- ded into his names in the point.C, {0 that let the lines AC & C B be incom- menfurablein power, hauing that which is compofed of the [quares of the lines A-C and C B.mediall, and that which is contained under the lines : | -A.C.and CB rationall .Then 1 {ay,that the line AB cannot in any other _ point be denided into hiswames but onely in the point C . F or if it le pofst- ble, let it bedewided into his names in the point.D, ; fothat let the lines AD and D B be incommen{ura- ‘blein power, hauing thatwhich is compofed ofthe. + fquares of the lines.A D and D Bmediall,and that which is contayned under the lines A D.and D B rationall .. Now fora{much as how much that which is contained under the lines AD ¢y D B twife differeth from that which is con- tained-under the lines.AC and CB twife, fomuch differ.the [quares of the lines ACG’CB addédtogether from the [quares of the lines.A D and DB added together... But that which 4s contayned under the lines AC and.C Btwife excedtth that which is contained under the lines AD and D.B.twife by a rational fuperficies ( for either of them isrationall ) . Wher- forealfothe fqwares of the lines AC and.C.B added together, exceede the{quares of the lines A Dand.D:B added together by arationall {uperficies, whenyet ech of them ts a mediall {i perficies : which is empofiible : Wherefore a.line.contaiming in power ara tionall and a medt- all, is in one point onely deuided into his names : which was required to be demonstrated. qi he Dp oC B of Euclides Elementess° Fol.2.6 As qT he 35. T heoreme. Lhe. 42, Propofition. A line contayning in power two medials, is 1m one point onely denided into his names. | | : — aN ; 4 * meavrenewmmpwis SF | | his names in the point C, {0 that let the lines AC and C B be :encommenfdravie J q : ) : ; 5: wi toe BES ’ 5 7 ts Ty alt Pe RD nvsA taht {SASOLM! mediall; and that alfo which i3 contained under the lines AC and C Breedtal, ‘ y y . m ; ‘ ; rive Av? >. f 2 4 ‘Ree af" A and neoreouer iacdminenlurabletoibat which is compofea of the [quares of the tines AC and es L OnjL7RED ICS be deuided intohis names but onclyin the pomt G. For if *. SENSE eee, it be pofsible;lerivbe denided into:his:names in the point D;fo that let #0t the line A C be one ana the fame,thatis, oo i equal with the line D B:but by [uppofition let thelime AC cc ake bethegreater . And take a rational line E F. And (by the | 43.0f the firft Upon the line EE apply arectangle paraite- | loerame EB Gequilltothat which 1s copoled of the [ymares of the lines AC anadGB: and likewife upon the lineH GC; which is equall to the line E F, apply the parallelogramme HK eaualltothat which is contained under the lines AC ¥ li. -@ and CB twife, Wherefore the whole parallelogrammze EK | asequall to the {quare of t he line AB. Againe vpon the fame line E F de[cribe the paralle» Demon f oie locramme E L equall to the fquares of the lines AD and DB.W. herefore the refidue;name- —— = ly, that which is contayned vader the lines AD and DB twife,is equall to the parallels. Aid EO; HPAES gramme remaining,namely,to M K . And fora[much as that which is copofed of the {quares ofthe lines A-C and € B, is (by fuppofition )mediall;therefore the parallelograme E G which 4s equall unto it, is allo mediall : andit is applied upon the vatiouall line EF . Wherefore (by the 22.0 the tenth) the line H E isvattonall and incommenfurable in leneth vniathe line EF . And by the fame reafon alfothe line HN is rationall and ‘acommenfurable in ase 5) c nif length to the [ame line E F . And forafmuch asthatwhich 13 compéfedof the fauaves of thé - < - ” ° y7 _ pl : : P lines AC and C B is incommenfurable to that which is contained under the lines AC and © Brwife (for it'vs fnppofed to be incommienfurable'to that which is cotained uniter the lines AC and G Bonce:) : thereforethe pavalleloeramme LE Cis incommenf{upable to. the paralle- locramme HK. . Wherefore the lime EH al[ors incommenfurable in levethtothe line H N, and they are rationall lines : wherforethe lines EH and HN arerationall commen{urable in power onely .Wherefore the whole line EN is a binomiall line, andis deuided into his she wines in the point H. And in like fortamay we prowe,that the fame. binomiallline E N is de- wtided into his names in the point M;and thatthe line EH is not one and the fame that ts ¢- quall with the line M N, as tt was prouedin the end of the demonstration of the 4. of this booke. Wherefore «binomiall line tsdenided into his names in two fundry pointes: which és imposible (by the 42.0f the tenth ) . Wherefore a line containing in power tivo medials is wot in {undry pointes deuided into his names . Wherefore it is deuided in one point oncly : which was required tobe Gemonsirated. : : J Ly. eSecane i? Ssxe kendes of binomial biHES 6 A binomial dine conifieth Of two partes Firji diffint- $20. Second ds ffs- ittoMe Third ABP RG LIB%- Thétenth Booke — gSecond Definitions. | ony f é% T was fhewed before that of binomial! lines there were fixe kindes, the JAE &/ definitions of all which are here now fet,and are called fecond-definitids. ras Ne All binomiall lines, as all other kindes of irfationall lines are cOceaued, eS 1 (ES yy cofidered,and perfe@ly vnderftanded onely in refpeGe of 4 rationall line 2L.\ 7 (whofe partes as beforeis taught, are certayne,and knowen, and may be diftinGly exprefled by number) vnto which line they are compared. Thys rationall line muft ye ever haue before your eyes, in all thefe definitions ,fo thall they alf be eafie inough. | A binomiallline(ye know) is made of two partes or names,wherof the oneis greater then the other. Wherfore the power or {quare alfo of the one¢ is greater then the power or {yuare of the other. The three firftkindes of binomiall lines namely, the firft the {e- cond,& the third,are produced,when the fquare of the greater name orpart ofa bino= miallexcedeth the {quare of the lefle name or part, by the fquare ofa line whichis‘com- menfurable in length toit,namely, tothe greater. The three laft kindes, namely; the fourth the fift,and the fixt,are produced,when the {quare of the greater name or part exccedeth the fquare of the lefle name or part, by the {quare ofa line incommenturable in length yntoit,thatis,to the greater part. A firft binomiall line is, whofe fauare of the greater part exceedeth the Square of the lefse part by the fquare of a line commenfurable in length to the greater part, and the greater part is alfo commenfurable in length to Cc the rationall line firft fet. As let the rationall line firft fet be A B : whofe partes are diftinéily knowen ; fuppofe alfo thatthe line C Ebea binomial line, whofe namiés ot pattés let be CD and DE . And let the {quare of the line CD the greater patt extede the {quare of the line D E the lefle part by the {quare of the line F G: which line F Gs iet bee commen- furable in length to the line 5 CD, hid is tie greater: Oe part of the binomiali line. And moreduer let theline © C.D the greater part be com menfurble in length tothe F +—4—+~—_+—+-+4—_ G rationall line firft fer, name- ty, to A B.So by this defini- tion the binomiallline CE isa firft binomiall line. A fecond binomiall line is, when the [quare of the greater part exceedeth the {quare of the leffe part by the fquare of a line commenfurable in length ‘ynto it, and the lefSe part 1s commenfurable in length to the rational ine firft fet. A$ (fuppofing ewer the rationall line) let C E be a bintomiall line deuided in the poyat D. The fquare of whofe greater part C_D let exceede the fquare of the lefle part D E by the fquare of the line EG, which line FG fet be c D ie Oe E i ec one oo eel Oe eee Oe ee ee eed comenttrable in length vnte phic, G the ine C D the greater | ices : part of the binomial line . And let alfo the line D E the leffe part of the binomiall line be commenfus rable in length to the rational] line firft {et A B . So by this definition the binomiall line C E is a fecond binomial line. A third binomiall line is, when the [quare of the greater part excedeth the 2 [quare - fe of Euchdes Elementes. Fol.265. fquare of the lefJe part by the fquare of a line comenfurable in length onto it. And neither part ts commen/urable ut length tothe rational line pene. As fuppofe the line C Eto be a binomial line: whofe partes areioyned together in the poyat D: and let the fquare of the line C D the greater part exceede the fquare of the leffe part DE bythe {quare ofthe line FG, and | let the lme FGbe @} | D commenfurable in length to the line CD the greater part of the binomiall. Moreouer, let neither the greater part CD, nor the leffe part D E, be commenfurable in lengeh co the tationall linewA B, then is the line C E by this definition a third binomiall line, a Sis By G A fourth binomiall line is, when the fquare of the greater part exceedeth the fquare of the leffe by the {quare of a line incommenfurable in leneth vn- to the ereater part. And the greater is alfo commenfurable in length to the rationall line. Aslet the line C E bea binomiall line,whofe partes letbe CD & DE,& let the {quare of the line C Dthe greater part exceede the fquare ofthe lineDEthe 4 ++—-+-++-+-+-++4—~ B leffe, by the fquare ” of the line EG. Cc $n Pt pet ere tN cee SS And let the line F Gheincommenfu- FF ———»——_____4 G. rable in length to the line C D the greater. Let alfo the line C D the greater pare be comtenfurable in length ynto the rationall line AB . Then by this definition the line C E is a fourth binomiallline, A fift binomial line is, when the fquare of the greater part exceedeth the Square of the le/Se part, by the {quare of a line incommenfurable nto it in length. And the lefJe part alfo 1s commenfurable in length to the rationall line geuen. As oe that C E bea binomiallline, whofe greater part Jet be C D, and let the leffe patt be DB. And let the {quare of the line C D excede the {quare of the line D E by the fquare of the line F Gywhich let be incémenfurable in length vn- to the line CD the greater part of D KE the binomialfline. And lettheline S i as | D E the fecond part of the binomi- | all line be commenfurable in length Po G vnto therationallline AB . So is the line C E by this definition a fift bis nemiall line. A fixt binomiall lineis, when the Jquare of the greater part exceedeth the fquare of the lefse, by the {quare of a line incommenfurable in len oth vato it . And neither part is commenfurable in length to the rational line veuen. Asletthe line C B bea binotniall line,deuided into his names in thé point D. The {quare of whofe greater part CD lerexceede the quaré of the leffe part DE by the fquare of the line F G, and let the line F G be incommenfurable in length to the line C D the greater part of the binomial line.Letalfo | Ti. iy. neither Fourth diffe 120%» Fifth diffind- tzon, Sixth diffint- $101. The tenth Booke neithersC D the greater part, nor DE... | D the leffe part be commentfurable in C length to the rationall line A Ba And F fo by this definition the line CE1sa fixt binomiall line. So ye fee that by thefe definitions, & their examples,and declarations, all the kindes Of binomiall lines are made very playne. ahs eek ata Sen Senet ee enn tee ea PRE nen} E G This is to be noted thathere is nothing fpoken of thofe lines, both whofe portions are comimenfurable in length vnto the rationall line firft fet, for that fuchlines cannot be binomiall lines. For binomiall lines are compofed of two rationall lines commenfi- rable in Power onely (by the 36. of this booke). But lines both whofe portions are commenturable in length to the rationall line firft fet are not binomiall lines. For that the partes of fuch lines thouid by the 12.o0fthis booke be commentfurable is length the one to the other.And fo fhould they not be fuch lines as are required to the’compofiti- on ofa binomial! line. Moreouer fuch lines fhould not be irrational! but rational! , for that they are commenfurableto ech of the parts whereof they are cépofed (by the 15. of this booke).And therefore they thould be rationall for that the lines which com pole theim are rationall. - fr 7 q Ibe 13. Probleme. he 48.propofition. Lo finde out a fi rt binomiall line. ee ee Yosape 4 Ake twa murabers AC and CB, let them be fuch,that the sumber which nae DRS) Ay 15 made-of them both added together , namely, AB, haue vato one of them, pofition. ‘ Sst Ve) 2 zel’ y, unto BC that proportion that a {quare number hath to a [quare OD <9) 4S number But unto the other namely,vato CA let it not haue that proportion ConStruftion, SO ~ 7-4 has a fauare number hath to a {quare number ( {uch as is euery [quare nam- ber which nay be dewsded into a{quare number and into a number not {quare) . Take alfe'a certayne rationall line, and let the frm SITE ry ree ors, = (® fame be D. And vuto the line D let ici oe the line EF be commenfurable in " : length. Wherefore the line EF is ra Bs Eo Sc os ome Sony E tionall. And asthe number A Bis PHS to the nuber AC, fo let the fquare na of the line EF be tothe [quare of an other line,namely of FG (by the coz Riva aegis ieienr cid svi ge vollaxy of the fixt of y teth) . Wher- fore the {quaxe of the line EF hath to the{quare of the line E G that proportion that number hath to number . Wherefore the square of theline EY is commenfurable to the (quare of the line F G(by the 6.of this booke) Demonfira~ ~ And the line E.¥ is rational VW ‘herefore the line E G alfo is rationall. And foralmuch as the 20 Me . | : 6 ap number A B hath not tothe number AC, that proportion that a {quare number hath to a {quare number neither fhal the {quare of the line EF haue to the fquare of the ine ¥ G that proportion that a{quare number hath to a {quare number. Wherefore the line EF is incom- menfurableinleneth tothe tine FG by thé 9.0f this booke ) . Wherefore thelines EF and ‘E.G arexationall commen{urable in power onely, Wherefore the whole line E.G is a binomi- all Line (by. the 36 of thetenth) I fay al{othat it is afirft biaomiallline . F or for that as the nuinver B Aistothe number AC, fois the fquare of the line EY to the (quare of the line E G-but the number B A is creater then the number AC: wherefore the {quare of the line Fas aifo greater then the {quare of the line G. Vuto the {quare of the line EE det the aie EE, J quares of ive lines EG and H be eqwall( which how to finde out is taught in the affumpt put 2 | after ~~ = — Ee ——— i 4 1 ; Hh NG eh i ie —_——— oor —— Se a ee = - -—- - a = Sn me — JG Ors Se ee ye ~ == , —_— 7 of Euclidés Elementess™ Fol.266. after thei3.ofthetenth) . And forthat as thenumberB Avistothe number AC , fois the fquare of the line EF to the {quare of the line E G< therefore by tomuerfion or enerfion of proportion ( by the corollary of the 19.0f the fift)as the number. Bis tothe number B Cfo is the [uare of the line E F to the {quare of the line H.But the number AB hath tothe num ber BC that proportion that a {quare number hath toa {quare number . Wherefore alfo the {quare of the lineE.¥ hath to the {quake of the linc that proportion thata {quare number hath to a{quare number . Wherefore the line EF is commenfurablein length tothe lineH (by the 9.0f this booke . Wherefore the line EE 4s in. power more then the liné FG, by the {quare of a line commenfurable in length tothe line EY . And the lines EF and F Gare ra- tional! commenfurable in power onely ..And the line EF. is commen{urable in lengthtothe yationall line D.Wherefore the ineEG 35a firfi binomial line. which was required to be doone. ‘ Lhe 14.Probleme. Lhe.49.Propofition. T 0 finde out a fecond binomial line. Spd Ake two numbers AC andCB, and let them be [uch that the number made of mM ee them both added together namely, AB have vutoB © that proportio that afquare Construction. 4 Livnumber hath toa fquare number, anduntothe number GA lebit not haue that proportion that a{quare number hath toa{quarenumber, as it was declared in the former propofition.T ake alfo a rationall line,andletthefame be , and vntothe line D let theline F G be commenfurable in length. Wherefore ¥ Gis arationallline. And as the number C Ais tothe number AB fo let the {quare of the tine GF be tothe fquare of theline FE (by the 6.of thetenth) Wherefore. the fquare of the line GE is commen{urable tothe {quare Dew of the Line F E.Wherfore alfo F Eis a rationatl line. And fora{muclvas thenuberC A hath |. _— e not unto the number A B that proportiothatu Square number hath toa {quare nuber; ther fore neither alfo the {quare of thelineGR SAK hath tothe fquaré of the line EE that propor. PD tion that afquare number hathto a fquare ss wumber . Wherefore the line GE is incom). Ba BAX SR ox ee ¢ menfurable in length unto the line FE ( bys the 9. of the tenth ): wherefore thelinesE Go #3, and EE arerationall commen{urable in po- wer onely. Wherefore the wholesline EG 8.4) ~ , binomiall line. I fay moreouer thatthe line HSER Ie EG isa fecand binomiall line: For for that *: by contrary proportion asthe number B Ats\s 2° to the number AC fois the {quareof the line\ rs NES SUES BE to the {quare of theline FG Butthenunmbér B Nis ereater then the number AC, sbherefore alfothe fqnare of the line Bisgreaterthen the [qware of the line F G .¥ntothe Square of the line ¥ let the [quaves of the lines F and beequall. Now by conuerfiom (bythe corollary of the 1.9 .of of the fift)as the number AB isto the number BC , fo is the fauareof theline EF to the [quare of the line Ws But the number AB hath to the number BC that proportion that a {quare number hathto afquare number . Wherefore the [quare of the line EP bath to the {quare of the line We; that proportion that a [quare num. ber hath toafquare number Wherefore(by the 9. of the tenth) the line EF is commenfu- rablein length'vnto the line Hi. Wherefore the line EF is in power more then the line FG by the fquareofa line commenfurablein length unto the line EF : and . - P rs OT RET « Fh , CeO iH ath We ate ’ I 5 4 7 | ’ : : Fah) ; i® : 8 : \ ’ 4 j i . i} | i HER : | Wait 3 ] i] - on | : iv : i ; ; | ot { a | Baa “) } will { nT Bt a : - \ iT] 2 : A Alb a ; tie! ‘i mil | A | » 7) Dp 1 ata | 4 | ) at y } Ta Hi ; \ ‘| Th ala) » i | My | ie WT ff . - A q \ | I : : \ 4 5 | 4 : TL 4 4 ' Naa ) we ! it a) } sy ERD ae y nie Hey (Mit! } SrA Chae WE | y T i} ! 7} ’ | ; 7 wi = ti ie \ Ni | lel Wied . weenie ; A | | 7 ah : \ - at th \ | ; | ; ; i Hes } P| a | i H 1 ikea! U U Pe , ‘ i } : Oe he i} i wie: i | ' } | A , i |) \ i) i} r 7 | , y | , al eae : } ; eat : ; A { | } th j } , ' t at : i] 1s) | { iy Pein ae r ie, nie } ik i! ue t : i ; : : j ha { A abl ie 0 @ | : Thy \ Ui | H | Bie Ow i 7 ip NW vi \ 4 a8. ie : se \ * h ( iy yi 1 oe Hy tat by “aa : :’ ae '¥ } i + a i | ag He f “ae eh Bit } P iu b hia , ATE ' 1 ‘ : 4 ye) : 7A ne ! ‘ \ < ; + oe , ite hy iit q { q vee WU } @ i ‘ | f s] a | oi ' ig \ 4 pA 4 ‘ yal I i TM P 7h » Rt is iW an! af a ‘ + an | Say tah "4 ' } 4a ; BR! neg ~~! ' h Sia we Hi 03%) WF bts * mune me WPI Ma) 4 ch | Dp vir 4 \ Ma) Pil he et Se a d Pit a \ v Tian + aL Piaf 1) ’ a ah | ' : eA iS if 4th : Yah a) BAT i : 5 ; = a ; r! 4 : men i j - | > RRS ae ie i iu a) eae , Pat bay ¢ wep’ r : "an ye OR Quik ts 7 a 1 - ‘ ; 7 , “y ; : } Lia als! ‘i 7 a ify |. a at : Uj : ‘ H if ina mile | ty } \ i Ae Ped Bae | - "hl ¥ rt et; a | } i ' oN Piatt the Une ae er a f \a ive ¢ aay : yy 4 t , 4 s ; ; Dh ira) s : ‘ya | sz iy a TT ' in a} of 9 La i | +f Adana |: ‘ q ' (3) ; it Ai |} { oe ht tar q ; Da ; ee a “ts ee Ne in : We ) y a“ YS ea : ; } ; On ae ae A. | f 7 ) . t ' q Le ! , hl : ‘ + ie t 4 .) el) PMO ai , He : el zi . a . H + HN pel ae $ if at ‘ ee Mas i 1) ie Wen Pe +o We {| he ca | { - i Ml dediitt Eby Hy!) : he ae i \ VAT hy Dy beh am ¢ he t rs " , in 1. i > q i H! i . at : q Cee a , La . } i bs = r pa = C onfiruttion. Demonfira- b 10M. Thetenth Booke EF ad DG are rationall commen[urablein. power onely and F G being the lefve name ts commtenfurable in length unto the rationall line geuen namely, toD Wherefore i Gis a fe- cond binomiall line-which was required to be done. y Dbeors. Probleme. Lhe\s0. Propofition. Jo finde outa third binomial line. Sea Pe Ake two numbers AC and C Band let thems be [uch that the number made of thems Ke bath naded tozether namely, AB, haue tothe number BC that proportion that a 4 \4u9(quare number hath toa [quare number But tothe number AC let it not hanethat proportion that a{quare number hath toa {quare number, as it was declared in the two for- mer. And take alfofome other number that is not a fquare number,and let the fame be D, and let not the number D hane either tothe number B A,or tothe number AC that propor- tion that afquare number hath toa [quart num- , ber. And take a rational line and let the the [ame be End.asthenunsber.D.is ta the number AB folet A Rie. the quare of thelineE beto the fauare of the line EG. Wherfore the /quare of the line Bus comen{us x Ke NE. Hu rableto the {quare of the line F G; buttheline E is | rational wherforethedine.F Gallo is rational.And BS for that the ae D hath not to p. nuber AB that * proportion thata(quare number hath to a fquare © wumber, neitherallofball the{quare of thé lineB\s Aves... ce Cuca B bade to thefquaxeof the line FG that proportion that.afquarenumberhath toa {quare nuber. Wher Ds wat. GL. y jore the line Eis tncommenfurable in leneth to the line F G (by the 9.0f the tenth.) Now a@aine asthe nitber A B is to the niatber AC, fo let the Square of the lin€F G be to the {quare of the line GH. Wherfore the fquare of the line F Gis commenfirable to the [quare of the line GH. And the line F Gis rationall Wherfore alfo the line G His vationall. And for that the number BA hath not to the nuber A C, that pro- portion that a{quare number hath:to 4° quare number. therefore neither alfo hath the {quare of the line F G to the (quare of the line Gdsthet praportion that-a fquare number hath toa {quare number. Wherfore the line F Gis incomenfurablein légth tothe line GH: Wherfore thelines® G ex G H are rational comenfirablein power only. Wherfore the whole line F H is a binomtaltine. fay moreouer that.it is athird binomial line.F or for that asthe niber D is tothe nuber A B,fois the [quare of the lineE tothe f{quare of the line F G:butas the nuber A B isto the number A C,fois the {quare of the line FG to the (quareofthelineG H3therfore.of cqualatie( by the 22.0f the fift)as the number Dis to the number AC,foisthe fauareof the line E tothe {quare of the line.G-H.But the niiber D hath not tothe nuber Ax C that proportiothatalquare nuberbath ton Square niuber: Wher fore neither alfa hath the {quar of theline E.to.the {quare ofthe line. GH that proportio that a {quare number hath to afquare number Wher fore the line Eis: incommenfurableinlégth to the line G H And for that a5 the number-AB is tothe number wAG, [0.13 the {quare of the line F G to the fquare of te line GH, therfore the {quareof theline F Gis greater then the fquare of the line G H. Vutothe {quareof the line FG, letthe quares of the lines H and K be equal.Wherfore(by ever fe proportia by the coxollary of the x9. of the fift)as the nither A B is to the number BC; fois the [quare ofp line F G to the {quae of the lineK.But the nitber AB hath tothe niuber | : BC =f@ of Enchdes Elementes. Fol.267. ‘BC that proportio that a [quare number hath to a fqnare number: Wherfore allo the [quare of theline FG hathto the (quare of the line K that proportion that a {quare number Aath to a {quare number.W. her fore the line F Gis commenfarable in length tothe line KW herfore the line Givin power more then the line G H Ly thé (quare of a'line commen|uradle wn lenothvatoit:And the lines FG and G H are vationall commen|urable in power onely and neither of them is commen{urable in length unto the rationall line E : wherfore the lyne F His athird binomiall line : whichwas requirca tobe done. | q The 16.Probleme. Lhe 5 1:Propofition. To finde out a fourth binomialt tine. Ya iat ‘Ake tyo nuntbers AC and CB, cy let the be fuchthat the nuber made of the Construction « ay Be eX fauare niuber).And take a rationall line, line D let theline E Ebe comen{urable in length. Demon fira- Wherfore EF is 4rationa line, and as the num- pie her B Aisto the number AC; [olet the {quare of. the line EF betothe (quare.of y.line F G.Where..~g, mie i aaslbicas sd fore i he {quare of the line EF 1s commenfura- wh Aa hey ce GY STN REE ‘ble to the [quareof the line F G, andtheline EF. 4 is avationall line..Wherfore allothe line F Gis.a rationall line. And for that the number B A hath notte the number AC that proportion that A. Arvcepvensss © cone B quare number bath to.a{quare numver, neither | alfo hall the [quare of the ine EF hauetathe{quare of the line F G.that proportion that 4 quare number hath toa [quare number. Whefare the line E F isincommen{urablein length tothe line E.G.Wherfure the lines E F and F Gare rationall commén|urablein power onely. Wherefore the whole line EG is a bincmiall line’: I fay moreouer that it tia fourth binomial lyne. For for that as the number B.A.is to. the number A C,fo is the [quare of the line EF to the fquare of the line FG. But the number B A is greater then eht number AC .. Wherefore allo the fquare of the line EF is greater then the {quare of the line FG . Vnto the fquare of the line E F let the {quares of the lines F G and H be equall. Wherfore by conuerfion (by the corollary of: the 1.9.0f the fift) asthenumber A B is to the number B C,fo is the {quare of the line E F to the [quare of ghe line H. But the num ber AB hath not tothe number BC that proportion that a{quarenumber hath toa {quare number: therfore neither alfa hath the (quare of the line E F to.the [quare of the line-H that proportio that a {quare nitber hath to. a(quare nuber Wherfore(by the 9.of the teth) the line EF isincomenfurable in length unto the line H.Wherfore the line E F is in power more the the line F G by the {quare of a line incommen{urable in length unto it. And the lynes E F and F Gare rationall commenfurable in power unely, and the line E F is commenfurable in length to the rationall line D.Wherfore the line EG is a fourth binomiall line: which was required to be found out. q.Uhe 17. Probleme. The sz. Propofition. T 0 finde out a fift binomiall lyne. T ake i, z —-= te yah tte i} ( + ‘ah Hy! f ‘ait | i i int ' i i hee ue | } fril Va : \ | yt? t i et : " 1a hy sau i | ya i y 7 Oa be eb ; { h ' ete inte ah th ft t, nt ww | itp | ; ene bibl ot d i | 1% t BEE ela wh j ah ; } Veh) | 4 wide f \ 4 a ; HD de if hla Oe + ARE ral f hi ae THe ; : ' ! Lith A : 11) ane ’ j 4 |) ae } i} i i) ae} i AU ae ql ‘| ay oT ) | th a } { : ’ ya ai q I re Te RL it : t H : Hive a t 4 art at aul ae F ‘he be welt ' vi : qe ue i i Ae | } , Ln oh) 1 } oni ede Tt Wied ni ; { Poi i , : i) { ) y } ; - ! { « , i } , i? : a | : : Ged A 7 : 1 ' y ; Te. : wit . | ay { i} | j vith 7 Hs a wit ] . 7 : ae my q : i ‘ J q } i We } He Hh h } | : - 14 a ib! : \ } \ Uy | i} An My AM} Vay i! $$$ —————— at = _ — —— SS Coustruttion, Demon flra- O3CHe Confirnttion. 7 be tenth Booke | 2 Ake two niml ers AC and CB} and let them-be futh, that thenumber AB rv haue toncither of the numbers A Cor C Bthai proportion'thata fquare num- , » er eth to $ fanare wumberasin the former propofition:\ And take a ratio. | Vs, 74 llineand let the fame be D.And-untothe line-D let the line F G. be. com + wen {urable in length. Wher fare the lineE Gis rational! Aundasthe number € Ais tothe number A B,fo let the fquare of the line G F be to the {quare of the line...P E F. Wherefore the {quare of the line G F és commen{urabletothe fquare oftheline = R20 Foi Ris G F E. Wherefore alfa the line F Eis ratio- ie e wall. And forthat the number CA hath Hi ka . not to the number AB that proportia that afquare number hath to a [quare, num- Y er ee ae Ml cas Ee ber, therfore neither alfo hath the [quare | ofthe tine G F tothe {quare of the line F E that proportion that afiquare number hath toa quare number Wherfare(by the 9.of the tenth the line G F is incommenfurable in leneth to tie line F E.Wherfore the lines E F and F G are rationall cominen{urable in power onl Wherfore the whole lineE G is a binomiall line. [ay moreouer that it is a fift binomiall line . F or for that as the number C A isto the number AB [01s the [quare of the line G F to the {quare of the line F E,therfore contrariwife,as the number B A is to the number AC, fois tive [quare of the line E F tothe {quare of the line F G : but the nnmber B A is creater then the numbers_AC. M ‘herfore allo the {quare of the line E F is eveater then the Square of the line F G. Vute the f quare of the line E F, let the [quares of the lines F G and # be equall. Wherfore by conuer fio (by the corollary of the 1.9 .of the fift) as the nuber A Bis to the al ber BC, fo is the [quare of the line E F to the ‘{quare of the line H But the niiber AB hath not to the number BC that proportio that a [quare number hath toa ‘[quare number Wherefore weitheFal{o hath the quareof the line E F to the {quare of the line H that proportion that a [quare number hath to a {quare number Wherfore( by the g of the tenth) theline E F is in. ‘commen mrable in length to the line H Wherforethe line E F is in power more then the line F G bythe fynare of a line ixcommen|urable in lencth unto it. And the ines E F and FG are rational commenfurablein power onely.And the line F G being the lefename, is com- menfuratlein length to the rationall-line genen,namely,to D. Wherfore the whole line E G ts apift binomiall ine: which was required to be found out. q Lhe 18. Probleme. he 53. Propofition. Lo finde out a fixt binomiall line. ‘CB that proportion that.a {quare nuber bors toa [quare mae number . Take alfo any other number : which ts not a fquare number, andlee — a s.. the famebe D’. And let not the num- ee = ber D haune to any one of thefe num- K Beg bers AB and AC that proportion that a {quare numberhath to a Adare WADE me ages... C.... B Let there be put moreoner arationall line, ae {- pong ana » = fe of Euclides Elementes. © Fol.2.68. and let the famebe E.And as the number D isto the number AB foletthe (quare of the lint E be to the [quareof F @. Wherefore (by the 6.of the tenth) the line Eis commenfurable in power tothe line F G,cy the line Eis rationall Wherfore allo thelineF Gisrationall , And forthat thenumber D hath not to the number A B that proportion that a{quare nitber ha th. 10 1 [quaré number, therefore neither alfo fhallthe fquare of the line E haueto the '{quare of the line ¥ G that proportion that a {quare number hath to a [quare number. Wherefore the line F G ts incommen| urable in length totheline E . A (gaine,as the number BA ts to the number AC, fo let the (quare of the line F G-beto the [quare of the line G H . Wherefore (by the 6 of tie tenth) the { quare of the line F G ts comm pfurable toth 4 {quare of the lineG A. And thefquare of the line F Gis rationall .W, herefore the {auare of theline G H4s alfora- Ponall.. Wherefore alfothe line GH is rationall. And for that the namber AB hath not to the number AC, that proportion that a{quarenumber hath toa lquare nember : therefore neither alfa hath the [quare of the line F.G to the (quarcof the liveG H,that proportion that afquare number bath toa fquare number. Wherefore the line F G 1s incommenf{urable in lenzth to theline GH . Wherefore the lines FG and GH. are rational commenfurable in power onely .Wherefore the whole line F H is a binomiall line .1 fay moreouer, that it wa fixct binomial line . For for that asthe number D is tothe number AB, fois the {quare of the line E to the (quare of the line FG. And asthe number B Aistothe number AC, fous the [quare of the line F G to the fauare of the line GH. Wherefore of equalitie ( by the 22. of the fift) asthe number Dis tothe number AC, [0 is the [quare of the line E to the [quare of the line GH . But the number D hath not tothe nuber AC that proportion that a{quare number hath toa (guare number .Wherefore neither alo hath the {quare of the line E to the |quare of the line G H that proportion that afquare number hath to a [quare number Wher- fore the line Eis incommen|urable in length tothe line GH . And itis already proued, that the line F Gis allo incommenfurable in length to the line E . Wherefore either of thefe lines F G avd GF is incommenfurablein length to theline E. And for that as the number B A isto the number AC; foisthe{quare of the line F G to the [quare of the line GH : therfore the fauare of-the line F Gis greater then the (quare.of the line GH . Vato the {quare of the line F G, let the {quares of the lines G Hand X be equall. Wherefore by exerfion of propor- tion, as the number A B is to the number BC, fo is the (quare of the line F G.to the {quare of the line K . But the number AB hath not to the number BC that proportion that a {quare number hath toa {quare number . Wherefore neither allo hath the {quare of the line F Gto ae quare of theline K that proportion that a [quaré number hath to a {quare number. Wherefore the line F G is incommenfurablein length unto the lineK . Wherefore the line F Gis in pomer more then the line GH , by the [quare of aline incommenfurable in length toit . And the lines F G and G H arevdtionall commenfurable in power onely . And neither of the lines F G & G H is commen[ugable in leneth to the rationall line geuen,mamely,to E. Wherefore the line F H is a fixet-binomiall line : which was required ta be found ont. | miieope Lo hes SEA Corollary added out of Flufates: ** AS By the,.6. former Propofitions itis manifest howto deuide any right line gewen into the-names nof eueryone aft be fixe forefayd binemmalliines.. Forifit be required todenide aright line ge yen into-a fir{t binomiall line,then by the 48.ef this booke finde out a firft binomiall Wne.And this tight line bcing fo found out deuidedinto his namies,you may by the ro. ofthe fixt,deuide the rightiine getten in hikefort: And fo in the orher fue following.” si wlthoughI herénotevnto youthis Corollary out of Fiayasjyet,in very con{cience and of gratefull goinde,I am enforced to\certifie you,thatsmany yeares before the tranailes of F/af/as(upd Enclides Geor metricall Elementes) were publifhed, thoorderhow tordeuide; not'onely the 6.Binomiall linesinto their names, but alfo to adde to the 6.Refiduals their due partes :'and farchermore to ‘devide all the o- ¢ KK.j. ther DemonSira- ti0M. At Corollary added by Fluffates. Wi Wi 4 - H if § "i y th 8 nt Dee? - ' \ nay if : i | \ He | : Hiiaine i { ; i Th ie viet | Hh) ; \) | 14) i reas hehe { ’ ire \ apie # \ i 7 hab ‘th nl \ é 4 ¥ Hi T HE Dae it Phas | Vat DY \ tapy ; | WA : y | 5 q ah ‘ bi : A) Ra he ; | ; ; : \ t ; : 7) 1 j , if ; 1) oe : f Tibet } F iets } : Hei it “2)> TR : WRIT Wile aly : ee ile Wie | +1 } ip : i Ny toa la } 1 | ANG | } ie } i ! , 7 - } ' { 1 . | ») bead q ial : q Ry iy : wn ih! ; APU Wie e Ciel) hy Fi} | hI i bt oi ait iPait I eth} : ) 0 : 4 | | FA ' i] } /, int) \-) j , i? i | ie : \ ry | ; i - \ q \ t ! ; ta ; at Da ait at Bl Th } Lil j r ik bia) \ | i ii} Mn tie \ ; | J An ie ’ vy, ie Me fit i ! ry i , i" i Wt eae oi | yj ‘ } : f ao cir! me : { } b 4 ' iv) ART | i il! } ; } | he alae i: - rt ar: } } ii ; i ; ae , \ the } j it } a0) age a0 0 a | } t t iy eh} \ tt +. L } Pe! / : a I ah) aime | , Tht cn wi 4} i) eA BH Pee H oy abe { A ae | lif rig vit aM if ea a yet et a ’ ye! eh } : ie it! Da Ri TH} ' Wits, wee Pah ; tlie At wet 1 | te Ab ‘ q 4 ) wa 7 Wal ne no tI) i » TT M4 9] ) | } his) | j ) W.Dee hes boo eC C4 lled Lyroceniim Matheusati~ ChB thes Affunspe ai was before soted, foilo De eth we g ir sof ly Wwetboue fa ether denion firation of the go Of th booke, Deman {tra- ti OMe 7 hetenth Booke ther strationall lines (of chis tenth booke) into the partes diftin®, of which they ate compofed ? With many other ftraunge conclufions Mathematicall,to the better ynderftanding of this tenth booke and o- ther Mathematicall bookes,moft neceflary, were by M./oh# Dee inuented and demonftrated : as in his booke; whofe title is T yrociniums Mathematicum (dedicated to Petrus Nonnius,An. Is$9.) may at large appeare. Where alfois one new arte, with fundry particular pointes whereby the Mathematicall Sci- ences, greatly may be enriched . Which his booke, I hope, God will one day allowe him opportunitie to publithe : with divers other his Mathematical and Metaphyficall labours and inuentions. € An Affumpt. If avight line be deuided into two partes bow foeuer : the reéZangle parale lelogramme contayned ynder both the partes,ss the meane proportional betwene the {quares of the fame parts. And thereétangle parallelogramme contained vnder the whole’ line and one of the partes, is the meane pros portional betwene the fquare of the ‘whole line and the fquare of the POE Be iYa part. Suppofe that there betwo [quares. AB and BC, and let the lines D Band BE fobe put that they.both make one right line . Wherefore (by the 14.0f the firft) the lines F B and B.G make allo both one right line . And make perfect the parallelogramme AC. Then I fay, that the rectangle parallelogramme D Gis the meane proportionall betwene the {quares. AB and BC: and moreouer, that the parallelogramme DC 15 the meane proportional betwene the [quares AC and C B . First the parallelugramme A Cis a {quare. For forafimuch as the line D Bis equall to the line B F and the line B E unto the line BG, therfore the whole line DE és eyuall to the whole line F G.But the line D Eis equall to-either of thefe lines AH & KC, and the line F G 1 equall to either of thefe lines A K and HC (by the 34.0f the firft) Wher- fore the parallelograme A C is equilater,it is alforectan- K gle (by the 29.0f the firft). Wherefore (by the 46 . of the firft) the parallelograme AC is afquare, Now for that as the line F Bis to the line BG, {ois theline D B tothe. line BE .But as line F Bis to theline B.G,fo(by.ther..? of the fixt)is theparallelagrame AB which is the{quare. of the line D B, to. the parallelogramme D G, and as the line D B isto theline B E, fo is the [ame parallelogranze DG tothe parallelogramme BC, which is the {quare of the line B E .Wherefore as the[quare AB. 15 to the pa- z ralleloeramume D G,fois the fame parallelogramme DG 4 E iw to the [quare B C Wherefore the parallelogramme D G GS C is the meane proportionall betwene the [quares AB and BC .1 fay moreouer, that the paral- lelogramme D C is the meane proporvionall berwene the fquares AC and CB. For for that asthe line A D is tothe line D K, fois the line K G tothe line G C (for they are ech equall toeche) . Wherefore by compofition by thet 8.of the fift) a3 the line A Kristo the line K D, fois the line KC to theline CG. Bat asthe line AX isto the line KD, (ois the [quare of the line AK, which is the {quare AC, ta the parallelogramme cotayned under the lines AK and K D, whichis the parallelogramme.C D.1 4nd asthe line K C1540 the line.C G, foal is the paraltelogramme D C ta the {quare of the line GC, which is the {9 uare BC. Wher efore as the {quate AGistothe parallelogrammeD Cforstle parallelograme D.C tothe{quare BC. Wherefore the parallelogramme D Cis the mesne propor tionall betwene thé fe quares 4C andBC : which was required to be dimonfirated.. > ol ile sudo ee rs of Euchides Elementess~ F0l,2.696 | q An Affurptss) \ 2s ogotsens : Maonitudes that are:meane proportionalls betiwene the felfe fame or -e- Ss : cane p | | quallmagnitudes are alfo equall the one to the other. - “Suppofe that there be three magnitudes ASB SC. ; | An Affumpt, And as NistoB, fo let B be to CS And likewife as oie : the fame nragnitude A is toD)folet D-be'to the fame *- : thagnitude Cs Then ¥ fay that Band Dare equall BR ey the'one the other. For the proportion of A -wnto Ge is double to that proportion which A° hath to BY by: the to . definition of the fifi) and lkewife the°felfe Ss fame proportioe of Ato Cxstby the fame definition)? | double tothat proportion which AhathtoD. But D ——+——— magnitudes whofe equemultiplices are either equall or the {elfe fame, are alfo equall. Wherefore as Kis to” Bfois Ato D. Wherefore (by the 9 of the fift) Band D are equall the oneto the other . So'fball rt alfo be if E there be other magnitudes equal to AandC,namely, iq | et oe a ont tte. a ne - oe fl E and F betwene which let the ma nitude LD be the = meaie pro portionall. | —— q The 36. Theoreme. Ihe 54. Propofition, Ifa fuperficies be contained ynder a rational line ex a firft binomial lines The fourth Se the line which containeth in power that {uperficies i: antrvationall line sep Mary by com 4 binomiall line. ee : = er V ppofe thatthe [uperficies A BC D, be contained under the rational line AB, and under 4 first binomial line A D.Thenl fay thatthe line which containeth < in power the [uperficies AC,ts an irrational line,and a binomial line.F or foral- Confirn teens St much as the line A D isa firft binomial line,it is in one only point denidedinta. a ioe. ae saa yo gat ee ean Pook es) Vb G ESut Se" ews his names (by the 42:0 this tenth) let it be denided into his names in the point B. And let A E be the greater name. Now is is wranifel? that the lines AE and E D are rational cem- ae KK.¥. mei Mt iH i) } 4 ret i qi TRY 7 Bible TA he HH AP th Oh ' Dea &f0Me: Ret T hetenth Broke. vxenfurable in power onel ty and that the'line A Eisin power more then the line E D, by the fquare of a line commenfurable in length to the line A £,and moreouer that the line A Ets commenfirable in lenath tothe rationalllinegenen A 3 bythe definition ofa first. Binemi- allline fet before the 48 propofition of this tenth. Dewice (by the 10.0f the fir ythe line E D into two equall partes in the point F. And forafmuch 1s the line A E is in power more ther _ the line E D by the (quare of a line commenfurable in eagth.vato the line AcE therefore if vpon the greater line, namely, upon the line.A E. te applied a parallelogramme equatl to the fourth part of the {quarc of the lefse linesthat. ist the {quure of theline EF, or wan- ting in forme by a fqwarestt fralkdenidethe greater lint, namely, A E tatotwo partes com. wacifurable in leneth the one to the.ather by the fecoud part of the 17. of the tenth.) Apply therfore upon the line A E aparallelograwsme equall t¢ thefquare of the line E F and wans ting in forme by a fquare by the 28.of thefixts and lernbe famehe that which is contained. under the lines AG and G EWherforethe lined Gisiammen[urableimlength Lo the lyar 8B Pe ae ee q . s ° 2 ee we GE.Draw by the pointes G,E,and F to either. of thefilines AB and D.C thefe parallel lines . GUE K,and F L (bythe 31. 0f thé firft) And (by the 14. of the fecond) unto the paralle- oneflrde logramme AH defcribe an equall [quare SN. And vsto the parallelogramme G K defcribe (by the [arae)an equal {yuare-N P. And let thefe lines MN x NX be fa put,that they both wmake one right line Wher fore (by the 54. of the first) the lines RIN and N.O-wnake alfo both one right line.« Make perfect the parallelpgramme S >. Wherfore theparallelogrammeS P is a [quare by thofe thinges which were demonftrated after the determination inthe fir ft af- fumptcoing before. And foralmuch as that which iscontained under the lines AG and G E is equallto the (quare of the line E F (by conftructioz) : therfore as the line A Gis to the E F,fo isthe lineE F to the line E Gby the 14. or 17,0f-thefixt) Wher fore alfo (by ther. of the fixt) as Wh oriers. - caee AH is to the paralelogramme EL, {ois the parallele- gramme E L to the parallelogrammeG K.Wherfore ie parallelogramme E L is the meane proportionall betwene the parallelogrammes AH andG K. But the parallelogrammé A H 1s equal to the {quare S Nand the parallelograme G K isequal to the (quare N P by epstruci:- on.Wherfore the parallelogramme EL is the meane prdportionall betwemethe [qugres SN and N P (by the 7.0f the fifth) But (by the fir{t a{fummt going before) the parallelogramme M R is the meane'proportionall betwane the {quares SN and NP. Wherefore the parallelo- gramme M Ris equall to the para llelagramme E L (bythe laft afjumpt going Lefore) But the parallelogramme M Ris equal to the parallelogrammed X( by the £3 of t he firft) and the pa- rallelogriimme EL is equail to the parallelograme F C by construciton and by the far ft of the fixt.Wherfore the whole parallelogramme E Cis equal to the two parallelocrammes MRO Or X, Aud the parallelo ramus AH aad GK Arg ental to the {quares SN ana os a 3 by conflructionV herfare = whole parallelagrarmmc A ¢ is equal to the whole {quare SB thas Zoi: ais ; iS, = f@ of Euclides Elementes. Fol.270, és, to the {quare of theline MX. Weereforethe line M X containeth in power the parallelo- To fir/? pare gramme AC. I fay morgouer that the line MX is a binomiall line.F or forafmuch as (by. the of +h ela 17.0f the tenth ) the line Gis comnenfurable in lengthto the line.E G.. Therefore (b fivation con the 15. of the tenth )the whole linea Eis commenfurable in lengthto either of thefe lines -~ - — se of Euchides Elementes. Fol.271. in length to the'line E D, tut the line A E is commenfurableinieneth to the lint AG; and the line E D 1s commenfurable in leneth to the line E F , therefore by the 13 of the tenthy the line AG is incommenfurable in length tothe lime E F . Wherefore( by the tof the fixt- and 11 .0f the tenth )the parallelograme A H 1s incomenfurable to the parallelogramme E L, that isthe {quare S N to the parallelogramme M R,that is;theline O N is incommenfurable tothe line N R that is the line M N to the line N X.And it is proued that the lines MN and NX are mediall lines commen{urable in power Wherefore the lines M Nand NX are medi-. The fourth all lines.commen|{urable ix power onely, Now I fay moreouer that they comprehend avational part cocludeds fuperficies.F or forafmuch as by fuppofition the line DE is comme{urable in length to either of thefe lines A B and E F therefore the line F E is commenf{urable in length to the line E K which isequall to the line A B (by the 12.0f the tenth) . And either of thefe lines E F and E Kis arationall line Wherefore the parallelograme E L,that is the parallelograme M R, is a rational fuperficies( by the 1.9.0f the tenth).But the parallelogramme M Ris that which is contayned vader the lines M Nand NX . But if two medtalllines commenfurable in power onely and comprehending a rational {uperficies be added together the whole line is trrational 71 and is called a firft biesediall( by the 37.0f the tenth) . Wherefore the line M X is afirft bime- ” Hy mg diall line-which was required to be demonfirated. : wieN The fift part concluded , q Lhe 33. Tbeoreme. he 56. Propofition. Ifa [uperfictes be contayned ynder a rationall ine and a third binomiall line: the line that contayneth in power that fuperficies is irrational, and is. a fecond bimediall line. : A ¥ ppofe that the {uperficies ABC D be comprehended under the rationallline f~\ AB and athird binomiall line AD, and let the line A D be f[uppofed to be de- SX" |\wided into his names inthe point E, of which let AE be the greater name. Then | aed 1 (ay, that the line that containeth in power the fuperficies A C is irrationall,and is a fecond bimediall line. Let the fame confiruction of the figures be in this that was in the two Propofitions next going before . And now fora{much as theline A D is a third binomi- alline, therefore the[e limes AE and E D are rationall commenfurable in power onely..And DemonSira- tion, a RK P “11 e cS B G Ss re. the line A Eis inpower more sh? the line E D by the {quare of a line comenfwrable in length tothe line A E, and neither of the lines A E nor E Dis commen|urable in length to the line AB bythe definition of athird binomiall line fet before the 48. 5 cats _ As in the ie mer Propofitions it wasdemonftrated, fo al{o may it in - PF ition be promea, rt ye | iti. The tenth Booke line MX containeth in power the [uperficies A C,and that the lines MN and N X are medi. all lines commenfurable in power onely . Wherefore the line M X is a bimediall line. Now re. freth to prowe that it is afecond bimediall line. F ora{much as the line D E is (by {uppofition) — B H.«K L et e incommenfurable in length to the line A B, that is,to the line EK. But the line E Dis corte men{urablein length to the line E F .Wherefore (by the 13 .0f the tenth) the line E E is in- commenfurableinlength to theline EK . And the kines F Eand E K arerationall. For by } 7 fuppofition the line E D is rationall , unto which the line F Eis commen{urable. Wherefore the lines F E and E K are rationall lines commen urablein power onely . Wherefore ( by the 21.of ihe tenth) the parallelogramme E L,that is, the parallelogramme M R which is con- tayned under thelines M Nand NX is a mediall [uperficies. Wherefore that which is con- tayned vader the lines MN and N X is a mediall [uperficies. Wherefore the line MX isa fecond bimediall line (by the 38. Propofition and definition annexed thereto): which was re- quired to be proned. qi he 39. C'beoreme. The 57. Propofition. Ifa fuperficies be contained ynder a rationall line,and a fourth binomial dine: the line which contaynethin power that fuperficies is irrationall, and is a greater line. : We V ppofe that the fuperficies.A C be comprehended under a rationallline AB anda 5 fourth binomiallline AD, c& let the binomiall line A D be [uppofed to be denided LOING into his names in the point E, fo that let the line AE bethe ertater name. Ther I fay, that the line which contayneth in power the fuperficies A Cisirrationall,andis 4 grea- Conftruction. ter line. For,fora{much as theline A Dis afourth binomiallline, therefore the lines\ AE and E D are rationall commen{urablein power onely. And the line A E is in power more then the line E D by the fquare of aline incommenfurablein length to AE. _Anad the line A Eis commen{uvable in length tothe line AB. Dewuide (by the 10.0f the firft) the line D E into twa equal partes in the point F . And upon the line A E applya parallelogramme equalt to the [quare of E F and wanting in fieure by a {quare: and tet the fame parallelocramme be that which is contayned under the lines AG & G E..Wherefore (by the fecond part of the 18.of the tenth.) the line A G 1s incomzmen|urable in length to the line EG. Draw unto the line AB, by the pointes G,E,F parallel lines G H,EK sand FL, and let the reff of the con Demonflran ftruttion be as it was im the three former Propojitions Now it is manifeft, that she lime biGih. MX contayneth 1a power the Juperficies AC Now refleth toproue thatthe line MX és - : ee . grvationas of EuchdesElementes, < Fol.272 irrational line,and a greater line. Forafmuch asthe line. AG tsinconmbenfarablein lexeth sotheline E.G, therefore (by the ‘xo the fixt, and r1.of the enh the parallelogramme A Hisincommenfurable to the paralleloeramme GK y that isjthefquare'SN to the founre. N Ps Wherefore thelines M Nand NX areincommepfurablein power! And foralmmuch as. the line-AE is commen|urable tn length to the vationall line AB, therefore the parvllelo-- | ~ a . “a Pet gramme AK isrationall . And itis equall to the [quaves of the lines M N and NX. Wher- fore that which is compofed of the [quares of the lines M N and N X added together is rati- onall. And fora{much as the line E D is incommenfirable in length to the line AB, that is, totheline EK, but the line ED is commenfurable inlength tothe line E F, therefore the line E F is incommenfurable in length tothe line EK. Wherefore the lines E K and E F are yationall commen|urablein power onely , Wherefure ( by the 21. of the tenth) the parallelo- gramme LE, that is, the parallelocramme M Ris mediall. And the arallelograme.M R is that which is contayned under the lines M Nand NX. .Wherefore that which is contay- ned under the lines MN and NX is mediall_And that which is compofed of the {quares of the lines MN G NX is proned to be rationall,¢> the line M N 1s demonftrated to be income wen[urablein power to theline NX. But if two lunes incommen|urable in power be added tocether, haning that which 1s made of the [quares of them added tovether rationall,c> that a is onder them mediall, the whole line is irrationall, and is called a greater line (by the 30.of the tenth) . Wherefore the line- M X is wrrationall, and is a greater line, and it contat- neth in power the fuperficies A C : which was required to be demonftrated. Ji q I be 40. I beoreme. The 58. Propofition. ° Ee Fi a.fuperficies be'con tained ynder a rational line-and a fift binomiall line: re Sea gehts 5 4 ~ sevhi yes : > ' . \* % sesh Ber Fe * 4 4 ~ the line whith contaynethin power that [uperficies 1s wrationall, and is a line contayning in power a rational and a medtall fuperficies. Teel ppofe that the fuperficies AC be contayned under the rational line AB, | SS a ee 4 fift {erie line ADs iS let. the fame line.A D. be fuppofed to ONC: | SOK’ name ..E ben 1 {ay, that the line mhich contayneth in power. the fuperficies AC 4s irrational, and is aline cantayning in powers rational and amediall fuperficies.. Letthe Jelfe [ame couflructions bein this, that were in.the foure Propofition next going before. And pr ismanifeltoa the lie dX cantayneth jn power the (eperfies A C.-.ddem sees praue that the line M X is a line tontayning in power a Pationall & a medial epee. 08 al(mouch ad — — be deuided into his names inthe. ay sth E, fothatlet the line AE be the greater os Theta Gh Bice DemonSfrae- £10. o -~* “fe Ww T betenth Booke afmuch as the line. Gis incommenfurable in length.to the line G E; therefore ( by thé. 4.0f- thefixt, and 10.0fthe tenth ) theparallelogramme A tis incommen|urable tothe parallrs 3° lagramme HE sthatis,the{quare of theline M:N tothe {quare of the line NX: Wherefore . thelines.M N-rnd NN Xare incommen| unable in power’. And foralmuch asthe line A D is afift binomiall line,and his lelfehamme-or part ts the line E D; thereforethe line E.D-is Cojte. 5 2 ? menfurable in length to the line.AB. But the line AE is sncommen|urable in length tothe line ED Wher efore (by the 13.0f the tenth) the line CAB is incommenfurable in lene to the line 4 E.} Uherefore the lines AB and AE arerationall commenfurable in powy onely Wherefore (by the.21 .of the tenth the parallelogrammeA K is medial, that is.thit hich is.compofea of the [quares of thelines MN ce NX added together. And ‘foralmuch 3° Be line D E is commenfurablein length to the line\_AB,that is, to the line EK , Sut the Lie D E us comme n{urable in length to the line E F , wherefore ( by the 12.0f the tenth) the line E Fis alfa commenfurable in length tothe line ER. And the line E Kis rationall: Wherefore (by the 10.0f the tenth) the parallelogramme EL, that is, the parallelogramme M R, whith is tontayned under the lines MN and NX is rational! Wherefare the lines _ mM Nand XX areincommen(urable in power, hauing that which is compofed of the [quares 1 of thm added together, “Mediall , and that which 1s contayned. under them, Rational! Waerefore (by the 40. of the tenth) the whole line MX 4 a line contayning in power a ra tional and a mediall [uperficies , and it contayneth in power the [uperficies AC : which was required to beproued, Pie | at | q Ihe. 41. Theoreme. Lhe s9,.Propofition. If a fuperficies be contayned ynder a rationall line, and a fixt, binomiall “tine, the be Swhich contayneth in power that ‘[uperficies js irrational @7 is called a line contayning in power two medials. ~~ -# ~ ‘+. sgn pole that the fuperficies A BC D Le contained under the rational line AB, and fe under a fixt binomiall line A D,andlet the line A D be fippofed to be deuided in. EAS to bis names in the point E.fo that let the line AE be thegreater name, Then 1 fa that the line that containeth in power the fuperficies A Cis wrratwonall, and is a line contay. Demoniirae —nin& 12 pomper two medials. Let the lelfe [ame conftruttios bet this that were in the forme . CANE cos propofitions.N ow it is manifeft that the line MX containeth in power the luperficies CAC, Ie and that the lime NG incommen|urable in power to theline NX. And forafmuch as the 2. UneaEis int ommén{urable in length to thekine AB, therfore the lines AE apd cA ore 4 ayy es Shay rationpa of Euchdes Elementes, . Fol.273. tationall commenfiurable in pomeronly.Wherfore(by the 21 .0f the tenth the parallelocrame AK thatis;that whichis compofed of the fquares of thelines MN and NX added together 3° is medial. Againe forafmuch asthe lime E D is incommenfurablein length tothe line AB, : therefore alfo the line E F is incomefurable in Vegeh to the line BR. Wherfore the lines E F and E K are rationall commenfurable in power onely. Wherfore the parallelogramme E L, that is,the parallelogramme M R whichis contained vnder the lines MN and NX 15 mite B H K j é G >. re diall. And fora{much as the line A Eis incommenfurablein length to the line E F, therfore the parallelogramme A K is al{o incommen{urable to the parallelogramme E L (by the firft of the fixt,and 10.0f the tenth.) But the parallelogramme A K is equal tothat which is com- pofed of the {quares of the lines MN and N_X added together. And the parallelogramme E Lis tquall to that whichis cotained under the lines M Nand NX Wherfore that which isconmpofed of the fquares of the lines MU Nand NX added together, is incommenfarable > to that which 3s contained under the lines M N und NX: and either of them, namely, that which 1s compofed of the {quares of the lines M N and N X added together, and that which ts contained onder the lines MN and N X,is proued mediall, andthe lines M Nand NX are proued incommen{urablein power. Wherfore(by the.ar.of the tenth) the whole line MX is a line contayning in power two medials, andit containeth in power the fuperficies AC: which was required to be demonfixated. OO An Affampt. Ifa right line be denided into two vnequall partes, thefquares which_are made of the ynequall partes are greater then the reézangle parallelogrammte coe tayned bnder the Mnequall partes ,twife. : sat | Suppofethat AB bea right-line, and letit be. déuidedinto.two vnequall partes $f the point C. And let the line AC be the greater part.T-hepd fay that thefquares of the lines AG and C B,are greater the that which ts contained under the lines A € thd GB stwife: Denide (by the x0 of the firft) theline ABintotwa é- Soros TAG » : quall partes ,in the point D.Now forafmuch aS ay oe se, i Pea the right line AB is deuided into two CALL ee ae partes in the point D, and into two vnequall dx | | | partesin the-point G; therfore (by the sof the v4: \ 4 (econd that which is contained under the lines AC.and G B,together with the [quare of the dine CD, is equall to the [quare of the line A D . Wherefore that which is doutaried — ack tpe * TZ betenth Booke’ the lines C dndt B;( omitting the fquare of the lineG-D,) is leffe then the {quare of the AD (by the 9.common fentence,and the fenenth of the fifth. Wherefore that which is con- tained under the lines AC andC B,twife,is leffe then the double of the [quare of the line A D (thatis,the twife the {quare of theline AD) ’ x by * alternate proportio,and the 14. of thefift. | Lookeafter Butihe{quares of the lines AC andCBare. ? ~ nd, B the Affumpe double tothe {quares of the lines A Dand DC conciudedat = (by the 9 .of the fecod).Therfore the fquares of thes marke:for 4 © anid C B are more then double t the {quare of A D alone; (leaning out the [quare of D = — CC) by the 8 of the fift.But the parallelogramme contained under the lines A Cand C B twife nine of this at eae 7 : ‘ r ; place. is proued lefse thé the double of the {quare of the line AD.Therfore the fame parallelograme contained under thelines AC and C B twife,is much lefve then the [quares of the lines AC and C B, If aright line therfore be denided into two vnequall partes, the [quares which are made of the vnequall partes,are greater thé the rectangle parallelogramme contained under the unequall partes twife : which was required to be demonftrated. * In numbers I heede not to haue fo alleaged, for the 17.0f the feuenth had confirmed the doubles co be one to the other,as their fingles were,butin our magnitudes, itlikewife is true and evident by alternate proportion;thus-As the parallelogramme of thelinesx c and c gw istohis donble, fo is the {quare of the line a D to his double (eche being halfe). Wherfore, alternately, as the parallelogramme is to the {quare, fo is the parallelograme his double to the double of the fquare.But the parallelograme was proucd leffe chen the {quare: wherfore his double is leffe then the {quare his double;by the 14. of che Afth. | e é This Affumiptis in fome bookes not réead,forthatin maner it femeth to be all one with that which.was put after the 3 g.of this booke:but forthe divers maner of demon= firating,it is neceilary.For the feate of inuentiOis therby furthered.And though Zante bert didin the demonttration hereof,omitte that which P, Montaureus could not fupe ~.... ply, but plainly doubted of the fufficiencie of this proofe, yet M.Dee,by onely allegae prapeitwn,S sion of thedue places of credite,whofe pithe & force, Theon his wordes do containe, ether follow- — ravtiteftored tothe demonftration f ufficiently, both light and authoritie, asyou may ona pereeitié,and chiefly fuch may indge; who can compare this demonftration here (thus furnithed) with the Greeke of Theon,or latine tranflation of Zambert. PA Pad 7 be vie of this A fSupape sin the next ° ~~ * & q The a T heoreme. The 6 oiPropo/ition. : The fift Se- Uhe fquare of a binomiall line applyed bnto a rational line, maketh the nary by come breadth or other fide a fir bmomuall line. potion, | oe resi : V ppofe that the line A Bbeas binomiall line, and let it be’ uppofed to be deuided into his names inthe poyntC, fothat Conftration, %tAsCobethe erearername> And takea® ® rationall line DE. Aad ( by the ¥g.of the frftyeamtothe lineD E ‘apply aveaas gle parallelograme DEF G equall to the Square of the line A.B and makine in breadththe line DG . Then t fay thatthe lineD Gis afirft binomiall line nto the line DE apply the parallelograme D H e- BOE TIOT ey ee quallto the {quare of the line A Cand A_- —e : Apalieiae oana! vato the line K Hwhich is -equall tothe: A AS ee a i "2° ~— as " . - 2 N 2 : — ene 5 : ‘ : / ‘ cies “4 - Po 2 " _" 18 » hone ie “ - > Gi eee eee, Oe ripe bail of Euchides Eleinentes: Fot.z-74. et igetade bly the parallelograme KK L equall to the fquare of the line BC... Wherefore the idue nani Ly that which ts contayned hundet phe tines A C & CB tivife ts equall to the re- (deve paseo the parallelogr ane F by thea. the fauare of a line cammenfurablej ix length tothe ereater. Wherefore the line D Missin power marethen the lineM G bythe ¢ {quare of a line commenfurableimléeneth vito the lize D M x And the lines D Mana M G are proned ra- Viowall comwien|urapletn power orely And the line DM is a cued the greater paneana geet re consonea firable th ledgah tothe rational line cenen DE. Wi peer ote 1 the definition of 4 firfre bii PEPE Bi lise fer before the 45 4 py apofition Oj ah booke,ike ize i? shes gr Afi firf tb iacur oH Fat A. is; ie ? COS; ‘ SANS Lene fy Demonfl rhe til. | to Conckuded that D G is a binomiall line. Se T he tenth Booke dine: which was required to be proued. This propofition and the fiue following are the conuerfes of the fixe former pro- pofitions. q¢ Lhe 43.1 heoreme. The 61. Propofition. The fquare of a fir[t bimediall line applied toa rationall line, maketh the breadth or other fide a fecond binomial line. ly ol Al ppofe that the line A B be a firft bimediall line, and let it be [uppofed to be de- ¥ @ ,_| tided into his partes im the point C,of which let AC bethe greater part . Take NY S| allo arationall line D E, and (by the 44.0f the firft) apply to the line D E the EG! parallelogriime D F equall to the {quare of the line A B, cr making in breadth Conltructions the line DG. Then I {ay, that the line D G is afecond Linomiall line . Let the fame con- Demon ftra- $209. I. 2. Ze Cenuciuded that DG ig a dinomiall dine. 4. 5 fiructions bein this, that werein the Propofition going before. And fora{much as the line A B is afir[t bimediall line, and is denided into his partes in the point C, therefore ( by the 37 of the tent) the lines AC andC Bare | mediall commenfurable in power onely, : coprehending a rationall {uperficies. Wher- | | | fore alfo the {quares of the lines AC and | CB aremediall. Wherefore the parallelo- | gramme D L is mediall ( by the Corollary | | | of the 23 .of the tenth )and it is applied up- pon the rationall line D E .Wherefore (by the 22.0f the tenth) the line M D is ratt- onall and incommenfurable in length to theline DE. Acaine forafmuch as that which is cotayned under the lines ACand C B twifeis rationall,therefore alfo the pa- rallelogramme MF is rationall, and itis applied unto the rationall line M L .Wherefore the line M G is rationall and commenfura- ble in length to the line ML, that is, to the line D E ( by the 20. of the tenth) . Wherefore the line D M is incommenfurable in length tothe line MG, and they are both rationall. Wherefore the lines DM and M G are rationall commen{urable in power onely . Wherefore the whole line D Gis a binomiall line.Now resteth to proue that itis a fecond binomiall line. Fora{much as the [quares of the lines AC and C B are greater then that which is contayned under the lines AC and CB twife.(by the Afcumpt before the 60.0f this booke) « therefore the parallelogramme D L is greater then the parallelogrrmme M F . Wherefore alfo (by the -o- ‘sy rae enentene eee ) | ' ! | ae a a | bam yo ie o) | Jirft of the fixt) the line D M is greater then the line MG. And forafnuch asthe {quare of the line AC is commen{urable to the {quare of the line C B, therefore the parallelogramme DH 1s commen{urable to the parallelogramme K L . Wherefore alfothe ime D K is com- ven{urable in length to the line K M.And that which is contayned under the lines D K and K M isequallto the [quare of the line M N, thatis, to the fourth part of the {quare of the line MG . Wherefore (by the 17.0f the tenth ) the line D M isinpower more then theline (MG, by the [quare of a line commenfurable in length vutothe line DM. : and the line UM G ts commenfurablein length to the rationallline put,namely, to D E . Wherefore the line DG is afecond binomiall line : which was required to be proued. qIbe —- a, es, ee —_ a err 2 —Ee————EE™- of Euclides Elementes, » Fol.275. nb ee q Ihe 44. EF heoreme. The 62, Propofition. The {quare of a fecond bimediall line applied vnto a rationall line: maketh the breadth or other fide therof,a third binomial hyne. b epeay, Vppofethat AB bewfecond bimediall line;and let A B-befuppofed tobe denided ‘ Cc» Paar + 8 . . A . BINA into his partes tn the point C [0 that let AC be the greater part._ ind take ara- UNE Zan : AY o aceonall line D E. And(by the 44.0f the firft) unto the line D_E apply the paralle- 3 logramnre DF equallto the {quareof the line AB, and making in breadth the line D G.ThenT {ay that thetine D Gis a third binomiull line. Let the felfe fame compirsc, tions be in this Pat were in the propofitions next gorng before. And forafimuch as the lized oo i B is afecond bimediall line, and is deuide into his partesin the point C; therfore (by the 38.of the tenth) the lines AC and CB « are medials commen{urable in power only, compreheding a mediall { uperficies.W her- fore t that which is made of the {quares.of the lines AC andC'B added ‘together, is wiediall, and Wis ejuall to the parallelo- gramme D.Laby confiruction, Wherefore the parallelogramme D Lis medial, and is applied-wnto, the rationall line, DE, wherfore( by the 22. of the tenth) the line M Dis rationall andincommenfurablein length tothe line D EB. ‘And by the lyke rea(on alo” the lineM Gis rationall qd os incommen|urable in le noth to the line M L, that is,to the line D E.Wherfore either of thefe lines D M and M Gis rational, and intommenurable in length tothe line D E. And foraf- snuch as the line A Cis incommenurable in length to the line C B, but as the line ACis.to Pe ine CB, fol dy the alfumpt coine vefore the 22:0F the tenth) is the [quare of the line AC th thut whith iscontained onder the'lines AC andC BWherfore the [auare of the lint AC Wincommenlurable to that whichis contayned under the lines AC and C B. Wherfore that MW 2hat whichis matte of the (quaresof the lines AC and CB added tovether, is incommen- Porable to thar whichis contained under the lines AC ACB twife, that is, LP Sivan gramme D L to the parallelogramme MF Wherfore(by the frst of the fixt,and 10.0f the tenth )the line D M is incommen{urable in length tothe line MG: And they are proued bath rationall,wherfore the whole line D Gis abinomiall ine by the Wefinition in the 36. of the tenth. Now refteth to prouc that itis a third binomial line. As in the former propofitions, fo al{o in this may we conclude that the line D\M is eréater then the line M Gand that the line D K ts commenfurable in length to the line. K M. “Awd that that which is contained vider the lines D K and K Mis equatl to the [quare of théline MN Wherfore the line D M is in power more then the line M G by the (quare of a linevommenfurablein leneth unto the line PD Myand neither of the lines D M nor M G is combmenfiirabletn lencth tp the rational line line which was required tobeproued. “3 % D E.Wherfore(by the definition of a third bigomialt line) the line DG isa third binomiall ~ ane . ~t& IVA) 44 ,* q Flere follow certaine annotations by M. Dveswnade vpon three places inthe cemontftration, which were not very éuident te yong beginners. oud = GiheXguares ofthelnes'a Candhe mane medials (asistauche afer thes Pofthis tenth) ard ther- fosefgraimucn as they are( by fuppofition) comméfurable th’one to the other:(by the 15.of the téth) gh KY LL. ij. nic Construction. Demonfra- tion, D G;conelu= ded a binomi=\ all line. a hs The tenth Booke il 1) TH a Ht che compound of them bothis eommenfurable to ech part. But the partes are medials, therfore(by the Tt a corollary of the 23.0fthe tenth) the compound fhall be medial. * For that m x is equall (by conftruétion) to that which is contayned ynder the lines ac andc 8, which is proued mediall : therfore(by the corollary of the 23.0f thistenth) m x is mediall, and ther- fore (by the fame corollary) his double M F is mediall.And itis applied to a rationall line,m 1 (beyng equallto pv s) therfore by the 22.0f the tenth, the line m Gis rationall and incommenfurable in length to m.1,thatis,top £- tt Becaufe the compound of the two fquares (of the lines ¢ and c 2) beyng commenfurable one to the other, is alfo to eyther fquare(by the rs. ) commenfurable,therfore to the {qnate ofa c: Butthe fquare ef-ac is proved inconimenfurable to that which is contained ynder ac & ¢ 2-once, Wherfore (by the 13.0f the tenth Jthe compound of the two fquares (ofthe lines a c and c x) is incommenf{u- rable,to that which 1s cotained vnder the lines @ c and c 8B once. Butto that which is twife contained vnder the fame lines a c and c 8,the parallelogarme once contayned,is commenfurable (for it is as 1. is to2.) therfore that which is made of the f{yuares of thelines. a-c.and‘¢ 8 isincommeniurable to the parallelogramme contained ynder a c and c g twife,by the fayd 13.0f this tenth, @ A Corollary. fares Flereby it ts enident that the fquares made oj the two partes of a Jecond bre adiedbyM, medtallline,compofed is acom pound mediall and that the fame compound 1$ 116 bet. commenfurable to the parallelogramme contayned bnder the two partes of the jecond bimediall hye. , ) The proofe hereof,is in the firftand third annotations here before annexed. ¢ he4s. Cheoreme.- The'63. Propofition. T he fquare of 4 greater line applied nto a rationall line, maketh the breadth or other fide a fourth binomiall line. ees xe ppofe that the line AB beagreater line, andlet it be{uppofed to be denided into wl Lis partes inthe point C, fo that let AC be the greater part . dud take a rationall mut line D E.And (by the 44.0f the firft)unte the line D E,apply.the parallelogramme DF equall to the [quare of the line.AB; and making in breadth the line D.G... Then. fay, that the line D G 1s a fourth binomiall line... Letthe felfefame. confiruttion be in this, that ConSirulion. syasin theformer Propofitions.« And for- mete 5 0} almuch as the line AB is a.greater line, SAGA Dewouftra>< 1s deuided into his partes in the point C+. ; tivits thercforethe lines A Gand C B are incom- menfurable in power, hauing that whichis made of the {quares of them added toge- ther rationall, and.the_parallelogramme whichis contayned vuder them, mediall. Now foralmuch as that which 1s made.of the {quares of the lines A Cand C B added together is rationall, therefore the paralle. logramme D L is rationall . Wherefore al- {othe line M Dis rationall and commen-~ 6 > as [urablein legth to theline D E (by the 20, . of this tenth) . Againe foralmuch as that , wich is cotainea under the lines AC and CB twife is ssediall, that is, she pavallelograme of Euclides Eletneimtess Fol.276. M F,and it is applied unto the rational line M L, therefore ( bythe 2%. of the tenth) the line MG is vationall and incomméen{urable in length to the line DE. T. herefore (by the 13. a. of the tenth). theline D.M isincommenfurable in length tothe linewMG . Wherefore the lines D M and MG are rationall commen{urable in:pomer ontly...Wherfore the whole Ze line D G is a binomiall line . New refleth-to proue, that it is:alfoca fourth binomiall line’. Even. as in the former Profofitions, fo alfo in this may we conclude, that the line DM is greater then the line. MG . Andthat thatwhich is.contayned under. the lines DK and KM is equall to the fquare of the line MN. Now forafmuch as the {quare of the line AC is incommenfurableto the lquarevof the line CB , therefore: the parallelogramme D H ts incommenfurable tothe parallelogramme K L. . Wherefore (by the r.of the fixt, and 10.0f the tinth) the lime DsK 4s incommen{urablein length tothe line KM. But if there be two unequal right lines, and if upon the greater. be.applied a pa rallelogramme equal to the fourth part of the fquare made of the lefesand wanting in figure 4 by a fquare, and if alfo the parallelogramme thusapplied deuide the line-wherupon-tt 1s ap- plied into partes incommenfurable in length, thegreater line fhall bes power more themthe le{ve,by the [quare of a line incomen|urable in length to the greater: by the18. of. the tenth) Wherefore the line DM. isin poner more then the line MG, by.the {quare of a line inco- menurable in length to DM. And the lines D M and DM G are proned to be rationall c0- menfurablein power onely. And the line D M is commenfurable imlength to the rationalt line geuen DE. Wherefore the lim D Gis a fourth binomial line ; which was required to be prone. ? q ihe 46. I heoreme. The 64. Propofition. T he [quare of a.line contayning in power arationall and a medial fupere ficies applied toa rationall line, maketh the breadth or other fide a fift bse nomiall line. —= Jom AV ppofe that the line A 3 be'a line contayning in power arationalland a medtalk if, | {uperfictes, and let it be{uppofed to be denided into his partesin the point C, fo ; (S| that let A C be the greater part, and takearationall line D E And (by the 44: Mele} of the fir) unto the line D E apply the parallelogramme D F equall to she fquare of the line AB, and makingin breadth i | the line DG. Then 1 fay, that the ine D G ts a fift binomiallline.Let the felfe [ame coftrudti-_ on be in this, that was in the former. And for- afmuch as AB ts aline contayning in power a vationall anda Mp er ana is deui- ded into his partes in the poynt C, therefore the lines AC ¢> CB are incomen{uratle in power, hauing that which is made of the fauares of thé |. ~ added together mediall, and that whichis. cotte Ao 3. tayned Vader them rationall .Nowfordfmuch. Boo. HO ™ * asthat whichis made of the {quares of the lines As. Lactitancamnd AGand CB added together is mediall , there- fore al{o the parallelogramme D L is mediall. Wherefore ( by the 22. of the tenth) the line DM isrationalland incommen{wablein length to theline DE . \Againe forafmuch as that which is contayned under.thelines A C and.C B. twife, that is, the parallelogramme | ME, is vationall, therefore by the. 20. the line MG is rationall & comen{urable in lang 2. ’ LLAy. te the G . Confirntion. Demonfira- $0 Lr) fe tls The tenth Booke roth lime D & -Veherefore ( by the 13: of the tenth) theline.D M is incommenfurablein length tothe line MG . Wherefore the'lines: DM and MG are rationall commenf{urable.. a power onely . Wherefore the whole line D.G-is abinomiall line » I [ay moreouer, that it safiftbinomiall . For,astn the former,foal- , fo in this may tt be proued , that that. whichis contayned vader the lines: DK and KM is e- quall tothe (quareof M Nthe halfe of the leffe: and that the line DK: is incommen{urablein \ length to the line KM» Wherefore (by ther8: of the tenth) the line D' Mis in power more thé the line MG by the {quare of a line incommen- fuvablew length totheline D M.And the lines DM and M G are rationall commenfurable in power onely; and the lefeline, namely, MG 4s commen|urable inleneth to.thevasromalh ine 4. oS geen DDE. Wherefore the line D Gis a fift binomial ling: which wasreguired to be demon firated: LD ® o The 47. T heoremess 3" Ehe 6s. Propofition. Phe jguare of a line contayning in power two medialls applyed bnto a raz tonal live maketh the breadthor other fide a fixt binomiall line. fe that the line A.B be a line contayning in power two medialls , and let it be | D F equall to the {quare of the line A B and making in breadth the lineD Gs Thenl fay Confiruction. that the line D Gis a fixt binomiall line. Lot thesfelfe' fame construction be in thys that wits ivi the former And foralmuch as DemonSra- theline A Bisrline Contaynineinpower b20%. two mediallssand ts denided into his partes in the poynt Ctherefore the lines AC co C Bare incommen{urable-tn power ha- wine that whichis madeof the {quares of them added together mediall , and that which is contayned under them , medtall, and moreouer incommenlurable to that' which is made of the [quares of them ad- iA . e B ded tagether . Wherefore by thofe thinges which have bene before proued,either of: ; thefe parallelogrames D Land M F is mediall, and either of themis apphed vpon the ration naliline D E.Wherefore(by the 22.0f thetenth either of thefe lines D Mand M G is ratio: 1. valland intommenfurable in length totheline DB. And forafmuch as that which is made of the [quaves of thelines A Cand C B, added together isincommenfurableto that which is contayned Onder the tines A Cand C B twife; therefore the parallelograme D L isin= tormmen{nrabletetheparallelograme ME: Wherefore ( by thet. of the fixt and 10 of the penth jibe line D MC is incommenfurable in length to the line MG. Wherefore the lines 2: DEM and MG apt rationall commen{wrablein power onely. Wherefore the whole line DG ae ck is A o. of Euchdes Blementet. Fol.277. tsa binomviall linet {xy alfa that is ajiset binonsiall line F or enemasimthe other propofitions it hath bene proned [0 al{o inthissmay it be proned, that thatwhithixcontayned under the lines D Kand KK Mis equallto the quare of the line MN, and thatthe line DK ts incom men{urable in length tothe lime K M,and therfore (bythe t 3e0f the.stenth) the line DM is in power more then the line MG by the {qnare of aline incommen{irablein length tothe line D M.And neither of thefe kines DM nor M'Giscommenfurablein length tatheratio- wall line zeuen DE.Wherefore theline D @ ts afintbinomialltines which was requixea to be demonflrated. Al . q Ihe 48. I heoreme. The 66. Propofition. A line commenf{nrable in length to a binomial line ,is alfo a binomiall line of the felfe fame order. ENE \V ppofe that the line A B bea binomiall line, andwnto theline A Bet the line \ Se by ICD be commen|urable in length . Then I fay that the line C D is a binomiall MVS Wine and of the felfe fame order that the line A Bis . For forafmuchas A Bisa WX!) i omsiall line let it be denided into his names in the poynt E,and let A E.be the greater DAME Wherefore the lines A Eand& Bare rationall commenfurable in power onely.Aand as the | line AB is to the line C D, (foby the 12.0f the fixt) SRL te SHEER a oes let the line A ¥ bc to the line C ¥, Wherefore(by the 19. of the fift) the refidue, namely, the lineE Bis toc | Se D the refidue namely ,to the line FD, as the line A Bis “een tothe line Ds But (by fuppofition) the line A Bis commen[urable iu length tothe line C D.Wherefore( by the 10.0f the tenth )theline A E is commeu[urable in length totheline C F,and the line EB totheline¥ D-And the lines AE and E Bare rational. Wherefore the lines C FandF¥ D are alfo rationall. And for that as the line A Eis to theline CF, fo isthe line EB to the line F D, therefore alternately (by the 16 .of the fift)as the line A E.is to the line EB, fois the line C F tothe line D . But thelines AE aud EB are commenfura- blein power onely wherefore the lines.C F.and FD are alfocommen{urable im power onely, ana they are rationall. Wherefore the wholeline C Dis a binomiallline. I fay alfa thatitas of the felfe fame order of binomial lines that the line ABis.F or the line AE is in power mare then the line EB either by the [quare of a line commen urable in length tothe line AE,or by the fquare of line incommenfurable in length to theline A E.If the line A Exbe in pomer more then the line E.B by the [quare of aline commen{urable in length to the line AE, the line allo CF (by the r4. of thetenth ) halbe in power more then the line FD. by the [quare of a line commenfurable in length ta CF . And if the line A E.be commen{urablein length to a rationall line cenen,the line C F alfo fhatbe commenfurable in. length to the fame. ( by the 12.0f the tenth). And fo cither of thefe lines A B and CD is a firft binomial line,that is,they are both of one and the felfe [ameorder » Butif the line EB be commenfurable in length tothe rational line put,the line ¥ D alfe apie commen|urableinlength to the fame. And by that meanes agayne the lines AB and CD are bath of one and the felfe fame order, jor.cither of them ts a [econd binomtall line But if neither of the lines AE nor EB becom- menfurable in length ta the rationall line put. neither alfo of thefe lines CF nor F Difhalbe commenfurablein length to the fame. And {ocither of the lines A.B and C Disa third bi- nomsiall line. But if the line AE be ix porer morethen theline EB .by the {quare of aline incommenfurablein lenoth tothe line AE, the line alfa GF fhalbein pomer more then the line F D-by the {quare of a line incommenfurable in length to the line CE, (by the a of os | LL. ii¥. the The fixt Sea nary Conftrutti0tte Demonfira~ tion, I's 26 2e The tenth Booke shetenth). And then if the line AE. be commenfurable in length to the rational line pes the line CFE alfo fhalbe commenfurablein length to the fame, and [0 cither of the lines AB; and sD fhalbe a fourth binomiall lines And if the line EB be commen|urablein length te therationall line genen,the line D alfo fhalbe commen|urablein length tothe fame. And fo either of the lines: A Band CD foalbea fift binomial line. But if neither of the lines AE nor EB be commenf{urablein length to the rationall line geuen neither alfo of the lines C For F-D fhalbe commenfurableinlencth tothe fame,and fo either of the lines A Band CD fhalbe a fixt binomiall line.A line , a commenfurable in length toa binomiall line,is alfo a tinomiall line of the felfe fame order:which was required to be proned. q Uhe-49. Dheoreme.: Ihe 67. Propofition. A line commenfurable in length to a bimediall line, is alfo a bimediall lyne and of the felfe fame orders ) PSL ppofe that the line A B bea bimediall line, And unto theline A B, let the lyne Oe C D be commenfurable in length. Then I fay that the line C D is a bimediall Nek ef line,and of the [elf order that theline A.B is. Denide the line A B into his partes WAM in the point E.And forafmuchas the line A Bis a bimediall line,andis denided into bis partes in the point E, therfore ( by the 37.and 38. of thetenth)the lines AE and E Bare medials commenfurable in power onely. And (by the 12.0f the fixt) as the line AB is to theline CD fe let the line AE be to the line CF. _ Wherfore (by the 1.9 of the fift) the refiduename A E B R " 2 z. 4 a ‘ reer er ee ly the line E Bis to thé refidue, namely ,to the line | F D,asthe line AB is tothe lineC D. But the F D “line A Bis commenfurablein length to the lyne CD .Wherfove the line A Eis commen|urable in length tothe lineC F and the line E B tothe line F D . Now the lines AE and EB are me- diall,wherfore ( by the 23.0f the tenth) the lines C F and F Dare alfo mediall..And for that as theline A E isto the line E B,fo isthe line C F to the line F D. But the lines AE and E B are commen|urable in power onely wherfore the lines C F and F D are alfo commenfara- ble in power onely. And it is proned that they are mediall Wherfore the lyneC D is a bimedi- alt line. I fay alfo that it is of the elfe [ame order that the line AB is.F or, for that as the line AE 1s tothe line EB,fois the line C F to the line F D but as the lineC F is to F D [0 is the Jquare of the lyne C F to the parallelogramme contained under the lynes CF andFED,b the firft of the fixt. Therfore as theline AE is to the line E B, fo (by the 11.0f the fift) és the f{quare of the line C F to the parallelogramme contained under the lines CF and F D+ but 8 AE is 10 E B,fo by the r.0f the fixt,is the [quare of the line AE, to the parallelogramme contained under the lines AE and E B, therfore ( by the 11.0f the fift) asthe [quare of the line AE is to that which is contained under the lines A E and E B, fois the Pausve of the line C F to that which is contained under the lines C F and F D. Wherfore alternately (by the 16.of the fift)as the {quare of the line AE is to the {quare of the line C F fo is that which 45 contained under the lines A E and E B to that whichis contained under the lines C F ce F D.But the fquare of the line A Eis commenfurable to the {quare of the line CF »becanfe A E and C F are commenfurablein length. Wherfore that which is contained under the lines A E and E B ts commenfurable to that which ts contained under the linesC F and F D. If therfore that whith is contained under the lines A E and E B be rationall,that is,if the line 4B TE eT ee of Euchdes Elementes..” Fol.278. AB be a'firft bimedzall line,that alfo whichis contained under thetinesC Fiand F D is rae tionall Wherfore alfo the line C Dts a fir ft bimediall line. But if. that which is. contained vi der thelines AE and E B be mediall,that is,if the line A B.bea fecond bimediall line, that allowhich is contayned under the lines C F and F Dis medial: wherfore alfothe line C D isa fecond biniediall line Wherfore the lines A B and CD are both of one and the felfe [ame order : which was required to be proued. | ay A Corollary added by Flufates: but firlt noted by P.cMontaurens. A line commenfurable in power onely.to a bimediall line, is alfo a bi meatal line and of the felfe fame order. Suppofe that A B be a bimediall line, either a firftora fecond;wherunto let the line G D be cémen- furable in power onely. Take alfo a rationall line E Z;vpon which (by the'4s.of the firft) apply arectan- gle aap equall to the fquare of the line AB; which let be EZ F C)and let the reciangie parallelogramme C F1H be equall to the fquareof theline GD . And forafmuch as vpon the rationall line E.Z1s.applyed ‘a rectangle. parallelogramme EF equall ro the {quare.ofa firft bimediall line, therefore the other fide rherof,namely, E.C, isa fecond bino- miallline,by the 6t,0f this booke.,And forafmuch as by {uppofition the fquares of the linesAB& GD are commenfurable, therefore the parallelogrammes « E FandClI ( which are equall vnto them. ):are-alfo commenfurable . And therefore by the 1.0f the fixt, the lines EC and C Hare commenturablein length. But the line E C is#fecond binomialbine .“Where- fore theline CH isalfo afecond binomiall line, by the 66.0f this booke ..And forafmuch,.as the fuper- . , ficies CI is contayned vider arationall line EZ ‘or C F, anda fecond binomiallline C H, therefore the line which contayneth itin power, namely, the ling wun. y= GD is a firit bimediall line, by the s5.0f thisbooke., @- And foisthe line GD inthe felfe fame order of bi- *” | imediall lines that the line AB is.s"The like demonitration.alfo-will ferue ifthe line AB be fuppofed to be afecond bimediallline .. For fo fhall itmake the breadth. E-C.a third binomiall line whereunto the line C H thall be commenfurable in length, and therefore CH alfo fhalh he achird binomiallline,by niednes' whereof the line which contaynethin power the'fuperficies CI, namely, the line GD fhall alfo be afecond bimediallline... Wherefore a line commenfurable either in length, or in power onely to a bimediall line, 1s alfo a bimediall line of the felfe fame order. | But fois it not of necefiitie in binomiall lines, for if their powers onely be com menfurable, itfol- loweth not of neceflitie that they are binomialls of one and the felfe fame order, but they are eche bi- nomialls eytherof-the three firttkindes, orofthe three laft-.- As for example. Suppofe that AB bea fir{t binomiall line, whofe greater name let be A G,and vnto AB let the line D Z.be céméfurable in pos wer onely. Then I fay,tharthelineD Z is notofthefclfe = _ fe 7 fame order thatthe line A Bis. For ifit be pofiblesler™ > ,=—___-___ oe the line D Z be of the felfe fame order that the line ABis.. S03 Whe Whereforetic line. DZ may like fort be dedided asthe’ 4 , line A Bis, by that which hath bene demonftrated inthe 4 .. 66. Propofition of this hooke : let it be fo deuidedinthe 4 poynit B. Wherefore iccan not be fo deuided in anyother,“ poyntsiby theg2.of this booke, And fot thatthe ine'A BS’ 9 iy is to. the line DZ astheline A G isto theline DE, but... thelines AG & DE namely,the greater names,are com- men{urable in' length the one tothe other (by the 10.0F * this booke )for thattheyare commen furable in length'to: one and the felfe fame rationall line, by the firlt definition of binomiall lines . Whereforethelines ABandD Z are, -. L-—— éommenfurable in length; by the 13. of thisitbooke. But. F , by fuppofition cheyare commenfurablein poweroncly ; which is impofible: F =o pe H K 4. AA Coroll sry addid by Fluffates. Note. = ae > a en ~ ° ; ne a ~ - ~- — - - = =e et ~ : - . ~ aes moll E : =e S75 ly = = = o_o > “ 3 = = - - : ae 7 = 5 z - = ; = = ————— ~ . . i Sea . 5 —> — A: =s-43' 7 ae oe i. a cae cient esti = oe - = sietaiianoae 44; i -«~ = . 5 se -= 79 = my ee — —s == = S as —~ —— = _— = = . _—— = . « — ~S ~~ aS Sa —-. a : = = — — = — = = — = a -- — - — 2 _—— - y x - ~—— = = nee as os — —~ =- — = = = : — - —-= _ - - - — - + + = — =— —_ - - : = d ‘ . + = == - = = == : a — — - == - ~ a - Sout; = Sea aS k= = — : : ———— - — — = oe = —— ~ - —— = = - —————_ —- re — a a ~ Faery " —- ~ SS. ——_--- = Ft + == — — ~ — a st , Ss SS OD ar 3 5 = — Sse se eee ae < = y LSS = _ == SE2s= = Ps === = = —— = oe — : - —_ ~ Py > . =~ - —- ——— —— ~ sre eos aa SS_ or = = = = - a --- - = — on ——= : = = SS = = Sa =. E : ~~ ns - : - — = —= <= = r = = —— ee ee a ee _— ---—- —_— —" = ~ ~ ——=--- = = = - —_----—-- = = = ——— > x _ = = . e =e = = = ~ . oe - -- - — = —se ~ ~ - a = — aa eee = a —— = - ———s — —_ = - — —— - ~- —~- eh . ——- eo Eee x = =$ ry - — ee ——- 2 — = —— -- “ = a “ —- — — - a mai - os mia Sa - ‘ > ——t a = 7 = ———* —— —-—-* Se ee SSS | SS I Oe Ghee = a = ~ — = = 2 = - ~ - — . —— - = — - i a : = =e ne — =~ = === <=> = —~ — —_*= +> = = = ee = - - —— SS ee <. ¢ —=— SS SS : <—— S — = may Pry eS —_ : : ts = - —-—- es. - — —— a a ~ == —— —--——-- —— SSS sd = Tes ee SS - > ; z - —~ J - —=s= oo STs _—— en ——— = —_ — = = = tates Le a ean —_ me ee SS aes =~ oS = SS = — —: ——— = , == == - ~ mS = —— — ———— -2 — i a - : = = - — ° ¢ er =—s =~ —————————————— ——S— . eer " > Pe 2X = = tah alls cn. » — > Aes nae == = : Pac a = —— <- , = ra -_ == — — sees ae Poe = eet ae ee - - - SS ee Se EOS ee Dema 1Slra~ tee He fore the line A Bas commenfura- lines CF and F D Whefore alternately (by the 16 of the fifi) 4s the fguare of the line A T betenth Booke The felfe fame dera snfttation alfo will ferue,if we fuppefe the line. AB to be a’fecond: binomial line ; for the lefie namesG B and E Z being commenturable in length to.oné and the felfe fame ratio- hall line, thall alfo be conmenfurablein length the one to the other. And therefore thelines AB and DZ which are in the {elt fame proportion with them, fhallalfo be commenfurablein length the one to the other : which is cntrary to theduppofition.,. Farther, if the fquares of the lines AB andD Z be applyed ynto the rational line C F,namely; the patallelogrammes\C.) and HL, they fhall make the breadthes C H and HK fint binomialftinés, of what order foeuer the lines AB & DZ (whofe {quares were applyed vnto the raional line)are, (by the 60.0f this booke) . Wherefore it i: manifeft, that ynder arationall D : line and a firft biaomiall Ine, are confufedly contayned all eran, 7 the powers of binomial ines (by the 54. of this. booke). c; Wherfore che onely contienfuradon of the powers doth # ore F not of neceflitie bryng forth one and the felfe famecorder of binomialllines . The elfe fame thyng alfo may be pro- C FT K ued, if the lines A B and) Z be fuppofed to bea fourth or figth binomiail line, whole powers oiely HOSdin Men hi a4ae a line commenfurable inlength vnto “AiGy theline DE: io op T ra “by the 16. of this booke), And fo thall the two lines). Butiftheline A G bein power more then he line G B by che ! ‘ofa line mmenfurable in length vate cheline A G3the line D E fhalf alld be in power mio: e then the line EZ by the iquare of'a line wncémenfiirable in length vnto the line DE} by tae ielte fame Propofiton. And fo fhall eche of the linésWB and DZ be ofthe three fait binomialt lines . But why it is not fin the third and fixt binomialllines, the téafon is : For thatin them neither O“the Names is commentrable in length co the rationall line put FC. ; dL be 68. Propofition. Whe ehaGansr nt. Awe GB by the fguar: ofa line inco q I heso. I heoreme. Aline com merfurable to a greater line is alfo a. greater. line, Dal ppofetha the lime AB bea greater line. And unto the line A Blet the line CD =" Of tbe fquares of thems ‘dded togetber yationall,and that which is contained ‘vuder thé mea dtall...And let the rest of the constenction bein this,asit wasin the former And for hates the line A B is to the lizeC D, fo- a Fist ot | 63 istheline A E tothe lise F ye the ling E Btothe line D, but Ag) | Eee. : em the line AB ts:commenurableto as | sitll the line C D by fuppofitia. Wher- G78? 2) 08 SS ae -D ble to the line GF andthe line EB to theline F D: And for thatas the line AB is to thé line C F fois the line EB to the line F D,T herfore alternately (by the 16.of the fift) as the line AE. 1s to theline EB.fois theline GF tothe line F D.Wherfare by. compojition alfo( by the 18.of the fift)as thiline A B is to the line E B,fots theline C.D 10 the line F D. Wheres fore (by the 22.0f the fict) as the (quare of thé line AB is to the [quare of the lincE B, [ois the [quare of theline CD to the fquare of the line F Di And in like fort may we prone that as the {quare of the line. A B is to the {quare of theline A E,fois the qstare of the line C D5 to the {quare of the lineC F. Wherfore (bythe 11.of the fift) as the {quare of the tyne AB is tothe [quares of the lies AE and E B, fois the [quare of the line C D tothe Squares of the 4S FS of Euclides Elementes. Fol.z79. isto the {quare of the line C D, {0 are the fquares of the lines AE and E Bio the {quares of the lines CF and F D.But the {quare of the line A Bis commenfurable tothe {quare of the line C D (for the line A Bis commenfurable tothe lineCD by [uppofitio) Wherfore alfo the [quares of the lines A E and E B are commenfurable to the fquares of the lines C.F and F D.But the {quares of the lines A E and E B are incommenfurable,and being added together are rationall. VWherfore the {quares of the lines C F and F D are incor smenfurable,ey being added together,are alfo rationall. And in like fort meay we proue that that which is contained under the lines AE andEB twife,is commen{urable to that which 4 contained under. the lines C F and F D twife. But that which is contained under the lines AE and EB twife, #8 mediall,wherfore alfo that whichis contained under the lines C F and F D twife is medial. Wherfore the lines C F and F D are incommenfurable in power hauing that which is wsade of the [quares of thems added together rationall, and that whichis contained under thé met diall Wherfore(by the 3.9. of the tenth) the whole line C D is irratiozall,¢’> is called a ored- ter line.A line therfore commenfurable to a greater line,is alfo a greater line. An other demontftration of Peter Montaureus to prove the fame. Suppofe that the line A B bea greater line,and vnto itlet the line C D be commenfurable any way, that is,either both in length and in power,or els in power onely. Then I fay that the line C D alfois a ereater line.Deuide the line A B into his partes in the point E.and let the ref of the conftruétion be in this as it was in the former.And for thatas the line A B isto the lineC D,fois the line A Eto the lyne C F,and theline EB to the line FD, therforeasthelineA Eistothelyne E = C F,fo is the line E.B to the line F D, th * but the line A B is commenfurable to the line C D.Wherforealfo the lyne¢ = D AEiscommenturable to the lyme C2 -——— ee F,and likewifetheline E B to theline F D.And for that.as the line A Eis to theline'C F,fois the line EB to the lins F:D, therfore alternately as the lyne A Eis to the line EB, fo is the line C F tothe lyne FD. Wherfore (by the 22.0f the fixt) as the {quare of the lyne A Eis to the {quare of the line E B, {0 is the fquare of the line C F to the fquare of the line F D.Wherfore by compofition(by the 18.of the fift)as that whichis made of the {quares of the lyaes A Eand EB added together is to the {quate of the lyne EB , fo is that which is made of the {quares of the lynes C F and F D added together to the fquare of the lyne F D. Wherefore by contrary sropertion as the {quare of the line E B is to chat which ismade of the {quares of thelines A Eand E Badded together,fo is the f{quare of the lyne F D ro that whichis made of the {quares of thelynes CF and F D added together. Wherfore alternately as the fquareoftthe line E Bis to the {quare of the lyne FD, fo is that which is made of the {quares of thelynes ALE and E B added togetherto that whicheis made of the fquares of the lynes C F and E D added together.But the {quare of the lyne EB is cémen- {urable to the {quare of the lyne F D,for it hath already bene proued that the lines E Band F Dare ¢é- méfurable. Wherfore that which is made of the {quares of the lines A E & EB added together is com- méfurable to that which is made of the {quares ofc P.& F Dadded together. Bur that which is made of the {quares ofthe lines A E and EB added together is rationall by fuppofitié. Wherfore that which is made of the {quares of the lynes C Fand F D added together is alfo rationall. Andas the lyne A.Ejis to the lyne EB, fo is the line C F tothelyne FD. Butas the lyne A Eis to the lyne EB, fo is the {quare of the line.A E to.the.parallélogrammecontayned vinder the lynesA'E and EB:therforeas thelyne C Fis to the lyne F D,fois the {quare of the lyne A E to the parallelogramme contayned ynder the lines AEand EB: & as the lyne C Fis to the lyne F D,fo is the {quare of the lyneC F to the parallelograme contayned vnder the lynes C F & F D.Wherfore as the fquare of the lyneA Eis to the parallelograme contained vider the lines A E and E B;fo is the fquare of the lyne C F to the parallelogramme cétay- hed vader the lynes C F and FD, Wherfore alternately asthe fquare of theline A E is to the {quare of thelynec F,fo is the parallelogramme contained ynder the lynes AE and EBtothe parallelogramme contayned viderthe lines c r andr p. But the fquare of the lyne A Eis commenfurable to the {quare of the lyne ¢ F,foritis already proued that the lynés A E and C F are comméfurable. Wherefore the parallelogrammecontayned ynder the lynes A Eand EB is commenfurable to the parallelogramme contayned vnderthelynes c F and Fp. But the parallelogramme contayned vnder thelines A Eand E B is mediall by fuppéfition.Wherfore the parallelogramme contayned vnder the lynes c rand F p al- fo is mediall.And(as it hath already bene proued) as the line A E is to the lyne EB, fois thelyne c F to the lyne FD.Buc the lyne A E was by fuppofition incommenfurablein power to the line EB. Wher- fore(by the 10. of the tenth) the lyne ¢ F is incommenturable in power to the lyne FD. Wherfore the vnc An other de~ mon stration after P.Mon- taureus. The tenth Booke jynesC Fand F Dare incommenfurable in power ,hauing that which is made of the fquares of them added together rationall,and that which is contayned ynder them mediall. Wherfore the whole lyne © Dis (by the 39.0f the tenth )a greater lyne. W herfore alyne commen{urable to a greater lyne is alfo agreater lyne ; which was required to be demonitrated. An other more briefe demonftration of the fame after Campane. Suppofe that « be a greater line, ynto which let the line z be bn cthevd com méfurable, either in length and power,or in power onely.And eH ODEN AE= + ake a rational line c Ds And vponit apply the fuperficies c 2 equall monsiration — eo the {quare of theline-a:and alfo vpé the line F x (whichis equall B after Cam- tothe rationallline cv ) apply the parallelogramine F c equall to pene. the fquare of che line 8 . And forafmuch as the fquares of the two lines A and are commenfurable by fuppofition,the fuperficies c &, fhalbe commenturable vnto. the fuperficies rie: and therefore by the ficit of the fixtand tenth of this booke, theline b & is commen- furable in length to'the line &. And forafmuchas (by the 63 . of this booke the line p & isa fourth binomiall line, therefore by the 46.0f this booke the line c gE isalfo.a fourth binomiallline : where- fore by the 97. of this booke theline s which contayneth in power che {uperfcies F G is a greater line. A. | mais © x | | URN eS CID lr S| ; D v y be ss. Dbeoreme. _- Ihe 69.Propofttion. 4 line commenfurable toa line contayning in power a rationall and ames diall:ss alfo.adine contayning in power.a rational and a medial. oS Say 5 Vppofe that AB be a line contayning in power arationall and amediall. And AI OX S| ontothe tine AB let the line CD be commen|urable,whether in leneth and power, or in power onely.T he I fay that theline C Dis a line cotayning in power Bas 4 rational & a medial. Duide the line A Binto his parts in the poynt E.Wher- Sore ( by the 40..0f the tenth ) the lines AFandEB i ave tsicorstvenfurable in power, hanine that whichis 4 E B mide of the fquares of them added together medi- | | al, and that which iscontayned under thé rationall.c Let:the fame confiruttion be in this that wasin the former And t like fort we may proue that the lines ) CF afd F D areincommenfurable in power, and that that which is made of thefquares of thelines A. E.and EB iscommenfurable to that which is made of the (quares of the lines ©Pand FD and that that alfo which is coutayned under the lines AEand E Bis comme. furable to that which is contayned under the lines C Fand FD... Wherefore that which is maae of the fquares of the lines C F and F Dis mediall,and that whichis contayned under the lines CF and F D is rationall . Wherefore the whole line C D is.a line contayning in power aratronalland a mediall:which was required tobe demonflrated. - — + ~. ——2 a or, , ¥ Sis te oh : Sri nn cee 5 ee Contraction. rr ao of tp eRe Se — = * ~ ", : ‘ ‘ x sR = on eS Se 9 97 oo ae —$——$—$$ of Euclides Elementess © _—«- Frai.28¢. Supofe that 4 8 bea line contayning in pow- a oes. p era rationall and a medial]: whereunto let the line eo ee G D be commenturable either in length and pow- er,or in poweronely. Then Lfay thar the line G D A ;—__+—_____ 8 is a line contayning in powerarationall anda me- diall.Take a rational line £ Z,vp6 which by the 45. of the firitapply a re€tangle parallelograme Ez FC equall to the iquare of the line 4.5: and ypon the line C F (which is equall to the line £2). applye the parallelogramme F C H/ equallto the fquare ofthelineG D: andlet the breadths of the fayd parallelogrammes be the lines EC and CH.And forafmuch as the line-48 is commenfurable tothe line G D at the leaitin power onely, therefore the parallelogrammes E F and F ¥¢ which are equall toitheir {quares ) thalbe commenfurable. Where- fore by the 1.0f the fixt the right lines. ¢ and C Hare c6méfurablein légth. And forafmuch as the pa- rallelogramme £ F ( whichis equall-co the {quare of the line -4.B which contayneth in power a ratio- nall and a medial) )is applyed vpon the rationall z Z,making in breadth theline E¢, therefore the line EC isa fifth binomiall line(by the 64.of this booke )vnto which line E Cthe line C H is c6méfurable in length, wherefore by the 66.of this booke the line C His alfo a fifth binomial! line. And forafmitich a3 the {uperficies C is. contayned ynder the rationall line £ Z ( thatis CF) anda fifth binomial Roa 3 therefore the line which contayneth in power the fuperficies C 7, which by fuppofition isthe lineéG D is a line contayning in power a rational] anda mediall by thes8. of this booke . A line therefore com- menturable to a line contayning in power a rationall and a mediall.&c. q Ihe sz. I heoreme. The 70. Propofition. A line commenfurable to a line contayning in power two medialls, is alfo a line contayning in power two medialls. | —. ] Vppofe that AB bealine contayning in power two medialls. And unto the line | A B let the line C D be commenfurable,whether in length cr power or in power LX > i onely . Then 1 fay, that the line C D is a line contayning in power two medialls. deuided into his partes in the point E . Wherefore (by the 41.0f the tenth ) the lines A E and E B are incommen|urable in power ,hauing that which ts made of the [quares of them added together mediall, and that alfo which is contained under. them medtall, and that which made of the fquares of the lines A E CaS 7 E B E Bis incommen{urable to that which a ——-—~+- 15 contained under the lines AE and . i 4 EB. Let the felfe fame conftruction —_ en be in this,that was in the former. And in like [ort may we proue, that the lines C F ¢ F D are incommen{urable in power,and that that which is made of the {quares of the lines .A E and EB added together,is — to that which is made of the {quares of the lines C F and F D added together, and that that alfowhich is contained under the lines A E and E B is commenf{urable to that which is con- tained under the lines C F and F D .Wherefore that which is made of the {quares of the lines C F and F D is mediall (by the Corollary of the 23. of the tenth) : and that which is contayned under the lines C F and F D is mediall (by the {ame Corollary ) : and moreouer, that which is made of the [quares of the lines C F GF D is incommenfurable to that which is contained under the lines C F and F D .Wherefore the line C D is a line containing in power two meatalls : which was required to be proued. MM..i. An An other dea monfiratio af= ter Campane, oSZ4Z8) Foralmuch as the line AB is a line contayning in power two medialls,let it be» Constrattions DemonSira- £207. c4n AlJumpt. An other de- moustration after ( 4it- eae. Thetenth Booke @ An Affum ptadded by atonranrens. That that which is made of the {quares of the lines CF and F D addd together, 1s ti commen{urable to that which is contained under the lines C F and F Dis ibus proued.F or, becaufe as that which is made of the fquares of the lines AE and EB added together is tothe Square of the line A E, fois that which is made of the [quares of the linesC F and FD ad- ded together, to the {quare of the line C F, as it was proued in the Propofitiins going before: herefore alternately, as that which is made of the {quares of AE and EB added together us to that which is made of the fquares of C F and F D added together , [ois the {quare of the line A Eto the {quare of the line C F . But before namely, in the 68. Propdition,it was pro- wed,that as the {quare of the line A E is tothe (quare of thelineC F, fois the parallelograme contained vider the lines A E and EB to the parallelogramme contained under the lines C Fand FD. Wherefore as that which is made of the { quares of the lines AE and EB is tothat which is made of the fauares of the lines C F and F’D [0 is the paralelogramme con- tained under thelines AE and EB tothe parallelogramme contained under the lines C F and FeD .Wherefore alternately,as that which is made of the {quares of the lines AE and E Bagsto the parallelogramme contained under the lines AE and EB, [ots that which w sane of the {quares of the lines CF and F D tothe parallelogramme contained under the lines CF and F D. But by [uppofition that whichis made of the {quares of the lines AE . and E B,is txcommenfurable to the parallelocramme contained under the lines AE Or EB. ? f S Whereforethat which is made of the [quares of the lines C F and F D added to. ether ,is in- ee ai : | ; ‘4 commenf{uravie tothe parallelogramme contained under the lines CF andF D : which was VEGU«TEA tO VE pi oued, “ An other demonftration after Campane. Suppofe that A B bea linecontayning in power two medialls : wherunto let theline G D be com- menfurable either in length,andin power,orin power onely . Then I fay, thatthe lise G Disa line cé- tayning in power two medialls. Let the fame conitruétion be in this, that was in th: former. And for- aimuch as the parallelogramme EF is equall tothe fquare of the line’ A B, and is applyed vpon aratio- Aalliine EZ,itmaketh the breadth EC a fixt bino- miallline, by the 6s.of this booke . And foraf{much as the parallelogrammes E F & CI(which are equall vnto the fquares of the lines AB and G D, which are fuppofed to be commenfurable)are commenfurable, therefore the lines EC and CH are commenfurable in Jength, by the firft of the fixt . But E C isa fixt bi- nomuall line : Wherefore C H alfo is a fixt binomial line, by the 66.0f this booke. And forafmuch as the fuperficies C lis contayned-vnder the rationall line > Panda fixt binomiall line C H, therefore the line which cétayneth in power the fuperficies C I,name- ly, the line GD is a line ‘contayning in power two medialls, by the 59,0f this booke .. Wherefore aline commenturable to a line contayning in power two medialls .&c. 6 +, 2 xt An Annotation. Hetherto hath bene [poken of fixe Senarys, of which the firft Semary contayneth the pro. duction of irrationall lines by compofttion : the fecond, the dinifion of then, namely, that thofe lines are in one point onely denided : the third,the finding out of binomiall lines ,of the first, I fay, the fecond, the third, the fourth, the fift, and the fixt : after thit beginneth the fourth Senary, containing the difference of irrational lines betwene them {dues . For by the ature x ~~ of Euclides.Elementess Fol.281. yatureof cuery neof the Livomigll lines are demsonflrated the diffexences of irrational lines. Thefineth catreateth of the applications of the fquaxes of eucry wrrational line,namely, what ET linesare the bheaathes of eer) Y uperficies fa applied An the fixt Senary is proved; that an line commenf{uxable to. any irrational line, 15 alfo an ixrationall lineofthe{ame ra. . pase.. Audeow [hall be (poken of the f exenth Senary wherein againeare plainly fet forth the elt. of the diffennces of the [aid lines betwene them felues, Andtheretscuenin.thofeirxationall lines, an artihwacticull px opr tionalitie .. E tothe line EF (by the 20.0f the tenth). Againe for afimuch as the fuperficies C Dis mediall, and is e- “rr or i ' | VE EE! eMart Fir? part of the firil cafe. Second part of the first cafe. The fecond cafe, First part of shefecdd cafe. Second part of the jecod cafes T he tenth Booke qual to the parallelograme H I;herfore the parallelograme H Tis alfo mediall; and 48 appli- ed vista the rational line E F , thatis, unto the lyne HG making in breadth the line H K. Wherfore the lyne H-K 1s rationall andincommenfurablein length to the line E F( by the 22 ofthe tenth.) And forafmuch as the {uperficies C D is mediall,and the [uperficies A Bis ra- tionall, therfore the [uperficies A Bis incommenfurable to the {uperficies C D.Wherfore alfa the parallelogramme E G 1sincommenfurable to the parallelocramme H I. But as the paral. lelogram:me G E isto the parallelogramme H I,fo( by the 1 of the fixt)is the line E H to-the lyne H K Wherfore (by the 10: of the tenth) the line E His incommen|urable in length to the line HK ,and they are bothrationall.Wherfore the lines EH and H K are rationall com menfurable in power onely Wherfore the whole line E Kis a binomiall line, and is. denided into his names in the poynt H. And foralmuch asthe {uperficies A Bis greater then the fu- perficiesC D but the {uperficies A Bis equallto the parallelogramme E G,and theuperficies CD to the parallelogramme H 1.Wherfore the parallelogramme E G is greater then the pa- rallelogramme H 1.Wherfore the line E His greater then the line H K. Wherfore the line E H isin power more then the line H K either by the [quare of a line commenfurablein length to the lyne E H,or by the {quare of a lyne incommen{urablein length to the lyne E H. Fir let it bein power more by the {quare of alyne come(urable in lecth unto the lineE H. Now the greater name, namely, E H is commenfurable in length to the rational line ceucn E F, as it hath already bene proned Wherfore the whole line E K is a first vinomialllyne. And thelne EF is a rational ljne.But if a fuperficies be contayned vn- pH X® der avationallline, anda firft binomiall lyne, the A C [ lyne that contayneth in power the [ame [uperficies,is | Agee | (by the 54.0f the tenth a binomiall line. Wherefore the lyne containing in power the parallelogramme E 1isabinomiallline .Whereforealfothe line contat- ning it power the fuperficies AD-is a binomiall : 24 line. a oe ee But now let the lyne E H be in power more then the line HK by the (quare of a line incommenf{urablein length tothe line EH: now the greater name that ts;EH is commenfurable in length tothe rationall line geuen E F.Wher. fore the line E K ts afourth binomiall line._And the line E F is rationall. But if-a fuperfi- cies becoutained under a rationall line and afourth binomial line, the line that containeth in power the fame {uperficies ts(by the 57. of the tenth)irrational,and is a greater line Wher fore the line which containethin power the parallelogramme E Iisa greater line. Wherefore alfothe line containing in power the {uperficies A D isa greater lyne. But now {uppofe that the [uperficies A B which is rationall,be leffethen the [uperficies € D which is mediall.Wherfore alfo the parallelogramme E G is leffe then the parallelagrame H I.Wherfore alfo the line E H is leffe then the line H K. Now the line H K isin power more then the lyne E H either by the [quare of a line comenf{urablein length to the lime K, or by the {quare of a lyne incommenfurable in length unto the lyne HK. Firk letit be in power more by the [quare of a line commenfurable in length unto HK< now the leffe name, thatisE H is commenfurable in length to the rationall line gewen E F, as it was before proued.Wherfore the whole line EK 15. fecond binomiall lime. And the line E F is a rattonall line.But if a [uperficies be contained under arationall lineand afecond binomiall Lyne,the lynethat contayneth in power the fame (uperficiests (by the 55. of the tenth) a fir bimediall line Wherfore the line which contayneth in power the parallelograme E 1-is a firs bimedtall line Wherfore alfo the line that comtaineth in power the [uperficies AD is afr bimedzall lyne. _ But now let the line H K be in power more then the line E H,by the {quare of a line im- commen oo 7S ee Er a — ss of Euchdes Elementes, Fol.2 82. tomenfurable tn len ethtothelyne H K now the leffe name, thatis; E His comen{urable ip length to the rationall lyne geuen E F.Wherfore the whole line ER isa fit binomial lyne. And thelyne E F is rationall But if a fuperficies be contayned under avationall bine, and a Sift binomiall lyne,the line that contayneth in power the fame fuperficies,2s(by the 53. of the tenth)a line containing in power a rationall and a medial. Wherefore the tyne that contay- neth in power the paralelocramme E Tis a line contayning th power arationall and a med:- all. Wherfore alfo the lyne that containeth in power the [uperficies AD ¥ 2 Lyné contayning in power a rational and amediall If therfore arationall and a medzall [nperficies beadded together,the lyne which contayneth in power the whole [uperficies,is one of thefe foure irrati- onall lines ,namely,cither a binomiallline,or a frft bimediall line,or a greater lyne,ora bine contayning in power a ra tionall and a médiall > whith wis required to ve demonftrated q The s4. Theoreme. Lhe 72. Propojition. If two mediall fuperficieces incomnienfurable the one tothe other be com pofed together : the line contayning in power the whole [uperficies is one of the twoirrationall lines remayning, namely, either a Jecond bimediall line, or a line contayning in power twomedialls. to the other be added tovether . Then I [ay, that the line which contayneth in jin power two medialls.F or the fuperficies A Bis either greater or leffe then the Za Juperficies CD for they can by.no meanes beequall, when as they areincom- menfurable). Firft let the fuperficies AB begreater then the [uperficies CD. And takearationall line EF) And E (by de 44.0f the firft) unto theline E ¥F apply the paralle- logramme E G eqnall to the fuperficies\ AByand making & ¢ in breadth the line EH sand unto the {ame line E F 3that 4s, to the tine H G, apply the paralleloeramme H T-equall tothe (uperficies CD, & makingin breadth the lineH K. And fora{much as either of thefefuperficieces A B GC D 2s medial; therefore al{o either of thefe parallelogrammes BoD. F_ “Gog EGandH lis mediall. And they are eche applied tothe rationallline EF, makingin breadth the linesE Hand Hk. Wherefore (by the 22.0f the tenth) either of thefe lines EH and H K isvationalland incommenfurable in leneth ta the dine E F «And forafmuch as the fuperficies AB is incommenfurable to the {uperficies C D, and the fuperficiés AB is equalltothe parallelogramme EG; andthe ‘[uperficies C Dto the parallelogramme HI + therefore the parallelogramme EG is incommenfurable to. the pa- rallelogramme H1. But (bythe 1. of the fixt) as the parallelogramme E G is tothe paralle- logramme H 1, fo is the line E H, tothe line HK . Wherefore (by the 10.0f the tenth) the line E His incommenfurable in tencth to theline HK. Wherefore thelines E H-and HK are rationall commen|urable in power onely. Wherfore the whole line EK is a binomial] line. And as in the former Propafition foalfoin this may tt be proued,thattheline E H is ereater then the line H K .Wherefore the line E H isin power more then the line HK, either by the {quare. of aline commenfurable in length tothe line EH, or by the (anare of a line incom- menf[urable in length to the line EH. Firft let it be greater by the [quare of a line. commen- Jurablein length unto theline EH. Now neither of thefe lines E Hand H Kis commen|t- rable in length to the rationall line geuen EF . Wherefore the whole line E K isa third bi- MM it. nomial! eine Construction. Iwo cafes in this Propoft< $70". The firft cafes The first pare of the first cafes SSS Sess eS - % oiaas ——- aaaeees - ate eee ape aan = ant = eo, > : 2 - : ae = 3 - SS ba : ‘ Aa Se" a = <= > : = Soe " = a thin - cies : i oo <= a “ > = — = = ——— 2 = —— SS - = = a Se en eee - ——- me > + ere = _ ——— = 3 x = aa A si = - j al ee BE nt - a= ear oa : eee : =: == ==" Ee —_— -- = = => S aenannatantiaae - = ~ aa - ~~ = -p SS aoe —= —— s = = — - ——_—_---— — ——-- —— SS me = ~~ — ro ar a—< . RESALE Sa = > ea Se bees Sis Ce —— = = —_— ——+* ~ — = = . = ae : < — —< : aon rine aia J. ~~~ 7 = . pana . x E : — ——- = - - -_—- — 2 paces oe ~ SSS : ~ = = SS SSE som owe F =e ———s = = x - : =~ = - = : —— or Se = e eras “ Se Ss = = 3 nnn — - — — ——= = z - ; = SS ee aaek a a SS : = a. - ———— on 227 oS ta commen —— - — a nr — ~ - - eel _ - = = ni = = = ECE a —— " ’ = n Sea wine ——— : > ~ a nS aS NS cm = gE a - — ——— —— = _ eT os SPS pe 2 ee Axett me eS 5 a ee dhe fecond Dart of the fi ft cafes efecona 2 Corollary, The tenth Booke nomial line. And the line EF is arationallline. But if a fuperficiesbe contayned under # rationall line ey a third binomial line, the live that cotaineth in power the fame fuperficiess. is (by the 56.of the tenth) a fecond bimedtall line. Wherefore the line that containeth in power the uperficies EI thatis,the {uperficies A D, 1s a fecond bimediall line . But now uppofe that the line E H be in power more then the line H K by the [quare of a line incommen{urable in length to the line EH - And foralmuch as either of thefe lines E Hand H K is incommen|urablein length to the rationall line geuen E F, therfore the line E K isa fixt binomiall line... But if a fuperficies be contained under arationall line and a fixt binomiallline, the line that containeth in power the [ame [uperficies, is (by the 59 .of the ‘tenth )ya line containing inpower two medialls Where- fore the line that containeth in. power the [uperficies.A D, EH, J is a line contayning in power two medialls . And after the : | | | felfe [ame maner, if the [uperficies A B be lefse then the {u- | SCE ee ae | perficies CD, may we proue, that the line that contayneth in power the fuperficies A D, 2s either afecond bimediall line, or a line containing in power two medialls . If there- foretwo mediall {uperficieces incommen|urable the one to the other be added together , the line contayning in power the whole fuperficies is one of the two irrational lines re- may ning namely, either alecond bimediall line, or a line cotatning in power two medialls: which was required to be proued. K aa | BD i « A Corollary following of the former Propofitions. — A binoiniallline and the other irratwnall lines following it, are neither mediall lines, nor one and the fame betwene them felues. For the {quare of a mediall line applied to a rati- onall line, maketh the breadth rationall and incommenfurale in length to the rattonall line, swherunto it is applied ( by the 22 .0f the tenth) . The {quare of a binomiall line applyed ta a vationall.line,maketh the breadth a firft binomiall line (by the 60. of the tenth). T he {quare of a firft bimediall line applied unto a rationall ine, maketh the breadth a fecond binomiall line (by the 61.0f the tenth). The {quare of a [econd bimediall line ne unto 4 rational line, maketh the breadth a third binomiall line (by the 62. of thetenth ) The {quare of 4 greater line applied to a rationall line, maketh the breadth a fourth binomiall line(by the 63. of the tenth ) . The {quare of a line comtaining in power a rationall Cy a mediall fuperficies, maketh the breadth a fift binomial line ( by the 64.0f the tenth ). And the {quare of a line containing in power two medialls, ap hed vntoarationall line, maketh the breadth os binomiall line (by the 65. of the tenth) . Seing therefore that thefe forefaid breadthes arffer both from the firft breadth, for that it is rationall, and differ alfo the one from the other,for shat they are binomials of diners orders : it is mamifeft that thofeirrationall lines differ alo the one from the other. $a Here beginneth the Senartes by fubftraction. T he 73. Propofition. If froma rationall line be taken away arationall line commenfurable in power onely to the whole line:the refidue 1s an irrationall line and is called a refiduall line. q Lhe ss. I heoreme. Suppofe Cee St TR nl ’ - 7 F — a i ee el of Euchdes iSlementes. Fol.233. AV ppofe that A B be a rat ionall line,and from AB take away a rationall line BC ‘| commenfirable in power onely-to the whole line A B.T hen I fay that the line re- WY). | mayning namely AC is irrationall and is called a refiduall line.F or ‘foralmuch ETEK! 15 the line A B is incom nen{urablein length unto the line BC , and(by the af- fumpt going before the 22. of the tenith)as the line AB is tothe line BC, fois the {quaireof the line AB tothat whichis contayned under the lines AB and BC: wherefore ( by the 10. of the tenth) the fquare of the line A Bis incommvenfurable to that which is contayned under the lines A Band BC.But unto the {quare of the line AB arecommenfura ble the (quares of the lines AB and B C(by the 1s.of the tenth ) . Wherefore the {quares of the lines A B and B C are in- commenfirable to that which is contayned under the lines AB and BC.But unto that which is contayned under the lines A B and BC is commen{urable that whichis contayned under the lines A B and BG twife. Wherefore the {quares of the ines AB and B Care income/ura- ble to that which is contayned under the lines A Band BC twife. But the {quares of the lines AB and BC are equall ta that which is contayned vader the lines AB and BC twife, and to the {quare of the line A C(by the 7.of the fecond) Wherefore that which is contayned under the lines AB and BC twife together with the {quare of the line A Cis incommenfurable to that which is cotayned under the lines A B and B C tivife.Wherefore( by the 2 part of the 16. of the teth)that which is cotayned under the lines A B and BC twife, is incomefurable to the fquare of the line A CWherefore (by the firft part of the fame,) that whichis contayned vn- der thelines A B and B C twife together with the {qware of the line AC, that is, the {quares of the lines AB and BC are incommen|furable to the {quare of the line AC . But the ‘fears of the lines AB and BC are rationall,for the lines A B and BC are put to be rationall: wher- fore the line A Cis irrational and is called arefiduall line: which was required to be proued. c B st An other demonftration after Campane. Campane demonftrateth this Propofition by a figure more briefly after . 4 c B this maner . Let the fuperficies EG be equall to the {quares of the lines A Band B C added together : which thall be rationall (for that the lines AB and BC are fuppofed to be rationall cémenfurable in power onely). Fré which fuperficies take away the fuperficies D F eo to that which | is contayned vnder the linés A B & B C twife,which shall be mediall(by the 23,0fthis booke) . Now by the 7. of the fecond, the fuperficies FG | is equallto the fquare of the line AC . And forafmuch as the fuperficies 1} EG isincommenfurable to the fuperficies D F ( for that the one is rati- onall and the other mediall ) : therefore ( by the 16. of this booke ) the fame fuperficies E Gis incommenfurable to the fuperficies F G . Wher- fore the fuperficies F G is irrationall . And therefore the line A C which contayneth it in power is irrationall ; which was required to be proued. D -Anannotation of P.cVontaureus. This Theoreme teacheth nothing els but that that portion ofthe greater name of a binomiall line which remayneth after the taking away of the leffe name from the grea- ter name isirrationall which is called a refiduall line, that is to fay, if from the greater name ofa binomialbline,which greater name is a rationall line comenfurable in power onely to the lefle name; be taken away the leffe name which felfe leffe nameis alfo com- men{urable in power onely tothe greater name (which greater name this Theoreme calleth the wholeline)thereftofthe line which remaineth 1s irrational, which he calleth . MM..Jiil. aire - The first Ses nary by {ub- ftracttone Demon ftras ti0st. An ether dés monslration after Cam- paue . Diffinttion of tie eight trra- siouall line. Demonttre- C20. Diffinition of T he tenth Booke 4 refiduailline.Wherfore all the lines which are intreated in this Theoreme, anid itt the fiue other which follow are the portions remayning «of the greater partes of the whole lines which were intreated of in the 36.37.3 8.3¢.40.41. propofitids, after the taking away the leffe part from the greater. * In this propofition is {et forth the nature of the eight kinde of irrationall lines which iscalled a refiduall line the definition whereof by this propofition is thus, A refiduall line is an irrationall line which remayneth When from arationall line geuen istaken away arationall line commenfurabletothe whole line sn power onely, a The 36. I heoreme. The:.74. Propofition. Tf froma mediall line be taken away a-mediall line commemfurable in power onely to the ‘whole tine, and comprehending together with the whole line a rationall fuperficies:the refidue is an irrationall line ,andis called a firft mediall refiduall line. aha Vppole that AB bea mediallline. And ~ | from the line AB takeaway a mediall Derren ert. - b MW ‘ “ , : a line BC commenlurable in power onely a : : tothe. whole line AB and comprehending toge- ther with the line AB arationall [uperficies , thatis,let that which is comprehended under the lines A B and BC be rationall.T hen 1 fay that the line remayning,namely,A C is irratio- pall and is called a firft medial refiduall line For forafmuch as the lines AB and BC are me- diall,therefore alfo the {quares of the lines AB and BC are mediall.But that which is con- tayned under the lines AB and BC twife is rationall. Wherefore that which is compofed of the {quares of the lines AB and BC, thatis , that which is contayned vader AB and BC twife together with the {quare of the line A C is incommen{urable to that which is contained under the lines AB and BC twife .Wherefore ( by the fecond part the 16. of the tenth) that which is contayned under the lines. A Band BC twifets incommen|urable tothe {quare of the line AC. But that which is contayned under the lines AB and BC twifetsrationall, wherefore the {quare of the line A C 1s trrationall. Wherefore alfo the line A C is irrationall: andis called a firft mediall refiduallline . I his firft mediallrefiduall lines alfo that part of the greater part of a first bimediall line , which remayneth after the taking away of the leffe part from the greater wherof tt hath allo bis name, and 1s called a firft mediallrefiduall lines which was required to be proued. Out of this propofition is taken the definition of the ninth kinde of irrationall lines, whichis called a firft-refiduall mediall line the difinition whereofis thus. A ficft refidnall mediall line is an irrationall line which remayneth , when from amediall line is the ninthirra- taken away amediall line commenfurableto the whole in power onely, andthe part taken aWway and stow all liste, the whole line.contayne a medtallfuperficies. An other demonftration after Campane. Let the line D E be rationall ,ypon which apply the fuperficies D F equall to that which is eqns ne = a ee a a er s of Euchdes Elementes. Fol.284. aed:vnder the lynes AB and B C twife,,andlet-the fuperficies ‘GE Beles) cen * qual.to that which iscompofed of the {quares of thedynesA Band BC:.. A. _ “ B , wherfore by the 7.of the fecond, the fupérficies FG is equal ro the fquaré © ofthe lyne A C.Andforalmuch as (byftippofition) the fiperficies E G is mediall, therfore(by the z2zof the tenth) the lyne D G is rationall.comens furableih. power onely.to-therational lyne DE.And forafmuchas by feR polrign the {uperficies EH is rational,therfore by the 20,o0frhe ténth,the lineD His rationalcomttienfurable in leneth ynto the rational line DE. Wherfore the lynes D G and D H arewationallicommenfurable in power only (by the affumpt put before the 13.0f this boke). Wherforeby the 73 of this boke;the lyne-G-H-is a refiduall lyne, and is therefore irrationall. Wherfore (by the corollary of the 2! Pens boke))'the fuperfictes F Gis irrational.And therfore thedine A Cowhich cétayneéth it in power is irra cionall,and is called a firlt medial reGduall tyne. pert VAs oo & RP ¢ The 57:T heoreme: Ihe 15. Propofition, If from amediall lyne betaken away a mediall lynecommenfurable in power only to the whole line sand comprehending together with the-whole lyne a medtall fuperfictes the refidue is an irrationall lyne, andis called a Second mediallrefidualllyne. CoV ppofe that AB be a mediallline,and from AB take away a mediall line CB <-> commen|urablein. power onely to the whole tine AB, and comprehending b0ge- > therwith the whole line AB a mediall {uperfictes namely, the parallelogr wig 4-3 AX) contained under.the lines:ABandBC:Then I : [4y that the refidue,namely;the linésA Cis irrationall,andis ~ A c 6 called a fecond medtall refiduall lige: Take aratiowallline D 1,and (by the 44.0f the firft) untothe lineD I apply the. pa- sia ig rallelogramme D E equall tothe (quares of the lines AB gy : | | B Cand making in bredth the line D'G. And unto the [ame line D'I apply the parallelogramme D.H equall to.thatwhich is cotained under the lines AB. BC twife, and makyngin 1 of loqnie breadth the line D F. Now the parallelogramme DH is lefce then the parallelogramme D E,for that alfo the {quare of the lines. Band BC are greater then that which is contained under the lines AB-and BC twife,by the fquare of the line’ A C by the 7.0f the fecond. Wherfore the parallelocramme remayning,namely,F E,is equalto the [quare of the line AC. And forafmuch as the Squares of the lines _A Band BC ave medi- all,therforealfo.the parallelogramme D £ is mediall,and is applied to the rationall line D Lmaking tv bvedth the line D-G.Wherfore (by the 22.0f the tenth )the line D G is rational and incommen|urable in length ta the.line D.1.Againe foralmuch as that which is contai- ned under the lines A B and BG is,mediall, therfore allo that which is contained under the lines A B and BC twife is mediall,but that. which is contained under the lines A Band BC twife is equall to the parallelogramme D H Wherfore the parallelogramme D H is mediall anda is applied to the rationall line D I making in breadth thé line D F.Wherfore thelineD F 15 rationall and tacommen|urable in length tothe line D I.And foralmuch as the lines A B and B Care comenfurable in power onely therfore the line A B is incomine/urable in legth to the line B G.Wher ‘ore (by the afjumpt going before. the 22. of the ténth, and by the 10. of thet enth the {quare of the line A B is incommen[urable to that which is contained under the lines A Bund BC. But vntothe Jquare of the line A B are commenfurable.the [quares of A Band BC (bp the t3.of the tenth And visto that which is contained Under the lines AB An other de- monfiratio af- ter Campane, Confiruttion,. Demonfira- tion, ELSA LENE, tionall.Wherfore the lines GD and DF are rationgll com. menfurable in power onely Wherforetheline F Gata refidu- all line (Ly the 73 .propofition of the tenth) And thé tine’ D E is avationallline.But a {uperficies comprehended under a rationall line,and an trrationall line is irrationall (by the 21 of the tenth andtheline which containeth in power thé fame J DEF} f wre ) Whenfore the paral Cork T be tenth Booke AB and BCis comm? [urable totlant whichis comtaimed under the tines AB and BC wife. Wherforethe fauares of the line? A B and BC dre incémmenfurableto that which 13 tontat- ned under the lines A B and BC twife.Bat untothe fquares of the lines A.B and BC is equal the parallelecrzme D E,andtothatwhich is cotained under the lines.A Band BC twife,ts equall the parallelogramuie. DH Wherefore the parallelograme D E7s income/urable tothe “ Avallelocramme D H.But as.thé parallelogramme D. Es to the parallelogramme DF Jo ie the live GD totheline D F Wherfore thelineG Dts cont men{urable in lécth tothe line DF? And either of thé i3ra- , ? la . . | cies is ivrattonall (by the afjumpt going before the leloordeme® & isivratronallBut the line-AC containeth in power she harallelogramme FEL her fore.the.line AC isan iriationall line andis called a.fecona wediall refiduall line, And this fecond medial refiduall lineis that part of the ereater part of 4b maediatl line which reHia yneth after the taking away of the leffe part from the greater: which was required to be pronca. Another demonfirtion more briefe after Campane. Suppofe tharvaw bevw mediallJine , from which take away.the me- diall. line GB comuienfyrable ynto tne whole line 4 Banpower onely aud confayning with ita mediall fupéerficies namely, that which 1s con- tayned ynder the lines 4.4 and B G. Flren I fay tharthe refidueW Gisan ~S D irrational! line,and is called afecond medialkrefiduall line, Fake ratio- | nallline DC ; vpom which apply a parallelogramme equallto thatwhich. | is campofed of the fquares of the lines 4 # and 8 G,which by the 45. of the frilerbe DCE, Agayne let thé parallelogramme APEY be équall to chat which is contayned vader the. lines 4 Band. 2. Grwile,Wherfore y f the fuperficies remaining D Fis equal to the {quare of the line 4G by. the. Sid 7 .of the fecond . ( Forthat which 1s contayned vnder the lines 4 and BG twife together with the {quareof the line4G 1s equallto that which is compofed of the {quares of the lines4.8 and B.G:) And forafmuch as the fquares of the lines 4 4 and B Garé medial , for chat they are defcri- bed of medial lines: the parllelograme D £ which 1s equal’vnto thé; fhall aiforbe medialli And forafmuch as that whichis cétained ynder thelines # B.and B.Gis by fuppofition médiall therfore thef{uperficies 2.2 which is double vnto itis alfo mediall. But the mediall fuperficiesD. £.excédeth — r Hor the mediall fuperficies 2 £ by a rational {uperficies( by the 26.0f this booke). Wherfore the exceffe, namely the fnperfities DF is irrationall,vnto\which the fquare of the line 4 G is equal]: wherefore the fquare'of the line 4G is irracionall,and therefore the line 4 G which contayneth it in power Is irratio= nall by the afiumpt put before the 21.ofthis booke,and is called a fecond mediall refidualbline. el OE eee Rt A®his propofitien fetteth forth the nattire of the tenth kinde ofirrationallines, which iscailed afecondrefiduall medialllinegwhich. isthus:defined. ) 3S OH fecond reftduall mediall lyne is an irtationall lyne Which remayneth,Wwhen from amedsal line istakemaw zy amediall lyne Commenfirrable to the Whole sw power onely and Phe part taken away & the whole tyne contayne'a medial faperficies: q Lhe 58, I beoreme... te Dede Propofition. If fiomaricht lineberaken awaya right ime tncommenfurable in power to the whole; and if that which ts made of the Jquares of the whole line and of the line taken away added together be rationall, and the.parallelograme cone ——=- - ee TE EEL Te of E uclides Eslementes. Fol.285, contained ‘vnder the fame lines mediall : the line remayning is irrational, and is called a leffe line. | (ENS EB OO SSA | : VY S"| which is compafed of the [quares of the lines AB and BC be rationall, and let Seed the parallelogramme contayned vader the fame lines AB and BC be mediall. Then I fay,that the line remayning, namely, the line-A Cisirrationall , c is called alefeliné.For” . | forafmuch as that which is copofed of the {quares = e . of the lines AB and BC ts rationall , and that which is contayned under the lines AB and BC twife is mediall , therefore that which is compofed ofthe [quares.o the lines A Band BC is incomsmenfurable to that which is con- tained under the lines AB and BC twife. Wherefore the [quares of the lines AB and BC, are incomenfurable to the [quare of A C,as it was faid in the 73.Propofition . But that which is made of the {quares of the lines AB and B Cis rationall .Wherefore the [quare of the line AC is irrationall:- wherefore alfo the line AC is irrationall ; and is called a leffe line. Andis therefore [0 called; for that itis that portio of the greater part of a greater line,which remai- neth after the taking away of the lefve part fro the greater : which was required to be proued. In thys Propofition iscontayned the definition of the-eleuenth kinde of irratio- nall lines, whichis called aleffe line, whofe definition is thus. eA lefve line is an irrationall line which remayneth, whe from aright line is taken away aright line incommenfurable. 1n power to the whole, and the {quare of the whole line,e the Square of the part taken away added together, make aratsonall {uperfictes, and the parallelogramme contayned of them 45 medial. This Propofition may after Campanzes way be demonftrated, ifyou remember well the order & po- fitions which he in the three former Propofitions vied, | q he 59. T’heoreme. Ihe 77. Propofition. If from.aright line be taken away aright line incommenfurable in power tothe whole line ,and sf that which 1s made of the {quares of the whole line and of the line taken away added together be mediall , and the paral- lelogramme contained bnder the fame lines rationall ; the line remaining is irPationall, and 1s called a line making ‘with a rationall fuperficies the whole fuperficies medial. V ppofe that A B be aright line, and from the right line AB take away a right line B C incommen{urable in power to the whole line AB, and let that which is ‘made of the fquares of the lines A B and BC added together ,be mediall, and the %| parallelogramme contained under the fame lines rationall .T hen I fay,that the line remayning, namely, the line AC, ts irratio- nall, and is called a line making with arationall Sars eae mee fuperficies the whole > gee mediall . For for- almuch as that which is made of the [quares of the lines AB and BC added tovether'is mediall,and that which is contained under the = AB an: (2M teh Vppofe that AB be a right line, and from the right line AB take away aricht eG. | line BC inco munen{uravle in power tothe whole line,namely,to A Band let that a e a ap oe j JEWGI STP Zo ' tiOt. Diffinstion 6 £ the eleneth ire ratsonall lines Demon ftra- 10%. Diffenition of the twelueth ir T he tenth Booke A Band B Ctwife is rational, therefore that which is made of the fquares of the lines AB and B Cadded together, is incommen|{urable to that which is contained under the lines A B and BC twife.. Wherefore (by the 16.0f the tenth) the refidue,namely, the [quare of the line ACis incommen{urable to that which is contained under the lines: AB and BC twife.But that which is contained under the lines A B and B C twifess rationall. Wherfore the [quare of the line AC is irrationall . Wherefore alfo the line A C is irrationall : and is called a line making with a rationall {uperficies the whole {u- perficies mediall : and is therfore [0 called for that that which is made of the (quares of the lines AB and B C added together is mediall,cy is a certaine whole [uperficies, part whereof 1s that which is contained under the lines AB cy B C,which is arationall [uperficies. For the {quares of the lines A Band BC, are equall to that which is contained under the lines AB and BC twife, andto the [quare of the line AC (by the 7 .of the fecond ) . Or itis therefore focalled for that the {quare thereof added to a rationall [u- perficies, maketh the whole [uperficies mediall, as fhall be proued by the tog. Propofition of this booke : which mas required to be proued. A, re B In this Propofition is declared the nature of the twelueth kind of irrationall lines, which is called a Jine making with a rationall fuperficies the whole fuperficies mediall, whofe definitionis thus. of line making with a rational fuperficies the Whole fuperficies medsall, 4s ar srrationall line which remazineth, whi fro aright line is taken away aright line incomenfurable in power to the Whole line, and the fquare of the Whole line & the fquare of the part taken away added together make ame- diall fuperficies and the paralelogramme contained of them is rational. This Propofition alfo may after Campanes way be demonftrated, obferuing the former caution. q Ihe 60.Theoreme. —- The 78. Propojition. If from aright line be taken away a right line incommenfurable in power to the whole line , and if that ‘which is made of the {quares of the whole line and of the line taken away added together be medial,and the parallelo gramme contayned ynder the fame lines be alfo mediall , and incommene furable to that ~which is made of the /quares of the fayd lines added toges ther:the line remayning is irrational , andis called a line making witha mediall fuperficies the twhole fuperficies medial. 3 Vppofe that AB bea right line, 5 SEE eS ES a ae == Se — = - = - ; — === - = —— An ASumpe of Campane. 1. Dee, Though Cam- panes lemma be truc,yct the maner of de- mon (irating at,(uarrowly confidered)zs aot artijiciall. Second Sena- rye Demon jlrati- on leadsng to an im pofsibs- bstie. The tenth Booke If there be fower quarities, & if the difference of the firft to rhe fecond, be as the difference of the third tothe fourth then alternately as the difference of the firft iste thethird , fo is the difference of the fecond to the fourth. | Thisis to be ynderftand of quatities in like fort referred the one to the other,thar is if the firft be greater then the fecond,the third ought to be greater then the fourth and ifthe firft beleffe then the fe- cond,the third ought to beleffe then the fourth: and is alfo to be vnderftand in arithmeticiall propor- tionality.As for example let the difference of a be vnto z as the difference of c isto p . Then I fay that as the difference of ais to C,fo is the difference of s to p. For (by this common fétence,the difference of the extreames is compofed of the differences: of the ex- treames to the meanes), the difference of a to c is compo- fed of the difference of a tos and of the difference of s to c. And (by thefame common fentence ) the difference of z to D iscompofed of the difference ofz to c , and of rhe difference of cto p*. And forafmuch as (( by fuppofition) the difference of ato z isas the difference ofc to p,ahd the difference of 8 to c is common to them both. Where- fore it followeth,that as the difference of ais to c,fois the difference of s to >: which was required to be proued. ¢ Lhe 61.I heoreme. Lhe 79. Propofition. Vato a refidual line can he ioyned one onely right lye rational, and come menfurable in power onely tothe-~whole lyne. y= Et AB bea refiduall line, and unto it let the line BC be fuppofed to be ioyned, fo ae that let the lines AC and BC berationall commen{urablein power onely. Then 1 ee (ay that unto the line AB cannot be ioyned any other rationall line commenfura- ble in vower onely tothe whole line.F or if it be poffible, let BD be fuch aline aa- ded unto it. Wherfore the lines A D and DB are rational commenfurable in power onely. A Bp > And fora{much ashow much the {quares of tee the lines A D and DB do exceede that which js Contained under thelines AD and D B twife,fo much alfo do the fquares of the lines AC and CB exceede that which is contained under the lines AC and CB twife,for the exceffe of eche is oneand the [ame,namely,the {quare of the line A B( by the 7. of the fecond.)Wher- fore alterisately (by the former affumpt of Campanus) how much the {quares of the lines A D-and D B doexceede the [quares of the lines AC & C B,fo much ‘Ne excedeth that which is contayned under the lines A D:and DB twife, that whith is contained under the lives AC and CB twife. But that which is made of the fquares of the lines AD and DB added together , exceedeth that which is made of the [quares of the lynes AC Ana C:Badded together by a rationall {uperficies : (for they.are either of them rational). Where- fare that which is contained under the lines AD and DB twife, exceedeth that which is contained under the lines AC and C B twife by arationall fuperficies. Butthat which is contained under the lines A D and D B twie, 1s mediall, for it is commenfurable to that whichis contained Under thelines AD and DB once, which fuperficiesis mediall (by the 21.0fthetenth awd by the [ame'reafon alfothat which is contamed Under the lynes AC and CB twife is mediall. Wherfore a mediall [uperficies differeth froma medtall fuperficies by a rationall [uper ficies which (by the 26 of the tenth)is smpo/fible. Wherfore unto the lyne Abs cannot be ioyned any other rationall line befides BC commenfurable a power onely, to the whole line Wherfore vate a refiduallline can betoyned one onely right line rationall and commenfurable in power onely to the whole lyne : which was required to be demonftrated. q The ———— * . arr? SST ———————— —— of Euchides Elementes. Fol.287. ¢ Ihe 62. IF heoveme. The 8. Propojition. Vnto a firft medial reftduall line can be toyned one onely medial! right byne, commenurable in power onely to the whole ine and comprebendyng wyth the whole lyne a rationall fuperficies. Cx V ppojeihat A Bbéa first mediallrefiduall line, & unto.AB toyne the Lyne BCs [a Ba that ler she lynes AC and BC be mediall commen|{urable in power onely cp let that ee which ts contained Under the ines AC avd BC berationall. Then t [ay that unio the line AB cannot be ioyned any other mediall lize c mmen{urablein power onely t0.the whole lyne,and com prehending together with the whole. lyne arationall {uperficies. For if it be poflible let the line B D be fach atine.Wherfore the lymes 4 D and D.B are mediall come menf{urable inp ower onely and that whichis con- tayned unde ‘the lines A Dand D Bis rational. R r P And forafmuch as how much the (quaresof the... lynes A D and D B exceede that whichis contay. | ned under the lynes A D and D_B twife,fo much alfo exceede the [quares of the lynes AC & B C,that which ts contayned under the lynes A Cand CB twife(for the exceffe of eche is one J and the fame, namely, the {quare of the lyne A B).Wherfore alternately(as it was fayd in the former propofition how much the {quares of the lynes A D.and D Bexceede the {quares of the lines.A C and C B,fo much allo that whith is contained under the lines \AD-and DB twife,excedeth that which is contained under the lines AC and CB wife.But that which is contained under the lines A D and D B twife,excedeth that whichis contained vuder the lines A C and C B twife by a rationall [uperficies, for they are either of them a rationall {y- perficies. Wherfore that which ismade of the {quares of the lines.A D cy DB excedeth that which is made of the {quares of the lines AC. cy C B.by a rational [uperficies which (by the 26.0f the tenth) as impoffible.F or they are either of them mediall ( for thofe foure lines were put to be mediall.) Wherfore vnta.afirft medial refiduall line can.be ioyned onely one right mediall line commen{urablein power onely to. the wholeline, and comprehending withthe whole line arationall {uperficies : which was required to.be proued. q Lhe 63. I heoreme. The 81. Propo/ition. Vnto afecond medial refiduall line can be ioyned onely one mediall right line, commenfurable in power onely to the “whole line, and comprehending with the whole line a mediall fuperficies. A Vppofethat A B be a fecod mediallre~ 4 2 cD BSS (iaual line,gy unto the line A Bioyne _' oe eeez'' the line BC, fo that let the lines AC and C B be mediall comenfurable in power one-. | by,and let that which is comprehended under | | thelines A Gand C B be mediall. Then I fay, | that unto the line A B can not be ioyned any 0- ther mediall right line comenfurable in power onely to the whole line, and comprehending to- gether with the whole line a medial {uperficies. For if; it be pofsble, let the line BD be fuch A line . Wherefore the lines A D cy DB are me- | diall commenf{urable in pomer Pee angie oe GN eG tk we 2 eet eee Oh AR SE TT RD Re Demon Zra- tion leading ta an abj wraitice.. ae Construction. Demon fira- tion léaling to an abfurditie, erdgnme E 'G,and v vutothat which. is contai- 7 betenth PB ooke which 1s contained under the lines CAD and DB is alo mediall . Take 4 rationall lime EF . Ana ( by the 44 of the fife ) unto the line E i: Appi y th be parallelogram Hho EG equall fo the {quares of the lines CAG and C B, 17 xd 113 aking 172 breadth th 1e line J EM = and from.that paral lelog?r ila EG take away the pears mme HG e- | qual to that which is contained under AC ana CB trife , an . making in breadth the line HM. Wherefore the paral agra remaynneg , namely , E age ( by the a of t be fet cond) eqtall to-th te fe juare of the line AB .W vole moe, aR co ei te y power ti: ee ara Helocram we EL. Againe,vnto the line EF wie? y (by the 44.0f th e firfh) the paralleloeramme E 1 equall to th ¢ fquares of thelincs A Dard D B, sibel “Airs breadth the € line EN Bw the (o VHA ES of the e liges AD ana D B are eahe ul to that whit Ai 1S COMtAL« ned Vader the linés A Dard DP Btn pile; and to the {a quare oj ‘the lin e AB. Where fore’ the parallel pevdmme ED aseyuall to that which 1s Re a unter t he ts pes A os we B tif l€> ana l0 th re ‘[quare OF | the re line AB. But the e parallelogr: mime E Lis equall to the [quare of t the ; ad ; ad an f ra, 1x isn gtnars +h j f , 7 . line “EDV IEE, 0} ae L parallelograne Me VEMAMNING, (AIR ely, ¥7/1,7 1S OG bes ht lLothat which ts ¢ - * ; : ie ae ae o . ; } us ak ‘ni Sat A [4A f * 547 77 ~* + 2 sv) ; COVA aA UZZACT tLPe ES A D ARGaA DB ti vt fa | y 4nd for Li? mud a as tp Ship a a a AHGL. b are nn . Hi . “4 j / ; ‘ es 7 "> | J ; p> . f L, | -s “~s aw f a’ at 7 i M “ . ’ } J i PIPE AEALL 5 EFEX ET OTE &. be iG hk of t: GE LEPES 20 4; nd CoD ae w7e4iall + a7 A they are eaua if : J Jd + ge . Eee Fi “ure + eres - A A r a 24 fi ; j f\ Saar 275 A/F 7 oe fb $> y r) as i 5! = é oe Eee ree a ea ; ‘ 4 . : : 7 : ° ; " 4 : ben inthe 75. Lrevohtiow).me dial rynditisapolied vate th vations tine ER epabine se vir Tw) f iB 7 ; Tre erre vy ‘set i, . > b is F 1;2 j 1 Sy ft. =e yy 1 ae J we \3 . ae ? ead PEGE HEL iv. Wh a fy) 4 ( by the we. OF the tenth ) De 4 line EM is YATi ag af 4% | Pore F J , be Roe ers f. ViPtrels AMG yf pg Sea in wisDR tothe line EP 2 Acaine) fora for a sthat which 1s contayned pediall, the orefore ( (oy the Corollary of t be ae af the tenth) | 7 | the lie ACuand CB twife 7. alfa mediall: avd itis eauall 0 sie seas a 8. et? G : wher efore all athe parallelogranme H G is ahs yi and is, tp. | n, L lin€@E PR. making in Laskey 2s ieee HM. Wherefore ( by the 22 of thet te th) aebe; iit te. Hd rite} ts abba and incommen nvable iz eneth totheline EF. And fo cafmuch 1s the liges A cand B are commen|furable in poser onely therefore the line AC iS incommenfure ibleta lenath to theline CB. But as theline 4Cistothe line CB ,fo (4 the Affiimpt going beforethe 2 2. of the tenth ) 7s the [quart of the line AC tothat which is cons tayned Under the lines AC & CB. Wherefore (by the r0.0f the tenth) the '[quare of the line AC ts incom bie bs ‘able to il which is can- tained under the lines AC andC B. But vnto the iy 7 ah f theline. AC are comipenfurable the {anaes of AC. CB,and vate that which is cantained under the lines AC and Byis conmen|urableth at whichis con tained Under the lin. es A ¢ ana CB tmife . Wy berefore the f qua res 5 of the lines AC CC B areincommen- f vider the J > . ‘ “+ a a 2 | sneer the lies AC ave CB is 4 PJ Vitrid dA w/4 tes sete Xs 447 6S (urable ta that wae 1s contained lines AC ana CB iwi} . Bat vanto the li ‘jquares ef thet Lines AC anaCB is equ ill the ‘pa rallelo- ned 2 under the lines AC & C E twife,is equalt PR CeERGe PRS 3 the d pasa Ai lelogramme GH. Where fore the pa- : | . 3 yallelogramme E Gis incom :men{urable to the par -allelocramme HG. But asthe paral. llelo- gramine E Gis to the parallelogy mme H G,fo ts the line EM w the line H M.Wherefore the ling E Mis incommen, furable in length to theline HM. Ant they are both risdonall lines. if VLerefore the lines EM and.M H are rationall commienfiralle jn dene oncly . Wherefore nek i isa refiduall line , and untositds joyned A rational line I: fei com menf{urable ip porber onely to the whole line EM. Intike fort allo may it be jroued, that unto the lint EH ; szoyned the line eH N, being allo rationall, ava commen|t able le in power onely to ti hewhole e ed he ee of Evclides Elementess Fol.288. lineEN. Wherefore vntoirefiduall line isioyned more then one opely line commenfuras ble in power onely to the whee line : which (bythe 7 9.0f thetenth) isimpofable . Wherefore wntoafecond mediall refidell line can be toyned onely one mediall right line commenfura- ble ix power oxely to the whde line, and comprehending with the whole line a mediall {uper- feces: which was requiredo be demonstrated. q The 64.K heoreme. Lhe é2z. Propofition. Vnto a leffe line carbe igyned onely one right line mcommenfurablein power to the -~wholeline, ‘and making together ~with the -whole byne that which is made of treir [quares added together rationall,and that which is contayned vnder them mediall. LV ppo[ethat 3 bea leffeline,and to AB ioynethelineBC, fothatle BC be uch aline asivequired inthe T heoreme.Wherfore the lines A C and C Bare \scomen|urabe 1 power ,haning that which ra) made of the {quares of them ad- ded together raionall,and that which is contained under them mediall.T hen 1 [ay that vuto.AB cannotheiayned any other [uch right lingF or tf it be joffible, letthelyne’ sees a +3 B.D.befuchaline. Wherfor thedines ADO DB are incommens. urablein power, hauing 3 | that which is made of the [qzares of them added together,rationall, and that which is con- tained under them: mediall: Aud for that how much the fquares of the lines.AD and DB excede the [quares Hi the lias AC and C B fomuch that which is contained under the lines A D and D B twife,excedeb that which is contained under the lines AC andG B twife( by thofethines whith were [paren in the 79 Propofition) But that which is made of the {quares of the lines A D-and D B alded tocether excedeth that which is made of the fe quares of the tines. A Cana GB added torether by a rationall {uperficies,for they are either of them ratio- pall by [uppofition.Wherfow that which is contained under the lines AD and DB twife, excedeth that which ts contsined under the lines A Cand C Btwife by a rationall fuperfi- cies: which-(by the 26.0f thetenth) is impoffible,for ether of them is mediall by {uppofition. Wherfore vato aleffe line cin be ioyned onely one richt line incommenfurable 72 power to the whole line,and making ogether with the whole line that which is made. of their {quares added together rationall,and that which is contained under them medial : which was re- quired t0 be demon|irated. | q The 65.I beoreme. The 83. Propo/ition. Vito a line makag with a rationall [uperficies the whole fuperficies. mee diall,can be toyned onely one right hneincommenfurable-in power to the whole lyne,and making together withthe whole kine that -which is made of their {quares acded together medialland that which is contained ynder them rationall, ie ppofe that A Blea line making with 4trationall fuperficies the whole fuperficies soy meaiall,and vntiit let the line BC be ioyned, {0 that let BC be [uch a line as is re- ‘% N Naty. power \e quired in the T horeme Wherfore the lines AC and C B are incommen{arable in Demon ra- tion leading to an abfurdstite Demon/flratio beading toa smapolfibilirie. T hetenth Looke power, hauing that which is made of thefauares of thelines AC and CB added tocether mediall, and that which is contained under the lynes AC and Brationall. Then fay that vnto the lyne AB cannot be toyned any other [uch line. For if'it be poffible, let the line B D be fuch alive. Wherfore the lines A D and DB are incommenfurible in power hauing that which is made of the [quares of the lines A D and D Badded tocetver mediall,and that which is contained under the ines A D and D B rationall.Now for that how much the {quares of thelines A Dand DB exceedethe {quares ofthe lines AC andC B, fomuch that which is contained vader the lines AD and DB tmifeexceedeth that which is contayned under the lynes AC and CB twife, by that which was {poken in the 79. propofition. But that which ts contained vader the lines -1.D amd DB twi'e , exteedeth that which is contained under the lines AC and C B twife by a rationall aprficies, for they are either of them ratio- nall by fuppofitio. Wherfore that which is made of the [quares of the lines A D and D B.ad- ded together excedeth that which 1s nade of the fymres of the lines AC and C B added to- gether by a ratiomall (uperficies, which by the 26 of the tenth is impoffible,for they are either of them mediall by fuppofition Wherfore unto the live AB cannot be joyned any other lyne hofj : A B c D eee belies BC making that which is required in the propofition: Wherfore unto a line making with 4 rationall {uperfictes the whole [uperficies medial can be ioyned onely one right line in- comen{urable tn power to the whole lineyand makinetocerher with the whole line that which is mace of their {quares added together mediall, and that which is cotained under them ra tionall : which was required to be proued. q Lhe 66. T heoreme. ’ The 84. Propofition. Vato a line making with a medial fuperficies the whole fuperficies medial, can Le oyned onely one right line incomnen[urablein power to the whole line ,and making together with the whole line that which is made of _ their [quares. added together mediall, end that whith is contained ‘onder thent medtall ,and moreoner making thit which is made of the ‘[quares of them added together mcommenfurabl to that which is contayned bne Peart om dey them. “h ws rN, i Vppofe that AB be a line making with amediall sit the whole fuperficies NIRA] uediall,and unto it let the line BC be nyned, fo that let BC be [uch a line asis OS vs required in theT heoreme . Wherefore the lines AC andC B are incommen|s- SaNSZONS rable in power ,hanine that which is mide of the {quares of the lines AC ana CB added together mediall,¢> that which 4s contained under thelizes AC and CB ediall , and moreoner that which is wide of the fauares of the lines AC ana CB: is incommenfurable to that which is ‘contained under the lines AC ana CB . Then I [ays that unto. the | line AB can be ioyned no other fuchline. | For if it be pofiible, let BD be fuch a line. Wherefore the lines A D.and DB arein- comen{urable in power, haning that which ismade of the fquares of the lines A.D and cee ; | DB ¥ ym o a A. ages 8 g D rn meee et ie He At < —_ ee bees — : = a ee ae ee Soe eerie a tn et aa eee ee es eS ee t we ae of Euchdes Elementess Fol.289, DB added tovether mediall, and hat which is contained ‘ander the lines AD and D B medial, and mertouer that nhich is made of the fquares of the lines ned E C,be the line ED : let moreouer. the fquare of the whole line E D, excede the fquare of theline adioyned E C by the {quare of the line F, which line Fletbe commenfurable in length to the whole lyne ED,and let the whole line E D be alfo commenfurable in length to the rationall line A B: then is a ¥ See eee came ene ont Sen, eee See ET aaaneeee Aen ates eee ‘the refiduall lyne C Diby this definition afirit refiduall line. A fecond refidualline is when the {quare of the whole excedeth the a of the line adioyned by the fquare of a line commenfurable dntoit in légth, and alfo the line adioyned is commenfurable in len eth to the rationall lyne. Asfuppofe the line C D to be a refiduall, and let the line adioy- ned vnto it be EC, and the whole made of them both, let be the line ED: & let the fquare of E D the.whole line excede the {quare of the lyne adioyned EC, by the fquare of the lyne F, and letthe g lyne F be comment{urable in length to the whole lyne ED,moreo- * uerlet the line adioyned E:C be commenfurable in length to the D ey rational line A B:then by this definition,the refiduall line C D is afecond refiduall line, A third refiduall ine 1s,,hen the [quare of the whole excedeth the fquare of the lyne adioyned ,by the {quare of a line commenfurablevnto st in legth and neither the whole line nor the line adtoyned is comenfurable in length to the rationall lyne. . | As(the former fuppofition ftanding){uppofe that the {quare of the whole lyne E D exceede the fquare of the lyneadioyned_ EC by the 2 {quare of the lyne F, and let the lyne F be commenfurable in length men Cc 7 B tine adioyned E C be commenturable in length to the rationalllyne § “= ---— A.B,then by this definition the refiduall lyne CD isa third refidu- if EuchdesElementess Fol.29¢. | RS OY ae YS oe f hal iy, Loksac kas hinie: Pa ees A fourth refiduall line is, when chefquare of the whole Wyne excedéth the Fourth diff fquare of the Lyne adioyned by the}quare of a bine mcommenfurable ynto nition, it in length, and the whole lyneis alfa nenfurable intength tothe r Lin engl, dnd fhe Whole tyne ts aij comumenfura Cin engl tothe rde . s s . tionall Lyne. Asthe refiduall lyne A B oe ee a rp fr beyng as beforeC D,& Nga > . thelyne adioynedEC, = + a & 7 d c R {~ Bone ae pee fee Rt AES eae eee ee OT andthe whole ED, ler | ak Sean AE ee thefquare ofthe whole © lyne ED exceede the {quare of the line adioyned E.C by the fquare of the lyne F,and lec thelyne F be incommenfurable in length to the wholeline E D,andlet E D the whole line be commenfirablé in length to the rationall lyne AB,thenis the refiduall line C D by this declaration’a fourth refiduall lyne: A fiueth refiduall ineis when the fquare of the-whole lyne exceedeth the pink js ~ - : ‘ trth fz ‘24; < fquare of the Lyne adtoyned ,by the fquare ofa lyneincommenfurable Ynto wen,” : it in length, and the lyne adioyned ts commenfurable in length to the raz tionall lyne. As the refiduall line beyng. c p, thelyneadioyned £ c, and the whole lyne E D, let the fquare of the whole lyne Ep exceede cn the fquare of thelineadioyned £ c by the {quare ofthe lyne F, and) 24+ E Jet the line r be incomment{urable.in length co the whole lynes pv, and Jet alfo g c the lyne adioyned be commenturable in length: to E.¥—_—_—_____. the rationall linea g,then fhall the refiduall c p be by this defini- tion a fifth refiduall lyne. A fixthrefiduall line 1s-when the fquare of the-whole line, exceedeth the sive diffini~ fquare of the line adtoyned ,by the {quare of a line incommenfurablebnto it tion, indlength, and neither the whole line nor the line adioyned 1s commenus rable in length to the rationall line. i Asfuppofe the refiduall line to bec p, and thelyne adioyned to be x c,andthe whole lyne compofed of them let bez. p, and let the p c {quare of the whole lync & » exceede the fquare ofitheline adioynied | 9S > by the {quare of thelyne r,which line let be incommenfurable in Fo. length to the wholelyne x p : moreouer let neitherthe,whole lyne

fe quare of the line BG to the {quare of the line G C that proportion that a fquare number hath to a fquare number Vi herfore(by the 9.of the tenth the tine B G is incommen{urablein length to the line GC.And they are both rationall Wherefore the lixes BG and GC arerational commenfurablein power onely Wherefore the line B Cis a refiduall line.I [ay moreouer that itisa firfirefiduall live, For foralmuch as the [quare.of the line BG 1s ereater then the {qGuare of the line G C (that it 15 ereater tt 1s manifest , for b y fuppofi tion the {a uare of the line B Gis to the {quare of the line G C,as the greater number, namely,E. D is to the num- ber D F)vnto the {quare of the line BG let the (quares of the lines GC andH be guall. And for that as the number D Eas to the number DF, fois the (quare of the lineB G to the fquare of the line <3 C, therefore by conuerfion of proportion ( by the correl ary of the 9.of the fifthyas the nui ber D Eis tothenumber EF, foisthe fquare of the lineB G to the fquare of the line\1. But the number DE hath tothe number EF that proportion that af{quare number hath toa {quare number,for either of them ts a [guare number , wherefore alfothe {quare of the line BG hath to the {quare of the lineH that proportion that a {quare UNI ber hath to a {quare number . Wherefore the line G B is commenfurable in length to the line H. Wherefore tbe live GB isin power sore then the line G C by the fquare.of a line com- men{urable in length to the line G.B : and the whole line, namely,G B iscommenfurable in length to the rational line A.Wherefore the lineB C is a firft refiduall line Wherefore there is founde out a firft refiduall line:-which was required to be done. Lhe 36. Propofition. q The 20. Probleme. To finde out a fecond refiduall line. | Ake a rational line, and let the fame be A,and unto it let the line-GC be com- P=) @ a wacnfurablein length idnd taketwofquare numbers D E and E-F sand let thens . ) aN (be [uch that the exceffe of the greater, namely, D F, be no {quare yumber. And SAE jas the number D F is to thenumber D E fo let the [quare of the line GC be to the Square of the line G B.Wherefore both the {quares are commenfu- — vable. And forafmuch asthe {quare of the line GC ts rational, A, therefore the (quare of the line BG isalfo rationall : Wherefore , . alfa the line B Gis vationall . And fora{much as the fquavesofihe © linesBG & GC haue not that proportion the one to the other 2 afquare number hath to.afquare.number therefore. the lines BG and GC aretncommenfurable in length, and they are both ratio- D .....F wii 8 wall. Wherefore the lines B Gand G Care rational cotmmenfura- —s lll it ae of Euchdes Elementes. Fol.291. ble in power onely. Whereforethe line BC is arefiduall line: Dfapmoreouer, that itis 2 fe- cond refiaval line.For forafmuch as the {quare of the line B-G is greater thé the {quare of the lineG C, unto the fquare of the line BG let the{quares of the linesG CG Hbe equall. And for that as the number D E%s to thenuber DF fo ts the [quare of the line G B to the [quare of the line G C, therefore (by conuerfion of proportion) asthe number D Eis tothe number E F, fois the {quare of the line BG to the {quare of theline H.. But either of thefe numbers D Eand£ F is.afquare number . Wherefore thelineG Bis commen|{urable in length to the line H. Wherefore the line BG isin power wiorethen the line GC, by the [quare of a line commen{urableinteneth tothe line BG : andthelineG C thatis 1oyned to the refiduall line 15 commenfuratle in length to the rationall line A: Wherefore the line BC is afecond refi duallline . Wherefore there is found out afecond refiduall line : which was required to be done . q Ihe 21.Probleme. Lhe 87. Propofition. Lo finde out a third refiduall line. ays Ake a rationall line, c& let the famebe.A: and take three numbers E,BC, and | ay C.D, not hauing the one to the other that proportion that afquare niaber hath 1S Dd to.a{quare number: and let the number BC haue to the number B.D. that AWN j proportion that a{quare number hathto a {quare number . And let the num- eoxte..2° ber BC be creater then the number C D . And as the number E is to the num- ber BC, fo let the {quare of the line A be to the fquare of the line F G: andasthenumber BC is to the number C D, fo let the {quare of the line F G betothe {quareof the line H G . Wherefore the [quare of the line A is com- Rigs Bs sou menfurable to the {quare of the line F G . But the [quare x of the line Ais rationall s Wherefore alfo the {quare of ~~ theline FG is rationall -whereforethelineF G is alfo rational ..And foralmuch as thenumber E hath notto Ev. +00s- the number BC that proportionthat a {quare number hath to a {quare number, thereforeneither alfo hath the Pee aD a0. Cc {quare of the line A to the fquare of theline FGthat —. proportion that a{quare number hath toafquarenumber . Wherefore the line A is incom- men{urablein length to theline F G. Againe for that as thenumber BC is to thenumber C D, fois the {quare of the line F G to the [quare of the line H G, therefore the [quare of the line F Gis commen{urableto the {quare of thelineH G. But the {quareof the line F Gis ra- tionall .Wherefore alfo the {quareof the line H G isrationall. Wherefore alfotheline H G is rational, And for thatthenumber BC hath not tothe number CD, that proportion that afquare number hath toa{quare number, therefore neither alfo hath the {quare of the line F Gtoshe (yuare of the ine HG; that proportion that a {quare auber hath to a fquare num- ber Wherefore the line FG isincommenfurableinteneth tothe line H G: and they are both vationall.Wherefore the lines E G ¢& HG arevationall comenfurable in power onely. Wher- fore the line F i 1s a refiduall tine .I {aymoreoner, that itis athird refiduall line. For for shat as the number E is tothe number BC, fois the fquare of theline A tothe {quare of the line F Gs and as thenumber BC is tothe number C D, foisthe fquare of the line F Gtothe fqeare of the line H G : therefore by equalitie of proportion, asthe anmber E isto the num- ber C D, foisthe {quareofthe line A to the {quare of the line HG : but the number E bath pit tothe number CD thatproportion that a {quare number hath to a{quare number, — ‘97e Construtltone Demonfira- $70M. Confiructione Denenflra- £20. The tenth Booke fore neither allohath the [quare of the line A to the {quare of the line H GC that proportion thata[quare number hath toa {quare number , therefore the line A is incommenfurablein length tothe line H G . Wherefore neither of the lines F G and HG is commenfurablein length tothe rationall line A. And fora{much as the {quare of the line F Gis greater then the{quare of the. 4 line HG ( that theline FG is greater then the line HG t is.manifelt, for by [uppofition the number BC ts grea- ie = ER reme te Ante ter'then the number C D) unto the fquare.of the line } FG let the [quares of thelinesH G eo Kibeequall. And a ehacseaitennenceeny for that.as the niber B.C isto the number C D., [ow the fquare of the line F G:tothe (quareof theline H.G,thera». Fi ++ 20+ fore (by conuerfion of proportion) as the number B.Cis tothe number B D, {ois the [quare of the line F G to the B.+».D..... Cc {quare of the line K. Butthénuber BC bath tothe num | ber B D that proportibn'that af{quare number hath to a [quare number . Wherefore thé fguare of the line F G hath tothe {quare of the line K that proportion that.a{quarepumber bath to a {quave number . Wherefore the line F G is commenfurable in length tothe line K. by hevefore the line F G is in power more then the line_H G, by the [quare of a line commen- forabletnleneth tothe line F GC, and neither of the lines F GandG H is commen urable in tenath totherationall line As wher yet notwithftandine either of the ines F GandGHes vatiohall. Wherefore thelineF His a third refiduall line . Wherefore there is found out a third refdnallline swhithivas required to be done. mh q Ihe 22. Probleme... The. 88. Propofstion. Lo finde ont a fourth refiduall line. easy nae Akeavationall liné and letthe fame be A: and unto it let theline BG be 2 a e 7 BY ey), ‘ ; : ’ ‘ Pa Teas commen{urable in leneth. Wherefore the line BG ts rationall And take CH p<) twowumbers D F and F Ey and let them be fuch that the whole number, 22 hé aS ‘vamely,D E haue to neither of the numbers D ¥ and FE that proportion ed Se) ray > : 9 = ; : =~ that afquare number hath tova fquare number. And asthenumber D Eis tothewumberE F,oo.4 folesthefauareof the line BG beto-thefquare of the ee : 2 - o lines. GQ: wherefore the fauare of thé lineB G2 com= B aetn{uraleto the (quaveofthe line GC, wherefore alfa», thefguarenf the line §s Qs vationalljand the lineG © ts ulforatiovall. And for thatthe number D Ehath not thenumber BY that proportion thata fquarenamber Disses. Fick bith to afquare number therefore the line BG ts incom~ ie menfurablein length tothe lineG C., Andthey are both rational -whereforetheline BC is avefiduall linet fay moreouer that stisa fourth refiduall line F or forafmuch as the {quare of the line B Gis greater then thé {quareof thelineG C, unto the [guare of the line BG let the [quares of the lines © Gand. Hbe-equall.And fortharasthenumber DE isto the number EB, fois the fqnare of the line BG to the fquareof the line G C, therefore by con- er (ton of proportion.as the number D E 1s tothe number DF; fois the {aware of theline BG tothe {quare af thehneH . But the numbersD Eand DF haue not the one-to the other that porportion thata fquare number hath toafquare number. Wherefore the line BG és duncouimen{ us ale in length tothelimela. Wherefore the line BG is in power morethen the Noe of Euchides Elementets? Fol.292. she line G C by the {quare of aline incommenfurable in lengtlto theline BG : and the whole line BG 13 commenfurable in length tothe rational line Ae. Wherefore the lineBC isa fourth refiduall line Wherefore there is founde out a fourth vefiduallline:which was re- quired to be doone. gq Ihe 23. Probleme. The 89. Propofition. Tofinde out afiftrefiduall lne. an commenfurable in length . Wherefore the lineC Gis rational. And take two numbers D F and FE, which let be fuch , that the number DE haue to neither of thefe numbers D F nor FE that proportion that a ARS Kg SS ee OLD — number D E, fo let the {quare of the line CG be to the [quare of the line BG Wherfore the [quare of the line CG A ae is commen{urable tothe (quareof the line BG: Wherefore the {quare of the line B Gis rationall, andthelineBGis ® : alfo rational.But the numbers D E and E F hauenot that 3 proportion the one to the other that afquare number hath to a{quare nuber.Wherfore the lines B Gand G C are ra- tionall commenfurable in power onely. Wherefore theline ~D....... yaaa z B Cis arefiduall line. I fay moreouer that it is a fift reft- =. duall line.F or forafmuch as the fqaare of the line BG 1s greater theh the [quare of the line G C,unto the {quare of the line BG tet the [quares of the lines G C and H be equal. Now ther- fore for that as the number D E is'to the number E F, fois the ‘{quare of the line BG ta the [quare of the line G C,therfore by conuerfion of proportion, as the number D E is to the nu- ber D F fois the {quare of the line BG to the [quare of the line H. But the humbers D E or DF haue not that proportion the one to the other that a [quare number hath to a {quare number. Wherefore theline B Gis incommen|urable in length to the line H. Wherefore the line B G is in power more ther the line CG by the {quare of 4 line incommen ifurable inlegth tothe line B Gand the lineC G whichis toyned to thé refidual line is commelurable in lecthr $0 the rationallline A. Wher fore thetine B Cis 4 fift refiduall line. Wherfore there is found out a fift refiduall line : which was required to be done, a q I he.24. Probleme. The 90.Propofition. Lo finde ont a fixth refiduall line. ay Ake a rationalline andlet the fame be A.And Meo take three numbers E, BC, and C D,. not ha- Agsbuing the one to the other that proportion thata Lquare number hath toa {quarenumber..And let not the number BC haue tothe number BD that proportin.> on that a{quare number hath toa (quarenumber.And let the number BC be greater then the number C Dey C) as the number Eis tathe number B C, folet the {quare E vesees, of the line A be to the (quate ofthe lyne F G. And a8 Co vevececeee R cess B the number BC is to the number CD, falet the ‘fquare 00.4 of vy Ake a rational line and let the fame-be A, and unto it let the line CG be /quare number hath toa fquare number. And as the number F E is to the Confiruflions ’ Demon hra~ fits Coufirnctione — — ——————— ee SS a : ——— ‘ ' = —— >= 4 —_ a —— = ——% 4 ee J — : —— ee : % ae = . ———s a BOSE FT Seaeeeesbad a ianeinedl —S = Wet tS ew eS eS Demonfira~ Cie, An other p7ere re die Waly 0 finde out tne fixe re~ Jrduall lines.” ‘Thetenth Booke of teline FE Gheto the {quare of the lineG H.. Now therfore for thas as the usimber Eis $6 the wuber BC, fo is the {quare of theline:A tothe [quare of the line F G,ther ‘fore the {quae of the line A is commenjureble tothe {quare of the line F GWherfore the (qudre of the line FG isrationall, and theline F Gis alfo rationall. And for that the number E hath not te the number BC that proportion that a {quare number bath to a{quare number therfore the line Ais incommenfurablein length to the line F G sAgainefor that as the number B Cis te the number C D,fois the {quare of the line F G to the {quare of the line G H, therefore the fquare of the line F Gis commenfurable to the {quare of theliaeG H. But the fguare of the line F G is vationall,wherfore the {quare alfo of the line G His rationall, wherfore the line GH is al[o rationall. And for that the number BC hath not.tothe number.C Dithat propay- tion thata{quare number-bath toa lquare nnmber,therfore the line F Gis incommen{ura- ble in length tothe line S Hand they are bath rational. Wherefore the lines? Gand GE dre rational commen{urable in power anely. Wher fore r P pan 6 BF pe Pac! the line F H isa refiduall line. Lfay.mooreouer thatitis..- A ae py Doe ati te Yr J . a fixt refiduall line.F or for that as the number. £ is to y ) ’ . Pi : « vf : } 7 F H 5 the number BC, {ots the fauare ofthe line A to the. ARERR VET sateen r > j- Eo 7 . 4 fanare of the line F Gand as the number B-C isto the : ‘nither CD [0 isthe {quare of the line¥ G.tothe {quare sachet ofthe line G H, therefore by equalitie of proportion as : the number E 1s tothe number C D, fo is the {quare of Be Oe theline A to the {quare of the line CH. But the num- ber E hath not to the number C D that proportion that. Coa.u.0+: en : eee a(quare number bath toa [quare umber, Wherefore the line A is incommen|yrablein length tothe lineG H, and neither of thefe lines kG nor G His commen{urablegn length tothe rationall line A. And foralmuch as the{quare of the line F Gis creaser then the [quare of the ive G H, vnto the {quare of the. line F G let. the the (quares of the lines G H and K be equall. Now therfore for that asthe number BC isto the pumber C D,fois the {quare of the kine F G tothe {quare of the line Gil, . therefore ky conuerfion of proportion as the number B Cis tothe number B D,fois the [quare of the line F GC tothe [quare of the line K But the number BC hath notto the number. BD that. pro- ortionthat afquare number hath to a {quare number, therfore the line F.G 43 incommen- firable in length to the line K. Wherfore the line F Gis in power more.then thelyneGH by the {quare of a line incommenfurable indength tothe line F Gs and neither of the lines F G nor G His commen|urable in length tothe rationall line A... Wherforethe line F 1 ts a fixt refidual line Wherfore there is found out a fixt refiduall line: which was required to be done. There is alfo a certayne other redier way to finde out euery one of the forfayd fixerefidu- all lines which is after this maner . Suppofe that it swere required to finde outa firitrefiduall line.T ake a firft binomiall line A C, cy let the greater name 4 ; thereof be AB. Andvunto the line BC let the lineB D be equall. Wherefore the lines ABand BC, thatisthelines s D A Band B D are rationall commenfurablein power onely, ° and the line A Bis in power more then the ine B C, that is,thenthelineB D by ‘the fquare ofa line commenfurablein length tothe line AB. And the line AB is commenfurablein length to the rationall line geuen.F or theline AC is put tobeafirft binomial line . Where- fore theline AD isafirft refidaal line: And in like maner may ye finde ont a fecond,a third, 4 fourth,afift,and a fixt refiduall line,if'ye take for eche a binomiall line of the [ame order. B C aoe Hy ’ T he | of Euclides Elementes. = Fol.293. bs q The 67. T heoreme. The. 91.Propofition. , Ifa J uperficies be.contayned yndera rationall line ¢o-afirft refiduall line: the line which contayneth in power that fuperficies,is a refiduall line. = yh V¥; ppofe that there be a rectangle [uperficies A Beontayned under a rationallline L, A.C anda firft refiduall line AD.T-hen1 fay.that the line which contaynethin power the [uperficies A Bis a refiduall line. For foraf{much'as AD isafirft re- was (poken of in the L ena of the 79 « pro- A D rE F@ pofition) .Wherefore cs & the lines AG and 3 2 G D are rationall come/(urable in pow- er only, er the whole line AG is comen- furablein length to | | the rationall line A. . + Boe bt Re QM Cand the lineAG is in power more then the line GD by the [quare of a line commenfurable in leneth unte AG, by the definition of a firf? refiduall line. Denide the line G D into two equall partesin the poynt B.And vpon the line AG apply a parallelogramme equall to the [quare of the line EG and wanting in figure by af{quare,and let the fayd parallelocramme be that whith is co- tayned under the lines A¥ and F G. Wherefore the line A F ts commenfurable in leneth to the line F G(by the 17 -0f the tenth). And by the poyntesE,F and G,draw onto theline AC. thefe parallel ines EVA,F l,and G K. And make perfect the parallelocrime AK. And for* asmuch as the line A Fis commenfurablein leneth'to the line F G, therefore alfo the whole line A\G%is commenfurable in length tocither of the lines AF and¥G ( by the ts. of the tenth). But the line A Gis commenfurablein length tothe line NC. Wherefore either of the'lines AF and F G ts commenfuravle in tencth to the line NC But theline A Cis ra: tionall,wherefore either of the lines A F and F Gis alfo rationall: Wherefore (by the 1.9. of the tenth ) either of the parallelogramizes Al and ¥ K 1s alforationall. And fora{much as the line E13 comme [urable in lenethto the line E G therfore alfo( by the r3.of the tenth) the line WG is commenfurableinlencth to either of the linesD E and EG . But the line. DG is vationall wherefore either of the lines D E and E G is rationall; and the felfe fame tine D G13 theommenfurable in length to the line A C(by the definition of a firft refdnall line or by the ¥3:0f thetenth ) For the line D' Gis incommenfirable in'lencth to the line AG which line AG is comenfurable in length to the line’ XC: wherfore either of the lines DE und EG'1s rationalland incommenfurable in length to the line AC. Wherefore ( by the 21.0f the tenth either of thefe parallelogrammes D H and Eis mediall .V nto the pa: rallelogramme AI let the [quare LM be equall,and vnte the parallelogramme F K let the Square X be equall,being taken amay from the [quare LMyand haying the aneleL.OM consmonto thein both. (-And to doo this, there szuft be founde out the meane proportionall be- twene the lines E Land F G . For the {quare of, the meane proportionalljs equall to. the pa- yallelogramme contayned under the lines ¥ \ and FG . And fromthe line LO cut of a line equall to themeane proportional [0 founde out, and defcribe the fanare thereof): Wherefore both the [quares L Mand X are about one and the felfe [ame diameter bythe 26. of the Sx 00.4. [ixth 6 Fourth Sente rye The firft bart of the Con- firatiion. The firft pare of the demone ftration, ieé wre Note, Aland FR concluded ra<- tional parallee logramme. Note. D Hana F K, parallelo- Lrammes mes diall, Second part of the conflruce $201. T hé tenth Booke Hi | Sikhced fixth)let their diameter be O R and deferibe the figure asit ts here fet forth ° Now then fore Mii dheilenonfira afmuch as the parallelogramme contayned under the lines AF @ FG is equal to the {quare a daa, of the line’ G ,therefore(by the 17-0f the fixth) asthe line A¥ isto the lineEG., fo1s the ay LineE G tothe line ¥ GBus as theline AF is tothe line EG, fois the parallelogramme AT | | A I tothe parallelo- Wi ee ; gramme EK . And Wii EE asthe lineE Gisto 1} Ae the line FG, fois eh ! : the parallelogramme Kal CG E K to the parallelo- | gramme F K.Wher- fore betwene the pa- grammes A land BK the parallelo- gramme EK 3s the A ee meane proportionall. ve it But (by the (econd part of the affumpt going Lefore the 54. of the tenth )betmene the {quares ee LM andN X the parallelogramme MN is the meane proportionall. And unto the paral- We Ate lelogramme A lis equall the [quare LM, and unto the parallelogramme F K.1s equall the fquare N X by construction. Wherefore the parallelogramme MN is equall to the parallelo- gramme EK (by the 2. affumpt coing before the 54. of the tenth) . But the parallelogramme E K is(by the firftof the fixth) equall to the parallelogramme D H,and the parallelogramme a M N ¢s(£y the 43.0f the first) equall to the parallelogramme LX . Wherefore the whole pa- An st rallelogramme DK is equall tothe gnomon\ T L. (which gnomou conjisteth of thofe pa- Att vallelogrammes by which ye fee in the figure pafveth a portion of acircle greater then afe- micircle) and moreouer to the {quareN X : and the parallelogramme AK is equal to the | fquaresL Mand N X by confiruction :and itis now proned, that the parallelogramme DK ) 3 1s equall to the gnomo VT L, and morcouer to the {quare N X.Wherfore the refidue name- | se ih LN, isthe ly.the parallelogramme A, Bis equall tothe {quareS Q whichis the {quare of the line LN. ee i i sag Wherefore the {aware of the line LN is equall to the parallelogramine AB. Wherefore the AM gi sei an line LN contayneth in power the parallelogramme AB. 1 fay moreouer that the line LN 4s Wily a refiduall line.F or foralmuch as.either of thefe parallelogrammes AlandF K isrationall, 1) ae . as it is before [ayd, therefore the {quares L MandN X which are equall unto them,that is, ch the [quares of the lines LO and QN are rationall Wherefore the lines LO andO N are Hh. : alfu rational. Agayne fora{much as the parallelogramme D H that is L.Xis mediall, there- Hee | fore the parallelogramme LX ts incommenfurableto the fquare N X. Wherefore (by thet. i i f of the fixth,and 10.0f thetenth)the line LO is incomenfurable in length to the lineO Ne ae) and they are both rationall. Wherefore they are lines rationall commen|urable in power onely. ae Wherefore LN is arefiduall line by the definition,and it contayneth in power the parallelo- mn | ne parallelogramme AB. If therefore a fuperficies be contayned under arationalllineand a ec ig i | frit refidual line, the line which comtayneth in power that {uperficies is arefiduall line:which Hi aa was required to.be demonftrated. HA . \ a’ feetag rrr nee ne metre == er tenn ° c B yo 2 oR ae Q q The 68. I’ beoreme. T he 92. Propofition. i If a [uperficies be contained ynder arationall line and a fecond refidual eae line : the line which containethin power that fuperficies, isa first mediall Se ce vefidnall line. a) oda RE es ee 99 Suppofe wd REET ag of Euchdles Elementess: Fol.2 94. iT Vppofethat AB benfuperficies contained vader aritionallline AC, anda fe- x] cod refiduallline AD. Thea l fay, that the line that containeth in power the %, | {aperficies A B is a firft mediallrefiduall line. F or let the line ioyned to the line OK! AD be DG. Wherefore the lines AG andG D are rationall commen{urable an power nely, and the . line that1stoyned to the refidnal line, namely the 7 line DG. 18 comen{ura- —~——— < SURE MPT ble imtength to the rati- onall line AC : aud the line AG 15 in power : | — nN > | | " 2 more then the line DG, by-the-fquare-of a tine |. comenfurable in length to thelined GC Desideig. ~ a awe Oe Gs PSSA TG Sige RK aK the line..D-G inte twa = wor Se | quall partes tn the point E\.Aad unto the lintA.G apply a parallelocramme equall tothe fourth part ofthe (quare of the line D G, thatis; eqnatlto the fguare of the line EC, and wanting infeure by afanaré, and let that parallelogramme be that which ts contained un- der the lines AF.and FG. Wherefore by therz sof thetenth) the tine A F is comimmenfuras blein lengthtothe-line FG: And by the pointés E,F and G, draw onto the line AC thele parallel lines BH, F I,andG K-sand forafmnch as the line AF 4s comenfwrable in lenoth to the line E.G; therefore the whole line A G3 commenfurable‘in lencth to either of thefe dines AF -and FG. But the line AG is rational and incommentuvabh in length to the line “a C . Wherefore either of thefelines\ AF and 2 Gvuare ratiovall and tncommen|urable in bength tothe linea. Wherefore either of thele parallelogravimes AT and F K is by the v21.of the tenth) medial . Againe,forafmuchas the line D Eis comrmenfurable in length to the line EG; therefore the line D:Geis commenfurable in length toeither of thefe lines D E \And.EG.. Battheline D Gis commen{urablecnlenath tothe rational line AC. Wherefore either of thefe-lines D Eand:B. Gis rationaltand commenfurablein leneth to the line AC Wherefore (by thest.o» of thetenth) tither of thefe parallelogrammes DT! and E ® is vat} enall, Kntothe parallelogramme AT defcribean equal ‘(quare LM, and unto the pavalle. logramme F K let the fquare NX be equall, asin the T'ropofition going before . Wherefore the {quares L Mand NX are-both about one anphthe fame. diameter. Let the diameter be OR, and defcribe the figure as is in the former Propofition expre(led . Now therefore for- almuch as the parallelogrammes Aland F Karepiediall and,* conmenfurable the one to the other,and the {quares of the lines LO. &r O.N which are cquall ta thofe parallelogrames, “are medial, therefore the lines & 9 and O Nvare allo mediall commenfurable in power. (And it is manifeft, that thelines L O andO N are comenfurable in power for their [quares are commen{urable, and thofe {quares,namely, the [quares of the lines. L.O-& O.N are com- Swnen{urable, for they are eqiall tel ee gener Aland F K, which are commen u- -vabléthe one tothe other : aid tat thofe parallelogrampses Al dnd F Rare comimenlurable “the one tothe other hereby it i manifelforthati wis before proued that the lines A F and FG are commen/urable in lencth > Wherefore by thet .of the fixt, and 10 of th e. tenth) the “parallelogram@mes. A land F Kare commenfurable the one to the other . Wherefore itis “wow manife by the way of rcfolution that the lines LO eo N are comenfurable in power). And foralmuch as the parallelooramnse contained onder the Eines FE 4nd FG isequallto the fasave of the line BG, therefore as the lint AF isto the line EC [0 is theline EG to “the line FG. But asthe line A Fist0 the line EG, [otsthe parallelogramme Al to the pa» vallelograxsine EK: awéasthe line E Gis to the line F G, [os thebaralleloovamme E K to 7 00 MY. the Fit part of the consrue- ti0M, The fir pave of thedemone firation. Al and FK concluded pae reall elogrames mediail , DH &E K; rationall, The fecond part of the confirnttion. The fecond part of the de» monslrations * A922] ytically the pre fe hereof’ followeth am oge other thinges. €¢ ¢¢ 1 ei 4) ta? i) int ; er ih ( Pie} ha Heeial r " { - qh i) EL a al | { eeu eaLa I ; } ain I} wy 1 Mt : ; | ea PIM os t HT ‘a i] Me? ee uf | } j : yi } i ’ 1 ¥ : i} $ i} | ia i} Ne | Bah i , 1 MBH Ft if hi) Page | i | nt { ii } | Tre y mei) tl t Hii \ i 1H LE ) ARR teed hi! i { ¥ } Sr inn : T ' in] | heat } 4 a mi) 4, +] / ae t 1 iil ‘hal } ' Wey ail . vi Hi i hy } Be | f { , Tt) ) () : i mt | ef nyk ai) : if i Wi Ra MR Nai ist 7 aes |) S: sth mm: | mt) i: . Hen ' a ie iW ’ we f fl ; a qh a Pe aay || ha | ih 1) } ane I Well The tenth Booke the parallelogramme F K . Wherefore theparallelocramme E K is the meane proportionalt betwenethe parallelogramines Al and F K: and the parallelogramme HAN. 1s alfo the meane proportionall be- | twene the {quares LM CP NX : and the paral- lelogramme AT is equall tothe [quare LM: ana the parallelugrame F K is equall to the{quare NX . Wherefore the pa- rallelocramme MN is | equall to the parallelo- | gramme EK . But the parallelogramme E K is equall to the parallelogramme DH sand the parallelogramme LX is equall te the parallelo. gramme MN . Wherefore the whole parallelogramme D K is equall tothe Gnomon¥ TZ, and to thefquare NX. Wher ef ore théxtfidue,namely,the parallelocramme A Bis equall to the[quare SQ ;, that is, to the [qnareof theline LN :Whereforethe line LN containeth The tine LN #h power the fuperficies AB, 1 fay morenuer, that the.line. LN is.afirft medial refiduall fo mdwhich Une. For forafmuch as the parallelogramme EK is rationall,andis equall to the parallelo- 15 the prince- _ graume MN, that 15, LX, therefore LX thatas the parallelogramme contained under the od i of al bonnes 8 O aid ON 4s ratéonall ~ But the [quare NX is mediall, for it is already proued that Sieoutie, the parallelogramme FK which ts equallto.the [quare NX, is mediall . Wherefore the pa- rallelogramme L X is incommen{urableto the fquare DUX But as the arallelogramme LX is tothe {quare N X, fois the line L Oto the line o N ( by the 1.of the he ) Wherefore (by the 10.0f the tenth) the lines LO-and-O.N are incommenfurable in leneth:Andit isalrea- ay proud, thatthey.aremediall commenfurable in power . Wherefore the lines L 0 & ON aremediall commenfurable in power onely containing arationall [uperficiess Wherefore the line LN tsa firft mediall refiduall line, and containeth in power the fuperfities AB, which iscontained ynder.4 rationall line and afecond refiduall line. If therefore afuperficies be contained vader avationallline and.afecund refiduall line: the tine which containeth tr power that fuperficies,tsa fir mediall refiduall line-which was required to be demonftrated. A q The 69. F heoreme. The 93. Propofition. Ifa fuperficies be contained onder a rational line anda third refiduall line: the line that containeth in power that [uperficies isa _fecond medtall refiduall line. | V ppofe that AB be a fuperficies contained under a rationall ine AC, or third refiduall line AD .Then L fay, that the line which containethin power the fu- perfictes AB isa fecond mediall refiduall line.Let thelinetoyned unto AD-,be AEE SND G . Wherefore the lines.A Gand G D are rationall commenfurable in power only, and neither of the lines AG norGD is commen{urablein length to the rationall line ofthe Con. AC. , and the whole line A Cis in power more then the line GD, by the {quare of « line com- firutiion. HEP upableinlength to the line AG, Let the reit of the confiruction be as it was in the for: The fit pare mer Lropofitions . Wherefore the lines AF anadF Gareconzmenfurable in lencth: andthe of tte demon- parallelogramme A 11s commen[urable tothe parallelogramme F K . And forafimuch asthe fration, Uines A F and F Gare commenfurable in length, therefore the whole line A Gis commen{u- | : «yable Thetivl bart ——_—$—_ — —-- — _—__ oS a a a te of Euchides Elementess Fol.29$ pable in leneth to either of the[e-lines A F and FG. But the kine AG. isrationall and:in- commenfurable in length totheline AC. Wherefore either of: thefe lines AF and F G-isra- tionalland incomenen{urable in length tothe line AC. Wherefore (bythe 21.of the texth) dither of theleparatlelogrammes Al and F K is medial. Againe;forafmuch as the line DE istommenfarablein leneth tothe line E G, therefore'allo the wholeline DG 18 commen x- rable in length to either of thefe lines D Eand EG :Buttheline DG isrationall commer- furable in power on- : | power only tothe line | AC . Wherefore ¢- | ty to the line AC. b oe gk ie Ne £0 Wherforealfocithr A 0 Fe | LA ofthe lines DE and See ee , phot EG is rationall and : | Rie SoA ie oe commenf{urable in | | | | ae | | Me : ther of thee paralle- 7 logrammes D H and Severe ee EK 1 mediall.. A- gaine forafmuch as | the lines A Gand DG are commrenfurable in pomer onely, thereforethey are incommen{ura- ble in length Bat the line A Gis commecsfurable in leneth to the line A F< and the line DG 45 commenfurabl tit length tothe line GE. Wherefore the line A+¥ ts incommenfurable lie length tothe line EG . But ast he line A F is ta the line E G, fois the parallelogrammeA I to the parallelogra mme EK .b Vherefore the para Helocramme ATL is incommenfurable to the grallelogramme EK. Vutotheparallelogramme AI def cribeanequall[quare LM. and unto the parallelocramme F K deforibe a i eats 1 {quare NX : and defcribe the figure as you did in the former Pro pofition . Now forafian ch as the parallelogramme con tained under the lines AF and F Gis equall to the (quare of the line E G, therefore as the line A F vis to the line EG, fois the line E G to the line F G. But as the line A F is tothe line EG, {ois the pa- rallelogramme Alto the parallelocramme E K : and as the line EG ts to theline FG, [ow the parallelagramme E K to the parallelogramme F_K . Wherefore as the paratlelograme Ad 4s to the parallelogramme E K, fois the parallelogramme EK tothe parallelogramme F K. Wherefore the parallelagramme EK is the meane proportionall betwene the parallelogrames Aland F:K . But the paraliclograrmme MN 1 the meane propartionall Letwene the [quares EM and NX. Whereforethe parallelocramme EK isequali tothe parallelogramme M N. Wherefore the whole parallelogramme DK 15equaltto the Gnomon VT Z, @ tothe {quare NX: And the parallelogramme A K is equalltathe(quares LM and NX Wherefore the refidue,mamelysthe parallelogramme AB 1s equal £0 the {quare. 98, that is, to the [quare of the line LN Wherefore theline LN containethin power thefuperficies AB I fay more- quer, thatthe line LN isa fecond mediall re{iduall line . For for that asit is proued,the pa- rallelogrammes Aland EK are medial, therefore the {quares that-are equall vate them, vamely, the {quares of thelines LO and ON are allo mediall . Wherefore either of thefe lines L.Ouand ON is mediall, And forafmuch as the parallelogranmme A 1 15“ commen{u- sabletathe parallelocramme F K, therefore the [quares that axeequallte them, narmely,the fquares.of the lines LO and O.N ace allo commen{urable. Againe,forafmuch as it is prosed, ‘Bhat the parallelogramine Al is.1#6 mnmenfuralde tothe parallelogramme EK, therfore the ‘fquare LM isincommenfurable tothe parallelogramme M N; that is, the [quare of the lize L.0. tothe parallelogramms contained under thelinesL 0. ON .Wherfere alfo the * line L.0 is tincommen|{urablein length to the line O.N., Wherefore the ines LO and OD are wicdiall commenfurablein pamer onely.I (ay morcouer, that they containe a meaiall [uper- peics . For fora[much as tt is proued, that the parallelogramme E K 13 mediall, therefore the = OO. iif. pavallelogr ame aa ee ee a ee eee. = - — a ee Nove. Aland FK, ea2E diall . Note DH andE K, wediall, NV ote, Al incommeéen~ furabletoE K, Second part of the ConiFru&ié The principall. line,L N feud. * Becanforhe Lines Afaud Fa Garé promed . commenfuratle in length, . *By the firft of the fixth and renth of the féewth. —— road ~~ == 3 a —— ge SS Lhe frft part of the con- feruttion, The fir part ofthe demen- firation, Note, AK rational, Note. D K mediall, wi land F K incommens4- rable. “Thefecond part of she Const rucliowe The fecand part of the de- Ywsinftration. Thetenth Booke: parallelogramme which is equal unto it;wamely, the parallelogramme contained onder the linesL Ound oO N.-is allo mediall Wherefore the line L Nisa fecond mediall refiduall line, And containeth in power the fuperficies AB. Wherefore the line that coutaineth in power the. fuperficies AB is afecond mediall refiduall line . If therefore afuperficies be contained under. 4 rational line and athird refiduall line, the line that containeth in power, that [uperficies, és@fecond mediall refiduall line : whith was requirea tobe demonstrated. “ee q Lhe.20.T heoreme. The 94. Propofition. If a fuperfacies be contayned bnder a rationall lyne, and a fourth refiduall dyne:the lyne which contayneth in power that Juperficies 35 a lee lyne. oe Se V ppofethat there bea fuperficies AB contained under a rationall Vine AC Ana. KAREXY 4 fourth refiduall line A D.Then I fay that the line which tontaineth in power SSO) teluperficies AB is aleffe line. For let the line toyned untpit be D G. Where- I sl fore the lines A G and D G ave rationall commen nrable in power only,and the line AG is in power more then the line D G by the {quare of a line inconien[urable in leneth to the line AG, and the line AG is commen(urable in lencth tothe line AC. Dewidethe. ine DG into two equall partes in the point E. And unto the line AG apply a parallelo- gramme equal to the [quare of the lige E Gand wanting infi- : gure bya lauare,and Bee ae Let that parallelo- | eraime be that which - 4s contayned Under the lines AF and F | G. Wherfore (by the | 8 of thé tenth) the ine A F isincomen- [urable-in length. to } the line F G. Draw by the pointes E,F > Gyunto the lines AC and DB thefe parallel lines EH,F land G K.Now foralmuch as the line AG is rational,and commenfurable in lencth tothe line A C,therfore the whole parallelogramme AK is (by the 19.of the tenth) rational. Aguirre forafinuch as the line D Gis incommenfurablein length to the line A C(for ifthe line D G were commen[urable in length to the line AC, then foralmuch as the line AG is commen urable in length to the fame line AC, the lines AG and D G fhould be commen{i- rable ip length the one to the other, when yet they are put to be commenfurable in power one. ly) and both thefelines AC and D G are rationall. Wherfore the parallelograme D K is wre all. Againe forafmauch'as the line A F isincommenfurablein leneth tothe lime FG, ther- fore the parallelogramme A I is incommenfurahle to the parallelogramme F K. Pnto the pa= rallelogramme AI defcribe an equal [quare L M,and unto the parallelograme F K de{cribe on equal fauare N X,and let the angle LOM be common to both thofe|quares. Whereforé the (quares L M and N X are about one and the felfe [ame diameter. Let their diameter be. O Rand defcribe the figure. And fora{much as the parallelograme contained under the lines AF ana F G tsequall to the fquare of the line E G,therfore proportionally as the line AF is. to the ine EG, i isthe line EG to the line F G, but asthe line A F is tothe line E G; fess the parallelogramnze A I to the parallelogramme E K(by the r.of the fixt.) Abd as the lime E Gistothe line F fos the parallelogrammie E K to the parallelocramme ¥ X. hi ten | ‘ m. . phe A A | G &b Be RE WR Puss of Euchdes Elementes. Fol.296. the parallelogramme EK is the nzeane proportional betwene the parallelocrammes Aland B\K ¢ wierfore asit was (aid in the former propofitions,the parallelogramme MN is equal tothe parallelocrnmme E Ks but the paraliclogramme D H is equall tothe parallelograme EK and the parallélocrammze CM N to the parallelogramme LX. Wherfore the whole pa- j rallelogramme D K 1s equal! to the enomonV T Z,and tothe | quare NX. Wherfore the re- fidue,namely tie pavallelogransme A B is equall to the refidue, namely, tothe {auare 8 2, that is to the (quare of the line L N.I fay moreouer that L Nis thatirrationall line which is called a le(Se line F or forafmuch as the parallelogramme A K is rationall,and is equall to the fquares of the lines L 0 and O N, therfore that which is made of the [quares of the lines L O and ON added together isrationall. Againe fora(much as the parallelogramme D K is me- diall, and is equall to that which is contained under the lines LO and O N twi e, therefore that which is contained Under the lynes L O and O N wwife,is alfo mediall. And forafmuch as the parallelogramme A 1is tncommenfurable to the parallelogramme F K, therefore the fquares which are equall unto them namely the {quares of the lines LO andON are incom men{urable the one to the other. Wherfore the lines LO andON are incommenfurable in power ,hanyng that which 1s made of their ‘[yuares added together rationall, and that which is contatned under them twife mediall,whichis commenfurable to that which is contayned vuder then: once. Wherfore that which zs contained under them once is allo mediall. Wher: fore L Nis that irrationall line which is called a leffe line and it containeth in power the [u- perficies Ab If therfore afuperficies be contained under arationall line and a fourth refi- duall line,tie line which containeth in power that fuperficiesis a leffe line: which was re- quired to be demonftrated. q The 71, T heoreme. Ihe 95. Propofition. Ifa fuperficies be contained ‘onder a rational line and a fift refidual line: the line that cotayneth in power the fame fuperficies js a line makin 1g with arationall {uperficies the ~whole fuperficies medial. V ppofe that there be a fuperficies A B contained under a ratiovallline AC and Z| 4 fift refiduall line A D.Thé I fay thatthe line that cotaineth in power y fuper- ficies AB, is a line making with a rationall {uperficies'the whole fuperficies wit SS I diall.F or Onto the line A D let.the line DG be toyned, which fhal be comefuras bleau légth to the rational tine\A C. And let the ref of the confiructio beasin the propofition next going before. And forafmuch as the line A G is incomenfurablein legth tothe line AC and they are both rationall,therfore the parallelocrame A K is medial. Againeforafmuch as the line D G is rationall and commenfurable in | length to the lyne AC; | oe 3 cer | a therefore the parallelo. eee PRT ce gramme D K ts ratio- wall. Vnto the parallelo- eramme AI defcribe an equall fymare LM, and untotheparalleloerame FK deforibe in equal Ba gsi and asin the propofition wext o0- — TN ee a cli a SY LE ae ee —s - LN (the chiefe line of this theoreme) founde. Demon fire 840i» Tbe line LN. Demouflra- £10%e fore -the parallelo- Tbe tenth Booke ing before,fo alfo in this may we prowe;that the line LN containeth in power the fuperficies AB. I fay moreouer that that lime LN isa line making with a rationall fuperficies the whole [uperficies medial. For foralzmuch as the parailelogramme A K is medial, therefore shat which is equall vnto it,namely,that which is made of the {quares of the lines LO and O Nadded together is al{o mediall., Againe forafmuch as the parallelocramme.D K is ra- tionall,therfore that which is equall unto it namely that which is contained under the lines L0 and O.N twife,isal[o rational. And forafmuch asthe line A F is.incommenfurable in length to the line F C,therfore( by the 1.0f the fixt,cy 10.0f the tenth) the parallelocrime A Lis incommen{urable to the parallelugramme F K,wherfore allo the{quareof the lysed O is incommen|[urableto the {quare of the line O N.IWherfore the lines LOand ON are iite commen|urable in power hauing that which is made of their {quares added together medi- all,and that which 1s contayned under them twife rationall. Wherforethe line LN is that srrationall line which 1s called a lyne making with a rational [uperficies the whole fuperfic cies mediall,and it Contayneth in power the [uperficies A B.Wherfore theline contayning 1A power the [upecficies AB, is.aline making with a rationall {uperficies the whole [uperpicies medial. If therfore a fuperficies be contayned under arationall lyne cya fift-refiduall line, she line that contayneti in pomer the fame fuperficies, is a line makyng with a rationall (4. j erficies the whole [uperjictes mediall: which was required tobe proued. The 96. Propofition. If a fuperficies be contayned "onder a rationall lineanda fixth refiduall line the line which contayneth in power the fame _fuperficies is a line mae King witha mediall /uperficies the whole fuperficies mediall; qt he 72. T heoreme. xd AV ppife that.A.B.bea {uperficies contayned under arationall line AC, Gr a fixt | Cy aah line AD, es “ fay that the Line which contayneth in power pr | SS 3 perfictes A B, ts aline making with a medial {uperficies the whole {uperficies me- | ase, all For vatothe line AD let theline DG be toyned. And let the rest be as in the propofitions go- ong before. And for- | afmuch asthe line. L No AF tsincommen{u= rablein length to the line FG , there-: gramme Al is iv- comen{urable tothe parallelogramme. F K»_And forafmuch —¢ a ae ne as the lines AG and | AC 4rerationall commenfurablein power onely , therefore the parallelagramme AK is me. dtall,and in like maner the arallelogramme D K is mediall. Nowmforafmuch as the lines AG andGD are chin taralle in power onely , therefore they are incommen{urable in length theone to the other But as the line AG is to the lineGD, fais the parallelogramme AK tothe parallelogramme D K , therefore the parallelogramme A K 15 incommenfurable to the parallelogramme D K.Defcribe the like figure that wasdefcribed in the former propo. fitions,and we may in lke fort prone that the line LN contayneth in power the fuperficies A B.1 fay moreouer that it is aline making with a medial [uperficies the whole [uperficies medtall R Qi mt ogi adh of Euclides Elementes. F-01297. wmetiall. F orthe parallelogtame A Kis medial, wherefore thatwhich ts équal unto it, namthe by;that which is made of the{qwares of the lines LO and 0 Naddedtogether 3s alfo medial. And forafmuch as the parallelogramme 1) K is medial, therefore that whithisequall unt at namely that which 13 contayned Under the lines LO and ON twifeisalfomediall. And Jorafmuch as the parallegramme A K 1s inconmmen{ arable tothe parallelogramme D.K, shereforethe {axares of the lines LO und ON areincommenfurabletothat whichis contas- ned cnder thelizes LO and ON twife.And forafwuch as the pavallelooramime Alisin- comimnenfurahle to the parallelogramme F K therefore alfotke [quar of the line LO is iz- commen|irabletothe|yuare of the line O N Wherefore the lines LO and ON are tacor- ; ; i. oe here oa f 4 Toe wigs yd. ; ie menfurablein pover hauing that whichis made of the (guares of the lines L Oand 0. Nme- + . é ; f z We > 7 4 ae Ss drall and that which is contayned Under thenstwife mediall, and moreouer that which ts eh ” . . in ? alae a ‘ — me ‘ eee made of the fauares of themrts incommenf{urable to that which 1s contayned under them Wherefore the line which contayneth in power the fuperficses A Bis a linemaking with a me- diall {uperficies the whole [uperficies mediall. If therefore a fuperfictes be contayned under.a vationall lineand afixth refiduall line, the linewhich contayneth in power the fame ‘(uper ¥ ciesisaline making which amediall fupa rfictes thewhele fsper, Lctesmediall: which was re- q The 73. T heorente. Ihe 97. Propofition. The fauare of arefiduall line applyed vnto arationall line , maketh the breadth or other fide a firft refiduall ine. ‘se Se ppofethat AB bea refiduall line, and al SF (et CD bea rativnall line..And vatothe Gy YING lime CD apply the paratlelogramme CE equall to the {quare of the line A Band making the breadth the Line CE. Thea l faythat the line CF asafirft refidualliine For yutathe line A B let the kine couenientty ioyned Le fuppofed to be B Gwhich felfelineis al{o.called alemetayned, as.we declared tn the end of the7.9. propofition ). Wherefore the lines AG and. GB areratiouall. commen{urable in 3 power ouely..dud vito theline.CD apply the pa- D £ rallelogramme CH equallto the {quare of the line A G,and vato the line K H (which is equall to the line CD) apply the parallelogramme K L equall to the {quare of the line B G.Wherfore the whole parallelogramme C Lis equall to the ee of the lines AG and B_And the parallelogramme C E is equall tothe [quare of the line A B,wherefore the parallelogramme remayning , namely,the parallelogramme F Lise. quall to that which is contayned under the lines A Gand G Btwife>Eor( by the7. of the fe- cond the [quares of the lines AG and G B are equall to that which is contayned under the lines AG & GB twife, and to the {quare of the ine A &.Deuide the line F M into two equall partesin the point N.And by the poynt N draw unto the line C Da parallel line N X .Wher- ore either of the parallelocramees F X and NL. is equallto that whichis contaynea Up- der the limeP AG and GB once.And forafmuch as the fonares of the lines AG anbG Bare waricnall, Ontowbichamnres the parillelocramme CL ts équall, therefore the pabaltelo- prance CD alfors rational S wherefore the line CM is rationall and* commenfurable in : | length ~ Zz a % = Be } | a be a ee ee ee a a! ’ | | ; ' ' ; ; % : + The fiueth Senary, Thefe fixe propofitions following are the conuerfes of the fixe fore mer propofi- 10S e Consiruttion. Demonjfires tion. * By the 20.0f the texth, See . a rs <= Se aia eet. = _— = = r = = — owe Ets = = . “= = = ~ ~ " — ae os = ~ - = = ~~ a ie —= ee ; = ee SS aS = aa RR ee SS et eee tre i eee cee Se eee = : 5 —_—_ — SRS = = ee = — iz oe ee = —— — : : cn Se ——— = = eee = 3 > oe = = x ~ > ——— =a = ee ae ne sete EX By she zie of the teath. *By the 22.0f the tenth. C Fcocluded 4 réfidual lines «, Comftractist. T he tenth Booke Sength to the line C Di Agayne foralmuch as that which is contayned under the lines AG and GBtwife, is ** mediall, thereforethe paralleloeramme equall unto. 4, namely the parallelogramme ¥ 1 isalfo mediall. Wherefore the line F M is rationall.and* incommen.- ferablein length tothe line CD. And fora{much as the {quares of the lines A Gand GB arerationall , and that which ts contayned under the lines AG and GB twife ,ismediall, therefore the { quares of the lines A. Gand GB are incomenfurableto that which iscontar- nea under the lines A Gand GB twife.But unto the Squares of the lines A G andG Bs equall the parallelogramme C Land to thatwhichis contayned under the lines AG and G Biwife 1s equallthe parallelogramme F L,wherefore the parallelogramme C Lis incom, menfurable tothe parallelogramme FL, . Wherefore alfothe line C Mis incommenlurable in length to the live M:and they are both rationall.Wherefore the lines C M and © Meare rational commenfurable in power onely:and therefore the lineC F isa refiduall line (by the 73.0f thetenth).1 {ay moreouer that itis.a firft refiduall line . Far forafmuch as that which és \contayned under thelines AG andG Bisthe 3 mene proportional betwene the {auares of the lines ALG aad GB by the affumept going before the § 4. of the tenth). Anaupto the guare of thé line AG Foivss eA ae 1s equal the paralielogramme C i, and vnto that | which is contayned under the lines AG andGB | as equall the parallelogramme N L , and unto the {quare of the line GB ss equall.the parallelograme KL. Wherefore the parallelogramme™N L is the sucane proportionall betwene the parallelogrammes | C Handk L.. Wherefore as C HistoN L, fois rete Sam N Ltok L. Batas C HistoN L ,fods the line sy CK tothe line N M Gas N Lis toK L, fois the line N M to the line K M.Wherfore as the line C K is tothe lineNM, foisthelineN M to the lineK M.Wherfore the parallelogramme contayned under the lines C K andK Mis equall to the fquare of the line N M,thatis to the fourth part of the {quare of the line F M. And forafmuch as the {quare of the line KG is commen{urableto the {quare of the line GB, therefore the parallelogramme C H is comenfurable to the parallelogramme KL. . But as C His tu K L,fois the line CK to the line K M : wherefore the line C K is commen|ura- ble in length to the line K M . Wherefore ( by the r 7.0f the tenth ) the line CM isin power more then the line F M by the {quare of aline commenfurable in leneth to the line C M.Buk the line C M 1s commenfurahle in length tothe rationall lineCD. Wherefore the line CF 35a fiirft refiduall line. Wherefore the fauare of a refiduall line applyed unto a rationall pene keth the breadth or other fide a first refiduall line : which was required to be demote strated. ' A B G --— ep ee i a i a ty eee HL e {epen daheneeeneemernacmememeteniemeetemetemnentatan tf ay ae + q The7z4. T heoveme. Ihe 98. Propofition. | The fquare of a firft mediall refiduall line applied to arationall line » Mae keth the breadth or other fide a fecond refiduall line, | | wn ave ppofe that A B be a firft mediall refidual line,and let C D bea rationall lime. And G70 the line C D apply SF ig ean CE equall to the {quare of the hne - eee: A Band making in breadth the line C F.T hen I fay that the line CE tsa fecand | — refiawalt © iad ee -. ies —" of Euclides Elementess Fol.298. refiduall line. Fer-unto the line AB, let the lyne couenienily ivyned be fuppofed tobe BG. Where. A ee - + Cc fore the lines AG anadBG aremediall commen- AAS Jurable in power onely comprehending a rationall fuperfictes. And vntothe line C D apply the pa- te N. KA rvalletlo gramme CH equal to the {quare of the line AG, and making in bredth the line C K, and vn- the line K H (which is equall to the line C D) ap- | ph the parallelogramme K Leguall to the [quare E % onbeoe of the line G B and making in breadth the line K M .Wherfore the whole parallelogrammeC Lise D guallto both the fauares of the lines AG and G B which are medtall cy commenfurable the one to the other. Wherfore the parallelograimmzes C Hand K L are mediall and commen{urable the one to the other Wherfore( by the 15.0f the tenth) the whole parallelogramme C L is commenf[urable to esther of thee parallelogrammes C H and K L.Wherfore (by the corollary of the 23 .of the tenth) the whole parallelogranzme C Lis al{ormediall. Wherefore (by the 22.0f the tenth) the line CM is rational and incom- men[urable in length to the line C D. And forafmuch asthe parallelogramme C L is equal to the {quares of the lines A G and G B: and the [quares of the lines AG and G B are equall to that which is cotained under the lines. A G and G B twife, together with the {quare of the line AB (by the 7.0f the fecond): and unto the {quare of the line A B is equall the parallelo- gramme CE Wherfore the refidue,namely,that which is contained under the lines AG and G B twife,is equal to the refidue,namely,to the parallelogrameF L. But that which is contai- ned vader the lines AG cy G Btwife is rational. Wherfore the parallelogramme F L is alfo rationall Wherfore the line F M is rationall and comen{urable in length to the lineC D(by the 20.0f the tenth.) Now foralmuch as the parallelogramme C L is mediall, and the paral- lelogramme F Lis rational,therfore they are incomenfurable the one to the other Wherfore alfo the lyne C M 1s incommsen[urablein length to the lyne F M,and they are both rationall. Wher fore the line C F ts a refidualllene.l fay morcouer that it is afecond refidualllyne. For denide the line F M into two equall partes in the point N, frome which point draw unto the line C D a parallel line N_X.Wherfore either of thefe parallelogvammes F X and N Lis e- quall to the parallelogramme contained under the lines AG and G B. And fora{much as the parallelogramme contained vuder the lines AG and GB is the meane proportionall betwene the (quares of the lines AG and BG. Therefore the parallelogramme N L is the meane pro- portional betwene the parallelogrammes C H and K L. But as C Histo N L, fois the line @ K to the line N.M,and as N Lis to K L,fois the line N_M to the line K MU herfore as the dine CK isto the lyne N M,fo is the lyne NM to the line K M.Wherfore the parallelograme contayned onder the lines C K and K M is equall to the (quare of the line N_M, that is, to the fourth part of the {quare of the lyae F M.But the parallelogramme C H is commenfura- ble to the parallelogramme K L.Wherfore alfo the lyne C K is commenfurablein length to the lyne K M.Wherfore (by the 17. of the tenth) the line C M is in power more then the line F M,by the (quare of a line commenfurable in length to the lineC M. _Andtheline F which is the line conueniently ioyned,is commenfurable in length to the rationall lyne C D. Wherfore the lint C F is a fecond refiduall lyne. Wherfore the [quare of a first mediall refi- dual line applied to a rationallline,maketh the breadth or other fide a fecond refiduall lynes which was required to be prowed. Demonfi ran 1i0Ms CF concluded avefiduall lite Deere = U . ' iJ 4 *s 1 | # 4 et Bye 0 Ay Ae ‘ a : H 4 4 Bie : { J ag ta - { - - a eh) om Cc. Pai ' SLD hee j , i ‘ sy i: uJ a! ei fils AMI - Wi if it { { * 7 re ‘ 1 ; u | if i ! 7 t ud Ua Wet. ig rh: ee al f ; n i 1 em ca? } te 4 mt. he : } ' iy : Hed aif rf it + UT hl Moe, a) j % ij 7 Py ae 7) f 1 ) ib ba at Ine, ‘ ar "y f ti , } Hy tT a eS eee es = re = oer —~= i" , — = = ai: Sis = — = oo Se — me. = es - ‘a a — = aes —— ee ao ~ — oe Ee ———— — - — = 5 ~ = — ——— ~ — a - - ; = . 2 Te —— — —— = = a = - — er = ars = = 3 = 1 - — > - = a eae a = = - a — — - - — - _— —— 4 ~ — — — = = = =~ . = == ===> = —-- ——~ —_—-- > -——- < : a = : = EEE — 5 << — eS r= -_— — = = = n = ore SSS ae = = - ; = - - wna : \gediall. Wherefore the line F M is rationall and of Euchides Elementes. < Fol.299. RLOING king in breadth the line C F.Then 1 fay,that the line C F is a fourth refiduall line. For vato the line AB let the tine conueniently toyned be {uppofed tobe BG.Wherefore the lines AG & G B are incommenfurablein power,hauing that which is made of their {quares added together rationall, and that which ts contained under them mediall . And let the rest of the conftruction be as in the Propofitiens going before. Wherefore the whole parallelo- gramme C Lis rationall .Wherefore the line CM : és al{o rational, and commen{urable inlengthto A a G the line C D. And fora{much as that which is con- ; tained under the lines AG and GB twife is medi- —\—-—— all, therefore the parallelogramme which is equall wntoit,namely, the parallelogramme F L, is alfo > hg is Pet the line CM is commenfurable in length tothe line C D . Wherefore (by the 13.0f the tenth) the line CM isincommenfurable in length totheline 5 Boga isp FM: and they are both rationall .Wherefore the lines C M and F M are rationall commen{urablein power onely . Wherefore the line C F is iduall line . 1 [ay moreouer, that it is a fourth refiduall line . For fora{much as the lines AGandG B areincommen{urable in power, therefore the {quares of them, that is, the pa- rallelogrammes, which are equall unto them, namely, the parallelogrammes CH and K L, are incommenfurable the one to the other . Wherefore alfo the line C K is incommenfurable in length to thelineK M ..Andin like fort may we proue, that the parallelogramme contay- ned under the lines CK and K M, is equall to the ‘{quare of the line NM, that is, to the fourth paut of the {quare of the line F M .Wherefore (bythe 18.of the tenth) the line CM 4s in power more then the line F M, by the {quare of 4 line incommen|urable in length tothe line C M .\And the whole line C M is commenfurablein length to the vationall line C D. Wherefore the line C F ts a fourth refiduall line .Wherefore the {quare of alee line applied unto a rationall line, maketh the breadth or other fide a fourth refiduall line : which was re- quired to be proued. incommenfurable in length tothe line CD. But oe ee ee See eee eee in| J See a wal 5 Ne Be ml cell q The 77. F heoreme. The tor. Propofition. T he [quare of a lyne making with a rationall [uperfictes the “whole fupers jicies mediall applied'ynto a rational line maketh the breadth or other fide a fift refiduall lyne. Nua Ppofe that AB be a line making with arationall [uperficies the whole {uperficies See catall,and let C D be a rational line.Andvntothe lineC D apply the parallelow ' : ; ees i gramme C E equall to the fyuare of the line A Band making in breadth the line C F. Then I fay that the line C F is a fift refidwall line. For vato the line AB let the line con- ueniently toyned be [uppofed to be B GC. Wherfore thelines AG andG B are incommen|ura- ble in power ,hauing that which is made of their [quares added together medial, and that which is contained under them rationall. Let the reff of the conftruétion be in this as it was in the former propofitions Wherfore the whole parallelogramme C Lis mediall. Wherefore the line C M is rationall and incomnren{urable in length to the line C D.And either of the PP 4. _ paral. Confiruttion. Demo Sire- t10Me CF prouedé reliduak line. CF prousda weidusil fine. Conjlrutiion. Bemo%ira- $80 Me wolength tothelin€C DeAnd forafmeuch as that TZ be tenth Booke parallelogramme F X & NL is rationall Wher- forcthe whole parallelogramme F Lisalforatio A Be wall. Wherfore alfo the line F Mis rationall and 5 » eri =) : ‘ es NW K ft commen|urable in légth to the line C D: And for- ROT hearer oie | atfmuch asthe parallelogramme. CL is mediall, and the parallelogramme F L 1s rationall, there- | foreC Land F Lare incommenfurable the one to the other, and the lineC.M is incommen|urable | in length to the line F M,and theyare both ratio- nall. Wherfore the lines C Mand F M arevatio- & wall Co zemen{urable tn power onely.Wherfore the D E cS Bas © Lyne CF isarefiduallline. 1 fay moreouer that 0 is a fift refidual line.F or we may in like fort proue,that the parallelocrame contained ondley thelinesC K and K M, is equallto the {quare of the line N M, that is; tothe ‘fourth part of the [quare of the line F M.And forafmuchas toe {quare of the line AG, thatis, the paral. lelograsnme C H is incommenfurable to the {quareoftheline BG, thatis to the paralleto- gramme K L,therfore the line C K 4s incommenfurable in leneth to the line K M Wherfore (by the 18 of the tenth) theline C M is in power more then the line F M ,by the fquare of a line incommenfurable in length tothe lineC M. And the line conuentently ioyned, namely, ihe lineF M ts consmen{arablein tenethto the rationall line C D Wherfore the line CF is afifi repidualt line Wherfore the line C F is.afift refiduall line Wherfore the |quare of aie making with a rational {uperficies the whole [uperficies medial, applied vnto a rational line suaketh the breadth or other fidea pift refiduall lyne:which was required to be dem onftrated. q The 78.1 heoreme. The roz. Pr opoftt ton. The fquare of a lyne making with a mediall [uperficies the whol® fuperfie cies medtall applied toa rationall line maketh the breadth or other fide,a fixt refiduall line. | V ppofe that AB be a line making with a medial (uperficies, the whole fuperficies NOOK Beatall,and let C_D be a rationall line. And unto the line C D apply the paral- oe <)\elogramme C E equall tothe {quare of the line AB and making in breadth WLS the line C F.Then Lfay that the line C Fis a fixt refidual line. For vnto the line A B let the line conuenzently toyned be B G. Wherfore the lines A G and BG are incommen- furable in power hauing that which is made of their [quares added together medial; & that which is contained under them mediall, and moreoner that which is made of their [quares added together és incommen{urabletothat which is contained under them. Let the rest of the con- firuction be inthis, as it was inthe propofitios go- tng before. Wherfore the whole parallelocramme GL 1s medial, (for it is equall to that which ts madeofthe (auaresof the lines AG cy GB added together which is fuppofed to be mediall) Where fore theline G24 is rationall and incommenf{ura- bleinlencthtathe line CD ::andinlikemanner the parallelogramme £ Leis mediall. Wherfore ale’ fe the line F Misvatiovallandincommenfurable® > > 2s — ~_— te sa aL ie ———————— ee Neen nnn ee UEEE EEE TEESE nnn of Euclides Elementes.. Fol.300. which is made of the [quares of the lines A Gand'G Badded together, 13 incommenfurable sp that which is contained under thelines AG andG B twife, therefore the parallelogrames equal to them ;namely,the parallelogrammes CL. and F L are incommenfurable the one to the other. Wherfore al{othe linesC M and-F M are incommen{urablein length,and they are both ratiinal Wherfore they are rational comen{urable in power only Wherfore theline C F isid.réfidualtlines! [ry moreoner thatiis a fixt refiduall line.Let the reft of the demon ftra- ion be as it was in the former propofitions.nd forafmuch asthe lines AG and BG are 1n- commenfurable in power ,therfore their {quares,thatis, the parallelogrammes which are e- guall vnto them,namely,the parallelogrammesC H and K Laretncommenfurable the ove to the other Wher fore alo the line CK isincommen{urablein length to the line K M Wher- fore(by the 18 .of the tenth the lime CM istn power more then the line F M by the fanxare-of a line incommen{urable in length to the line C Mv'Aud neither of che lines CM nor FM ts commenfurable in length to the rationall line C D Wherfore the line CF is afixt refiduall line. Wherfore the fquare of aline making with a medial fuperficies the whole {uperficies me- diall applied ta a rational linesmaketir the breadth or other fide a fit refiduall line : which was required to be acmon|t rated. q The 79. Fheoreme. The 103. Propofition. A line com menfurable in lenoth toa refiduall line:is it Jelfe alfo a vefiduall line of the felfe ame order. i ppofe that AB beavefiduall line, unto which let the line CD be commen{u- 7 rablein length . Then I [a), that the line € Dis al(oarefiduall line, ana of the bn) Sha | 7 . : {elfe fame order of refiduall lines that the line AB is . For forafmuch as the ‘I line A.B is arcfiduall line, letthe-line conueniently ivyned unto it be fuppofed to be BE .Wherefore the lines AE 4na.B E are.ra-, tionall commenfurable ize power onely «As the line. ‘A... 8 3 B AB isto the line OD; {0 (by the 12. of the fixt) let ~ the'line BE beto the line D F< Wherefore (bythe © D F t2.of the fift ) as one of the apetecedentes 15 to one of Phe confequéntes, [0 are all the antecedentes to all. the confequentes Wherefore as the line A bis tothe... a lineC D, [ois thewhole line AE to the whole line C F,andtheline BE tothe line:D F Wherefore (by the 10.0f the tenth) the line A Eis commenfurable in length tothe lineC F, and thelineB BtathehneD EF .Bubthéline AE is rational. Whereforetheline C F is alfo rationall . Andin like (ort the line DF is rationall, for that the line BE, to whom itis conrmibi{urables isallorationall And for that as thelineB Eis tothelineA E, foisthe line D F totheline CF . But the lines BE and AE are commenfurable in pomer onely :Where- fore the lines C D and D F are commenfurable in power onely . Wherefore the line CD is a wefidiiall line. L sty mortonerstharatisarefiduall line of the felfe fame order that the'line A Bis. For forthat aswe hauebefore faid,astheline A E istothelineCF, fo is the line BE tothelize DF, thereforealternately, as the line AE ts to the-line BE; {01s the lize C F to the line DE But the line AL isin power morethenthe line EB; either by the (quare of aline commenf{urable in length to the ine AE, onby the fquare of a line incommenfurable in lenethtorheline AE .1f AE bein power morethen BE by the {quare of a line com- menfurablein length to AE, theathe line CF fhaltalfo (by the 14.0f the tenth) bein power more then theline D F, by the [quare of a line commen|urablein length tothe line CF, and foif the line AE be commenfurablean length tothe rdtionall line put, foralmuch asthe line wes PP... AE es —— $$$ = eee ———— __ ss . ’ ? 4 - c 22 — " - - a — - - — 7 7 - a a a Sy " pay — - bate . os ee ; ~ 4S CF treueda réfiduall, The fixt Se- nary. Construttion. Demonfira- tt0M CD cocluded a reiiduall littee Thetenth Booke wi 6 -—. z c af a na then tf the line AE be commenf{urable in length to the rationall line, the line € F hall allo in like fort be comimenurable in length tothe fame rationall line: and (pe; ther of the lines AB andCD isa fourth refiduall line. And if the live B E be comefurablein| éeth tothe rationall line, the line DF fhall allo be comenfurablein legth to the fame line:and Socither of the lines AB & Mp) 5+ CDsafifirefiduallline. Andif neither of thelines\ AE nor BE be commenfurable in eee ae length to the rational line,in Like fort neither of the lines C F nor D F hall be comenf{urable ae ant in legth tothe fame rational line. And [ocither of the lines AB @ C-Dis 4 fixtrefidual line. Wherefore the line C Disa refidwall line of the felfe fame order that the line AB is. Aline therfore commenfarable in lencth toa refiduall line, isit [elfe alfo a refidualllene of the felfe fame order : which was required to be proued. “a As before touching binomial! lines, fo alfo touching refiduall lines, this is to he noted, that aline commenfurable in length toa refiduall line, is alwayes a refidual} line Note, of the felfe fame order that the refiduall line is:vnto whom it is comenfurable , as hath before in this 103.propofitid bene proued.Butifaline be cOmen{urable in power orily to a refiduallline then followeth it not, yeait is impoffible, that that line fhould bea refiduall of the felf fame order that therefidual line, is vnto whom itis comment{urable in power onely . Howbeit thofe two lines hall of neceffitie be both either of the three firftorders of refiduall lines or of the three laft orders : which is not hard to Froue, if ye marke diligently the former demonftration, and that which was fj poken of binomi- all lines astouching this.aatter, yf Lhe 80.T heoreme. The 104. Propofition. | A line commenfurable toa mediall refiduall line jis it ‘felfe alfo a medial res | fiduall line and of the felfe ‘fame order. | | Vppofe that AB be amediall refiduall line, unto whome let theline CD be commenfurableinlength and in power,or in power onely. Then I fay that CD ‘saifo a mediall refiduall line,and of the felfe fame order. F or foralmuch as the MSA 2S) line AB is amediall refiduallline, let the line consuenititth ioyned unto st be BE: wherefore the lines AE and BEare mediall com. Siena ee Conftruclion. E men{urable in power onely As A BistoC D, fa by the 22.0f the fixth ) let BE betoD FB. And in like fort as in© D , | | Demonftra- the former fo alfointhis may we proue,that the line AE * aga 1s commenfurable in length and in pomer,or in power one. | by unte of Euclides Ellementes. - Fol, 30 dyunto theline CF ,¢& the line BEtothe lineD EF Wherefore ( Wythe 23. of the tenth the line C F is a mediall line, and the line D Fis alo a'mediall line, for that tt 13 commen|ura- ble to the mediall line BE. Andin Ike fort the lines C F and D¥ are commenfurable in power onely: for that they haue the ‘[elfe [ame proportio the one to the other,that the lines AE and E.B haue,mbith are commen{uruble im power onely. Wherefore the ling © Dis a mediall refiduall line.J {ay moreower that it is of the felfe fame order that the line A Bis.F or for that as the line A Eis tothe line BE,fois the line C F to the line D¥ . But as the line AE is to the line BE,fo is thefquare-of the line AE. ta the parallelogramme contayned under the lines AE and BE(by the first of the fixth) : and asthe line CF isto the line DF, fois the {quare.of the line © F tothe parallelograinme contayned under. the lines C F and DY, Wherefore asthe [quare of the line A Eis to the parallelogramme contayned under the lines AE and BE, fois the [quare of the line C ¥F to the parallelogramme contayned under the lines C F and D F.Wherefore alternately as the [quareof the line A Eis to the [quare of the line C F,,fois the parallelagramme contayned Under the lines AE and BE,t0 the parallelo- gramme contained under the nes C F and D ¥. But the [quare of the line A Eis commen- furableto the {quare of the line CE (for the line AE is commenfurable tothe line CF). Wherefore alfo the parallelogramme contayned under the lines A E.and BE, is commen{u- rable to the parallelogramme contayned under the lines C F and D F . Wherefore if the pa- rallelogramme contayned under the lines AE and EB be rationall , the parallelogramme alfo contayned under the lines C F and F D [hall be rationall. And then either of the lines A Band C Disa first mediall refiduall line . But if the parallelogramme coptayned under she lines A and BE be mediall,the parallelogramme allo contayned vuder the lines C F and FD fhall be allo mediall( by the corollary of the 23 .of the teth)- and [0 eather of the lines ABandC Dis afecond mediallrefiduall line Wherefore the line CD 1s amediallrefidu- all line of the felfe [ame order that the line A Bs. A line therefore commen{urable to ame. diall refiduall line,ts tt felfe alfo.a mediallrefiduall line of the felfefame order, which was re- quired to be demonftrated- | | This Theoreme is vnderftanded generall y, thar whether aline be commenturable infength & in power,orin power onely to a mediall refiduall line, itis it felfe alfo a. me- diall refidwall- lines and of théfelfe fame order, which thing alfois'to be vnderftanded of the three Theoremes which follow: An other demonftration after Campane. Suppofe thar be 2 mediall tefiduallline, vnto whome let the line & be commen furableimm length orin A power oncly, And takearationalllineC D, ynto which as © : apply the parallelogramme ¢ E equall to.the {quare of. B.. ety ¢ line 4,and ynto theline FE ( which 1s equall to the lineC D apply the parallelogramme FG eqnall té the fquare of the line 8.. Now thénthe. parallelogrammes, CE ahd FG fhall be commenfurable for that the lines A,B arecommenturable in power: wherefote by the r. of the fixth and ro. of this booke,the lines Band F G* are commenfurable in length . Now then 1f 2 be a firft mediall refiduall line , then is the line D £ a fecondrefi-_ dual line bythe 98:0f this booke : and if the line 4'be 4 fecond medial refiduialitine,then is the line D § a thitd.: ) EB pS refiduall line by the 99.0f this booke Butif D Ebeafe-- @ k D cond refiduall line’, G £alfo fhall bea fecond refiduall linte( by the 103,0f this boke). Andif D's be'a'third re- PP.iiij. fiduall CD prened 4 metal. Confirnuttione Demonflra- tion, eee a — or == = — a * 7 na ~~" +4 —_— ~ ——- — > — - - Se Fs . a aqua a mn = = = ———s — —_ ae Si : ———s 7 ~ - ~ es = — — ~ a = = _— - . - : —— EA = — _ = —* = SSG Sa wares = = LS = cS = = a ee Conftration. Demon fira- tim, Conffriction. T he tenth Booke fiduall line,G £ alfo thall (by the fame) bealfo a'third refiduallline, Wherefore i and 93.0f this booke,that 2 is eithera firft medial refiduall line ora fecond me ding asthe line .4 is fuppofed to be: which was required to be proued. t followeth by thegz, diall refiduall line,accor q Lhe sr: T beopeines The tos. Propofition: A line commenfurable to a leffe line. ts it felfe alfo.a leffe Tine. Ef (aaa ppofe that A B be a le(ve line,unto whom let the line C D be commen|{urable.T hes o. SOY I (ay, that the line C Dis alfo aleffe line. For let the fame conjtruction be in this, ND NB that was in the former Propofitions. And forafimuch as the lines AE and EB are Ancommenlurable in power,therefore (bythe22. ~~ of the fixt, aud ro of the tenth ).the lines CF c} ED are incommen{urable in power. Againe (b y the 22. of the (ixt) as the [quare of the line AE a8 to the [quare of the live’ BE foisthe[quareof . & D F the line C F to the [quare of the line D F Wher- fore by copofition as the fquares of the lines AE and BE are tothe ‘[quare of the line BE, foare | the [quares of the lines CF and D-F, tathe [quare of the line DF : and alternately, as the fquares of the lines AE and B E are to the [quares. of the lines C F. and D F, faisthe fquare ofthelineBE tot he {quare of the line DF. But the '[quare of the line B-E 7s commen{urable t0 the [quare of the line D F ( for the lines BE and DF axe commen|urable ) . Wherefore that which is made of the [quares of the lines AE and BE added togetber,is commenfurable to that whith ismade of the quares-of the lines € F and DF ndded together But that which is maae of the fauares of the lines A E and B E added together, is rational . Wherefore that which is made of the [quares of the lines CF and D F added together, is al{orationall . A- aine, for that as the {quare of the line AE is tothe parallelogramme contained.under the Voor bE ana BE, [o7s the ‘[quare of the line CF tothe parallelogramme contained under the.lines CF and D EC as we declared in the Propofition mext going before)': therefore: ab- ternately, as the fauare of the line AE is tothe | quare of the line. G F;fois,the parallelo- gramme contained under the lines AE and BE ,to the paralleloeramme contained under the linesC F and DF . But the {quare of the line AE is commen|urable to the [quare of the line CF, for the lines AcE: CE -ape-coupmen (unable Wherefore.the parallelocramme con- tained under the lines AE and BE, is commenfurable to the parallelogramme contained under the lines C F and D F . But the parallclogramme contained under the lines ABanad BE is mediall . Wherefore the parallelogramme contained vader the lines C.F-and. D Fis alfo mediall . Wherefore the lines CF and:D Fave incommien{ur able in power, hauing that which is made of their fauares added together Vationall, and the parallelogramme contai- ned under them mediall . Wherefore the lineC Disa lefedine. A linethertforecommen|ita rableto ale(feline;isit felfealfaa leffe tine: which was required to béproued.”*~* : A B £ -_ oe | An other demonftration. a Suppofe that Abe aleffe line, and unto Aletthe line B-be commenfurable: whether:its length and power,or in pomer onely. ThenkfaythaBivaleffeline. Take ay ationall ligeG D: And unto the lize@D apply (by the 44 0f the firft) the parallelogramme CE equall to the {quare of: the line A,and making in breath the ine GF. W, herefore.(by theot 00. 7 opas | er ition a cl lan aT. 2 F . “ a ———— eee << hh hl —_ S$ SS Se of Euchdes Elementes. » Fol.z02%. ion the line CF is a fourth refiduall line Pato the hve F E upp (by the [ame)the parallelogramme E H equall tothe fquare of the ~A_______. line B,and making in breadth theline F H. Now fora{much asthe line Ais commen|urable tothe line B; therefore alfo the {quare of °°" the line Ais comenf{urable tothe {quare of theline B. But untothe ~~ ¢ FY {quare of the line Ais equallthe parallelogrammeCE, e> onto the | fqsare of the line B is equal the parallelogramme E H IWherfore the arallelogramme C E is comimen{urable to the parallelogramme E H. But as the parallelogrammeG £ is to the parallelozramme EH, | fois the lineC F to the line F H Wherforethe line C F 1s commen- | | furable in length tothe ine F H: But the lineC Fis ‘a fourth ref duall line. Wherfore theline F His alfo a fourtirefiduall line (by | the 103.0f the tenth)-and the line F E 1s rationall:Bat if a fuperfi- | cies be contained under arationallline,and a fourth vefiduall lyne, | the line that containeth inpower that (uperfictes 1s (by the 94.0f the tenth) aleffelyne. But the line B containeth in power the [uperficies po E H Wherfore the line Bis alefeline:which was required to be pro- wed. ; og U he 82.:T heoreme. ~The 106. Propofition. Aline commenfurable to a lne making with a rationall fuperficies the whole /uperfictes médsall is it felfe alfo alynemaking-with a rational fue perfictes the whole fuperficies mediall. CV ppafe that AB be.alinemaking with a rationall [uperficies the whole fuperfi- & C. YS cies mediall,unto whom let the line C D\ be commenfurable T hen I fay that the | CSI line C D is a line making with a rationall fuperficies the whole {uperficies me- SIS diall.Vnto the line-A B let the lineconuentently ioyned be BE. Wherefore the lines AE and EB areincommen|urable in power,bauing that which as made of their -[quares added together me- diall,and the parallelogramme contained under themra- < D F tionall.Let he conftruction be in this as. tt was.i0 the for. | mer propojitions.And in like [ort may we proue that asthe E Demon ftra-~ tio’, Confirution. Demon tra- | : | , : : | ISS, tion. line A E1s to the line B E,fois the line CF tothehne DF, and that that which is made of a the [quares of the lines AE and B E added together is commen{urableto that whith ts made of the [quares of the lines C F and D E,added together,and that that which 1s contained vn- der the lynes A E and E B,is in like fort commnen{urable to that which 1s contained under the lines C F and D F.Wherfore alfo thelinesC F and D Fare commenfurable in power, haning that which is made of their {quares.added together mediall,and that which is contat- ned under them rationall. Wherfore thelineC.D.is a lyne making witha rationall [uperfi. ctesthe whole [uperfictes mediall, Wherfore a line commen{urable to aline making with a rationall [uperficies the wh ole{uperficies medial, is it elfe alfoa lyne making with a rational uperficies the whole [uperfictes medial - which was required tobe demonftrated. xs An. other demonftration. Suppofe that Abe a line waking with 4 rationall [uperficies the whole [uperficies ante ; ‘ AP Confiruttion Demon ftra- EL0He Conjiruttions Demonitra~ 610, _ CD apply the parallelogramme C E equal to the {quare of the line A T hetenth Booke and unto it let the lyne B be commenfurable either in length andin Then I fay that Bis a lyne making with a rationall fuperficies the whole fuperficies mediall.Take a rational lineC D,and vnto the line and makyne in breadth the lyneC.E. Wherfore (by the r01.propo- Sition the lyne C F is a fiftrefidualllyne Againe unto the line F E i | apply the parallelogrammeF G equall tothe [quare of the line B,and (7-4 makyig in breadth the lyne F H.Now foralmuch as the line Ais co- men{urable to the lyne B, therfore the {quare of the lyne Ais comme- farable to the fquare of the line B. But unto the {quare of the lyne A is equall the parallelogramme CE,and unto the fquare of the line B as equall the parailelogramme FG Wherfore the parallelogramine C E 1s commenfurableto the parallelogramme F G. Wherefore theline | C F is alfo commenfurableinlength tothe line F H.But thelineC EF as a fift refiduall line Wherfore alfo the line F His afift refidual line. And the line F E is rationall.. But if afuperficies be contaynedun- |} der arationallline and a fift.refiduall lye, the lyne that contaynecth D &£ G in power that [uperficies,is (by the 9 5.0f the tenth )a lyne making with a raticnall (uperficies —, ~ the whole fuperficies mediall But the lyne B containeth in power the parallelocramme F G. Wherfore the lyne B ts alyne making with a rational {uperficies the whole Juperficies medi- all: which was required to be demonftrated. y Lhe 83. T heoreme. Ihe 107. Propofition. A line comenfurable to a line making with a medial [uperficies the whole Juperfictes medrall, is it felfealfo a line making with a mediall /uperficies the whole fuperficies medial. AV ppofethat AB be a line making with a medial [uperficies the whole fuperf- cies mediall,unto whome let the line C_D be commen{urable. Then I fay that the line C D is alfo a line making with a mediall [uperficies the whole fuperfi- eG ores mediall.F or unto the line A B let the line conneniently ioyned beyB E.And let the roft of the conftruction be in this as it was inthe former pro ofitions. Wherefore the lines AE and BE are incomen{urable in power ,hauing that at At $5 made of their (quares added together mediatl,and that which is contained under them alo mediall and moreouer that which is made of their [nares added tocether istncommenfurable to that which is containedvn- ¢——_B der them. But the lines A Eand BE (as wehaue be- fore proued) are commenfurableto the lines C Fo DF, and that which is made of the [quares of the lines AE und BE added together,is sore Seton that which is made of the [quares of the ines C F and F D added together,and the parallelogramme contained under the lines AE and BE is commenfurable to the parallelogramme contained Under the lines C'F and D F.Wherfore thelines C Fand D F are oe in power hauing that which is made of their {quares added together mediall,and that which is contained under them alfo mediall,and moreouer that which is made of their {quares added together, is incommen{u- rable to that which is contained under them Wherfore the line C D is a line making with a mediall {uperficies the whole fuperficies medial, Aline therefore commenfurable i a lyne ei ao i. yh toast making Pp RB power ,or in power onely. = — ae - ———— ede eee . - . 4 ns oe eli Mon ~ a A ae ee Serene st Ss senshi of Euchides Elementes. Fol.303. wiakine witha medial fuperficies the whole Super fities mediall,is it felfe allo a line makyng with a mecdiall Superficies the whole fiperficies medi all: which was required to be pro ays A This propofition may alfo be an other way demonftrated;s ‘the three fornier propofitions were. If vpon a rationall line you apply;parallelogrammes equall to the fquares ofthe lines AB and CD, the breadthes of which parallelogrames thal! be eche a fixth refidual line, and therfore the lineswhich cé- tayne them in power ,namely,the lines A Band C D fliall be both fuch lines as is required inthe pro- ~ ae Sieh is caly to conclude marking the order ofthe demonftration in. the three former pro- pofitions. ¢ Lhe 34.1 heoreme. Lhe 108. Propofition. If from a vationall fuperficies be taken away a mediall fuperficies, the line ‘which containeth in power the fuperfictes remayning, is one of thefe two ir rational lines namely either a refiduall line, or a lefse line. eS ~~ iV ppofethat BC be arationall [uperfictes ,and from it take away a mediall {u- TIONS : LS): perficies, namely, BD .ThenI {ay,that the line which containeth in power the KR eperficies remayning namely, the [uperficies E C, is one of thefe two irrationall SOLAS) lines, namely, either a refiduall line, or aleffeline. Take arationall line FG. 4 » And vpon FG defcribe (by the 44.of the firft) a rectangle parallelogramme GH equallto the [uperfictes BC. And from the parallelogramme G H take away the parallelogramme GK eguall to the [uperficies B ).Wherefore (by the third common fentence) the fuperficies remayning, namely, EC, is equallto the parallelogranmme remayning,namel),toL A . And foraf- much as BC isrationall,and BD is mediall,and BC is equall to the parallelogransme G H, and B D.to the parallelogramme GK : therefore GH is rationall, and G Kis mediall:and the parallelogramime GH isapplied unto therationall. line F G. Wherefore (by the 20.0f the tenth) theline F H is rationall and commenfurable ix length to the line F G . And the paral- lelogramme GK is alfa applied vato the rationall line F G. Wherefore (by the 22.0f the tenth) the line F K is rationall and incommenfurable in length tothe line# G .Wherefore(by the Affumpt of the 12.0f the tenth the line F Hus incommen- furable in length totheline PK. And they are both vationall. Wherefore the lines F Hand F K are rational commen{urablein power oncly. Wherefore the line K His arefiduall line: ana the line conueniently toyned vaste it is K F . Now the line F H isin power more then the line K F, either by the [quare of a line commenfirablein length tothe line F H, or by the fquare of alineiscommen|urablein length to the line F H.FirSt let it bein power more then ithe line F Ky by the [quare of aline commen|urablein length to the line,F H, and the whole line F Hi 1s commen{urablein length tothe rationall line put nanely,to.F G. Wherefore the line K His afir[t refiduallline. But if a {uperficies be contained under arationall line, and apirfi pefiduall lize, the linethat. coptaineth in power that {uperfictes, is (bythe or. of the tenth) arefiduall line .Wherefore the line which containeth in power L H,that is,the fuper- ficies EC, isarefiduallline . But tf the line HF bein power more then the line*F K, by the fquare of alineincommen|urablein length to theline F H, andthe wholeline F H is com- men{urablein length tothe rationall linegeuen F G. Wherefore the line K H is a fourth refiduall ine . But aline containing in power a fuperficies contained under a rationall line and a fourth refiduall line, is aleffe line (by the 94.0f the tenth) . Wherefore the line that éontaincth in power thefe uperficies L H,that is, the [uperficies EC, 18a leffeline . If there- forefrom a rationall fuperficies be taken away a mediall [uperficies,the line which containeth in Bf F KA Sexenth Se. NAY» ConfirnGions Demonftra- t10Me _ - S . —— — = 7 \. SS ee Ss SS = =; i . ~ — — o ~ op = See = ——-—.- —aeeecaneet sus - . “ —— ‘ Conffruction. Demon fra- $i0H. Ze The tenth Booke in power the [uperficies remayning, is one of the{etwo irrationall lines, nanvely, either a refe- duall line,or a le(se line : which was required to be proued. q The 8s. I'heoreme. Lhe ro9. Propofition. Tf from a mediall fuperfictes be taken away a rationall fuperficies , the line which contayneth in power the [uperficies remaynin 12 15 one of thefe two ire rattonall lines, namely either a firft. mediall refiduall line , or aline mas king with arationall /uperficies the whole fuperficies mediall. wl’ ppofe that BC bea A hmediall fuperficies ana » | | [rou it take awiyarats FF oo a ee re —klFal alla “46 et ) MA ade OAL} uperficies namely , yea =a" _ ee eee EC is one of thefe twoirrationall lines ; either a firft mediall refiduall a D Bee tine,or a line making with a rational | fuperficies the whole fuperficies mediall.T ake a rationall line F G, and let the rest of the con firuction be iz this as it was in the former propofition . Wherefore it followeth that the line I FL 1s rational and incommenfurablein length to the line F G ( by the 22. of the tenth ). And that the line K F is (by the 20. of the tenth ) rationall and commenfurablein lencth to the line ¥ G.Wherefore the lines FH and F K are rational commenfurablein power onely. Wherefore K His arefiduall line. And the line conuentently ioyned unto it is F K. Now the line F His in power more then the line ¥ K , either by the {quare of a line commenfurable in length to the line ¥ H, or by the fauare of aline incommenfurable in length vntoit . If the line F H be iz power more then the line ¥ K by the (quare of a line commenfurablein length to the line ¥ Hand the line coueniently ioyned unto it namely, F K,ts comen{urable in legth to the rationall line ¥ G . Wherefore the line K His a fecond refiduall line. And the line F Gisarationall tine.But a line contayning in power a fuperficies comprehended under a ra- tionall line and a fecond refiduall line ts (by the 9 2.0f the tenth )a firft mediall refiduall line. Wherefore the line that contayneth in power the [uperficies LH , that is, the [uperficies C EB 4s a fir[t meatal reftduall lue. But if the line H F be in power more then the line F K by the {quare of aline tncommen{urable in lenath to the line F Hand the line conueniently ioynedy pamely ,the line F K 1s commen{urablein length tothe rationall line put , namely, tob Ge wherefore the line K H is a fift refiduall line. Wherefore (by the 95.0f the tenth) the line that contayneth in power the [uperficies LH,that is,the [uperficies EC, is a line making with a vationall {uperficies,the whole {uperfictes mediall:which was required to be promed. The 110. Propofition. ! tie i : | | remayning , namely the {uperficies | aed Li q lhe 86. I heoreme. If from a mediall fuperficies be taken away a medtall [uperficies income menfurable to the whole fuperficies the line which containetl in power the Juperficies which remameth, is one of thefe two wrationallilines, namely, either afecond.mediall refiduallline, or a line making “vith a medtall fw perficies the whole fuperficies medial. ES , f | ae see — ro ee eee — - + . of Euclides Elemente’. Fol.304. ~ Lay . - : , » . . “Sin the former defcriptionssfo here alfo takeawayyfré the wredtall fuperficies © {) B Ca medial [uperfictes B Djvand tet\ BD besncomuenfarable. tothe mhole B/D Ss fuperficies BC. Then ifay,thabthelinewbich containeth im power the fi FY 7\ ¥ x >”. "7 “et * &, eet. ae : a ' . a "2 ' Va I \ perjicses EC, is one of thefetwotrrationall lines, namely, either, a fecoud me- ARVIN VES. ; ; oa \: : “3 debs A 4) re (dual line,or a linemaking with a mediall{iperficiesthe whole fuper- ficies mediall For forafmuch as ecitherofa hefe [uperficieces ss at —_ - BCanlB D is mediall, and BC is incomimenfarableto BD}: Ext ist AMIS A Sy ; it followeth (by the 22.0f the tenth) that ditherofthe/e limes Yio poo A Wa 4 BH ana F K isvaticnall and incommedifurahle tialength 10% > AY betta hood the line F G. And foralmuch as the fupebficies BG is incom > RESO aeai meen|urable to the fuperficies B D, that.ts, thefiuperficies G H; Shy a] Laat to the {uperficies G K, therefore (by the firftofthe fixctyeri0. AEASpaerop oe) cy of thetenth ) the line F H isincommenfurablein hkuevh PO Wresafs O/C AGS AV DOr D : line F KWherfore the lines HE aid F K arerationabcomens A\ 28.6 Si furableto power onely. Wherefore (by the 73: of the tenth thew WH kine K Hisarchduall line,aud the line coweniently toyned Untoit isk K New the lime WF +s on power more themthe line F K ycither by the fquare of aline comenfurable in leneth ta the line F, or-by the (quare. of a line sncousmen{urable tr lencth wonton. Tf the lneH F bein power more then the liek K, by the fquare of atinecimen(arableimlength tothe liner Hy, =F: and neither of the lines HE nor F_K is.conamen{urableto the rationallline pat FG SvWker- fore theline K His a third refiduall, But the line GF thatiss the line RL, 3 vationallsAnd arechangle {uperficies conthined vader arationall line andathird refiduall lines 18 trratio- pall, and the line which containethin power that fuperficiesyis (bythe 93. of the tenth a fe- cond medtall refiauall line Wherefore the tine that coutaineth ix power the [uperfities BH; that is, the {uperficies E Cisaecond mediallrefiduall line . But if the line H F be in power more then the line F K, by the [quare of alineincommenfurable in length to the line FH, 20. ms and neither of the lines HE gar £K is commenlirablein length tothe line FG. Wherefore © the line H K PS. a fixt refi. aunll line. But a linec optatning 12 power A fé p erfici es contamedarn. derarationallline ded a fixt refidnall line, 15 (by the 96 of thetenth) a line making wish amediallfaperficies the whole fasperficies medial . Wherefore the line that containcth in ower the /wverficies EH, thatis, the fuperficies E C, 33.4 line ma SES PEE ES jie the whole fuperficies medial. If therefore froma mediall fuperjicies be taken away 4 mediall{wperficies, zncommen|itrable to the whale [uperficies, the line that contameth in | Power the fiperficies which remaineth, is one of the two irrational! lines remaining, namely, | either afecind mediall refrauall line, or a line making with a medial [uperficies the whole [uperficées medial : which was required to be proued, RNGACA 3 | | g The 87. T heoreme.. = The IIt. Propofition. Confiructiome Se de tionall line D.C. And (by the 44.0f the first) untothe line CD apply a rettanele ~ parallelograsemeC.E equall tothe fquare of the line A Band wraking in bredth theline D E. And forafmuch as A Bisa vefiduall line, therfore (by the 97\of the tenth) theline DE arse tng 54 firit refiduall kine.Let the line conucniently toyned unto it be EF .Wherfore the lines D’ 4, impofsibi-~ Q 2.4. F and btee, —— = —_ ——— me eae a ee ee i = > ~ ate o> = ee oA Corollary. irrational lines whith follow st being applredto a rationall line,do make the breadthes ont T hetenth Booke ¥F and E are vationall commenfurable in power on- ly, andthe line D F isin power morethen theline F.. A MK 8 E bythe fquare of a line commen furableinlength to the line DE ¢> the line DF scomenf{urableimlegth Pus 8. © F to.the rational line put DC. Again forafmuchas AB is by pofition a binomiall line,therefore.( by.the' 60. of the tenth) the line D Eis a first binomsaltlinesDe» | wide it into his names in the point G. AndletD G: be. | the greater name. Wherfore the lines: DG and GE | are rational commen{urable inpower otiely Amd the, = line D G isin power more thew the line GE by. the {quare of a line commen|urablein length tothe tyne’ D G,and the line D G is commen{urablermteneth to therationall line put DC. Wherefore the line D F is commen{uvableintencthtothetine D G.Wherfore (by thers. of thé tenth) the whole line D F is commen{urablein téath tatheline remiinine namely, tothe line G F . Ana foral mich as the line PFs comefurable tathelineF Gbutthe line F Dis rational, Wherfore the line F Gisalterationall. Andyforalmuch astheline F D iscommenfurable in length tothe line FG butihe live DF tstncommenfurable iwlenath to theline F E.Wher- fore thelineF G isintominen{irableintength totheline FE, (bythe 13.0f the tenth) and sheyare both rationall lines: WherforethedlinesGF and FE are: rationall commenf{urable in poeroirely.W herforelbythe.73..0f thétenth) theline E Gisarefiduall line, buritis alfo rational! ( as-befove bath bene proued )swhichss impo!cble,namely,that one ty the fame line Jhauld be bath rational and irrationall.. Wherforearefianall lineis not-one and the fame pitina binomial linesthatis ss nota binomiall tine: which was required to be demonfirated. Lok odtenla dtorisgey A Ooretlary: eR VA refiddall lyne and the other fine irrationall hnes following it are neither mediall lines nior-one and the fame betwene themfelues,. tbat ts, oneis. vtterly of € diuers kinde tro an other F or the {quare of amediall line appliedto ara tiowall linesamaketh the breadth rationall and incommenfurable in length to the: rationall.hyne,.whereun- to it is. applied (by the 22. of the tenth) The {quare of arefiduall line applied toa rationall line maketh the breadth « firft refiduall line (bythe 97.of the tenth). The{quareof afirft mediall refiduall line applied to avationallline, maketh the breadth afecond.refiduall lyne (by the g8.of the tenth) T he (quare of a fecopd mediall refiduall lineupplied vntoa ration nall line; maketh the breadth a third vefiduall line ( yy the 99.0f thetenth) T-he {quare.of aleffeline applied to arationall line,maketh the breadth a fourth refiduall line (by the 100. of the tenth) The {quare of 2 line making with a rationall faperficies the whole [uperfictes mediall applied to a rationaltline,maketh the breadth afift reftduallline (bythe ror. of the tenth) And the fquure of 4 line making with a medial [uperficies the whole [uperficies medt- all applied to a rational line,maketh the breadth a fixt refiduall line (by the 102.0f the teth) Now forafmuch as thefe fore[aid fides which arethe breadthes differ hoth from the first breadth for that u is vational;and differ al{o the one fro the other for that they are refiduals of diners orders and kindes,it is manifeft that thofe irrationall lines differ al[o the one from the other. dnd ferafmuchasit hathbeneprotedin the rrx.propopition, that a refidwat line is,nei ohedndshe fame witha binomial line,and it hathalfo bene proned that the {quares of arcfavalllineand of thefiucirrationalllinesthat follow wt being applica to a rational line do maketheir breadthes oneof the refiduals of that order of which they were, whofe fquares spexe applied to the rattanall line,likewvifealfo the fauares of a binomiallline, and of the fime of of Euchdes Elementes. Fol.z05. of the binomials of that order of which they were,whofe [quares were Applied to the rationall line. Wherfore the irra tionall lines which follow the binomiall ine, and the irrational lines which follow the refiduall line differ the one from the other,fo that allthe irrational! tynes are 13.in number namely ,thefe. i A mediall line. : A binomiall line. A first bimediall line. A fecond bimediall line. A greater line. A line containing in power a rationall (uperficies and a medial fuperfictes. A line contayning in power two medial fuperficieces. Arefiduall line. A first mediall refiduall line. A fecond mediall refiduallline. A lefse line. A linemaking with a rationall fuperficies the whole [uperficies medcall. A line making with a mediall fuperfictes the whole [uperficies medial. me te te Bt NS, So NV. Re OM RB Ww WB = q The 83. Theoreme. Lhe 112. Propofition. T he [quare of arationall Ine applyed nto a binomiall line, maketh the breadth or other fide a refiduall line., whofe names arecommenfurable to the names of the binomial line ex in the felfe fame proportio:<7~ moreouer that refiduall line is in the fe'fe fame order of refiduall lines, that the binos miall line is of. binomtall line:. a V ppofethat A be a rational line, andBCa binomiall line whofe greater name The determi. ~ | let be CD. And unto the feware of the line A let the parallelogramme contayned nation hath under the lines B C and BE (fo that EF be the breadth) be equall. Then I fay Jundry partes Wa! that E F is arefiduallline , whole names are commenfurable tothe names of the ht e bizomiall line C,which names let be D 3 P and B , and are in the fame proportion Sans with them «and moreouerthe line bisiz & ae SVE Lo x C the felfe fame order of refidual lines thet the . line B C is of binomiall lines. Vaio the © Morin vie i ek {quare of theline A let the parallelogranme contayned under the lines D and G bee- gual. Now forafmuch as that which is sotayned under the lines BC cE Fis equal to that Confiruttion. whichis contayned under the lines BD and G, therfore reciprocally (by the 14.0f the fixth) asthe line C Bis to the B D,fois the bne G to the line FE .But the line BC is ereater then the line BD, wherefore the line G 3 greater then the line EF. Vntothe line G lettheline DemonSira- EH beequall Wher efore( by the r1.0f the fift)as the line C Bistothe line BD fois the line tion. H Eto the line F E.Wherefore by deuiton (by the 17 .of the fifth ) as the line C D 1s to the lizeB D,fo is the line UF to the line ¥ KE. 4sthe tine H F sstothe f E folet the line F K be tothe line This isan Af- x «(how thisisto be done we will declare at the end of this demonstration) . Wherefore ( by im tT ak the 12. of the fift ) the whole line Ll K* isto the whole line K F as the line F K is to the line wae Sled und K E.For as one of the antecedentes is to one of the confequentes, [0 are all the antecedentes t0 demonstrated. all the confequentes . But as the line EK 1s thelineKE, fois the line C Dtotheline DB Se 2a me ae (for E K isto. K as H Bis to F E,ardH F is to F Eas C Dis DB) Wherfore( by the tl. «. continual pa | ae sf of proportion. | “¥ FEconcluded arefdaall. A. line, whith is jomwmhat. pre- posteronfly,in ve/bect of the order prepoun ded both ti 16 be proboftza, and alfoin the determinat?0a ae The tenth Booke of the fiftas the line HK is to the line K B.,fois the line C.D.to.the line DB, But the {quaré of the line CL) is commen|urable tothe {quare of the line D B: wherefore ( by the 10.0f the tenth) the {qnare of the line 11K 4s comenf urable tothe {quare.of the line ¥ K But thefe three lines. K,F K,and E K are proportionall in cotinuall proportio(as it hath already bene pro- ued) Wherefore (by the fecond corrollary of the 20.0f the fixth)the [quare of the line H K is to the fquare of the line ¥ Kas the line HK is to the line EK: wherfore the line H K is com- mcn{urable in length to the line E, K Wherefore (by the 15.0f the tenth) the line H Eis com- menfurable in length tothe ine ¥K . And foralmuch as the [quare of the line Nis equall to that which is contayned under the lines EH and BD, but the [quare of the line A z: rA- tionall,wherefore that which is contayned under the linesEH andB Dis rationall. Andit is applyed unto the rationall line BD. Wherefore( by the 20. of the tenth) the line EH is ra- tionall and commen{urable in length tothe ine BD. Wherefore allo the line EK which is commen{urable in leneth to the line WE ts * . rationaland commen{urable in length to the i eR, lineBD.Now for that as the line C Dis to B D ein DB, penbean Fates ee EK ( for it was before prosed , that as CD Fs Snead ei tees isto DB, foisH Eto FE, andasH Fis to G FE, fois PK toE K)batthe lines C D and D B arecommenfurable in power onely wherefore (by the ro of the tenth) the lines F K and K Eare alfo commenfurable in power onely. And for that as the line C D ts to the line DB, fois the line F K ta the line E K, therefore by contrary proportion as D Bis to C D,fois EK to F Kan alternately as D Bis toEK, foisC DtoFK: but the lines BD andEK are commenfurable im length as.it hath already bene proued ) .Wherfore alfothe linesC D and EK are commen{urable in length.But theline C D is rationall: wherefore al{o the line F K is vationall . Wherefore the lines FK and EX are rationall commenfurable sn power onely. Wherefore the line F Eis a refiduall line:whofe names ¥ K and KE are commenfurable to the names CD and B.D of the binomiall line BC and in the fame proportion as is proued. L {zy-anoreouer that ites a refiduall line of the felfe [ame order that the binomialllinets . For ihe line CD is im power more.then the line BD either by the {quare of a line commenf{ura- ble in length.to the line CD, or by the fquare of aline incommen{urable in length. Nowif the line CD bein power more then the lineBD by the [quare of a line commenfurablein length unto the line C 1 ,then (by the 13. of the tenth) the line VK 4s in power more thez the line E.K by the (q- are of a line comenf{urable in length to the line EF K. And foif the line CD be comenfurable in legth to the rational line put,the line F K alfo fhalbe comen{urable in légth to the fame rational line-wherfore then the ine B C is a firft binomiall line, cy the line F Eis likewife afirft refiduall line. And if the line BD be commenfurable in length to the raiionall line,the line EK is alfo commen|urable tn length to the fame , and then the line B.Czsafecond binomiallline,ana the line FE a fecond refiduall line. And if neither of the lines D nor D.B becommenfurablein length unto the rationall line , neither of the lines E Kivor EK arecommenfurablein length vatothe fame ,.and then the line B Cisathird binomiall line,e> the line © Eis a thirdrefiduall line. And if the line C D bein power-more then the line BD,by the {quare of a line incommenfurable in length tothe line C D,the line EK is alfo( by the 14.0f the tenth)in power more then the lineE K by the [quare of a line incommenfurablein length tothe line EK. And [oif the line Dbe commenfurablein lench toa rationallline put,the line ¥ K alfois commenfurabletm length to the {ame, where- fore the line BC is a fourth binomiall line,and the line FE is 4 fourth refiduall line. And if is the line B D be commenlurable in lex eth to the rationall line , the line EK is likewife COMA= ~ intii{urablein length tathe fame,and then the line BC is afifth binomiall line and the line EF of Euchdes Elementess Fol.306. EF a fifth refidwall line. And if neither of the lines C D nor D B.be commenfirable in length to the rationall line, neither alfo of the lines F K nor E K is commen{urable in length tothe [ame,and then the line BC is a fixth binomiall line, and the line¥ Ea fixthrefiduall line Wherfore the line F Eis arefiduall line,whofe names namely, ¥ Kand& Kare comme- fura ble to the names of the binoiniall line namely,to the names C D and D By and arein the felfe {ame proportior . and therefidualllineEF isin the felfe fame order of refiduall lines, that the binomall line B C is of binomall lines. Wherefore the {quave of a vationall.line applyed unto a binomiall line maketh the breadth(or other fide)a rcfiduall tine,whofenames are commen{urable tothe names of the binomiall line, and in the felfe [ame proportion , and moreouer that refiduall lineisin the felfe fame order of refiduall Lines , that the binomiall line is of binomiall lines: which was required to be demonstrated. x Here is the Affumpt (of the foregoing Propofition) confirmed. Now let vs declare how as the'line H F is to the line F E, [0 to make the line F K to the line EK. The line D is greater then theline.B D by [uppofition . Wherefore alfa the line H F is greater then the line.E E (by alternate proportion and the 14.of the fifth). From the line H F take away the line F L equall to the line F E . Wherefore the line remayning name- ly,H L,is leffe then the line H F for the line H F 3 D - ssequall to thelines HL CP -h Fi. cAs H Listo Be sik Sa H F, fo (bythe t2,0f the fixt}leeFEbetoFK, HL f E K Wher fore by contrary proportion (by the Corolla. ry of the g.of the fifth) asH F isto HL, fois F.K to F E .Wherefore by conuerfion of pro- portion (by the Corollary of the r9.0f the fifth) as HF is to LP, that is, tothe line equall vutoit, namely, to FE, fois thelineF K to the line EK. uM. Dee,ofthis Aflumpt, maketh (aoptopall:@s, thatis, ef cquifinely,) a Probleme vniuerfall, thus: Two unequal right lines being propounded, to adioyne unto the lefe,aright line which take with the le/e( as oneright line ) {hall haue the fame proportion,to the line adwyned, which, the greater of thetwo propounded, hath totheleffe. The conftruGion and demonftration hereof, is worde for worde to be taken, as it ftandeth here before : after. thefe wordes : Theline H F is greater thenthe line F E. q A Corollary alfo noted by: I. Dee. Ie is therefore enident that thus'are threeright lines (in our handling ) in continual proportion: st 1s to Weete, the greater, the leffe.and the adioyned, make the firft the leffe with the adioyned, make the fecond : and the adioyned line is the turd. This is proued in the beginning of the demonftration,after the Affumpt vied, sz An other demontftration after Flaffas. *. Take a rationall line A, and lect GB be a binomial! line, whofe greater line let be GD: and ypon the line G B apply (by the 45. ofthe frit) che parallelograme BZ equall to the fquare of the line A,and making in breadth the line G Z. Likewife vpon the line D B (by the fame) apply the parallclogramme Bl equall alfo to the fquare of the line A, and making in breadth theline DI: and put the line G ZT equallto theline DI. Then I fay, that G Z is fuch a refiduall line as is required ‘in the Propofition. QQ. uj. Forafmuch Coufirnttione Demonjira@ ‘tion, a ConStruttione : q \ a i : 0 ealk a Od I ith : t te ) ' iP aa 4 aT leas ‘ ul) : 4 : ‘ F th 2 ‘ i 4 ; EE Oy SE a — Wa Demonfirie Sion, Docag T hetenth Booke | Forafmuch asthe paralle- logrammesB Z 8 BI are NS equall, therefore ( by the 14.0f the fixt) reciprocal- ly as the line’G B is to the 4 line BD, foisthelineDI incised ie feria beallalialac or the line G-T,(which is equall ynto it) vnto the line G Z. Wherefore by dinifion, as the line GD is to the line D B,fo is the line T Z to the line ZG (by the 17, of the fifth }. Wherefore the line T Z is | : ‘ greater then the line Z G . ( Forthe line G Dis the greater mame of the binomiall line G B) . Wnto the line Z G pug the line Z C equall.. And as the line T Cis to the line T Z, fo ( by the 11.0f the fixth ) lee the line Z @be to theline Z K . Wherefore conttarywile (by thé ‘Cotollary of the 4. of thefifth) the line T Z isto the line T C, as the line Z K isto the line Z G . Wherefore by conuerfion of proportion ¢by.the 19 -of the fifth).as the line F:Zis to'thédline,Z C(thatis, to ZG, whichis equail vatodt) fo is theline Z Ktotheline KG. But the line.T Z isto the line Z G,as the line G Dis to the line DB.Wher- fore (by the 1t1.0f the fifthy the line Z K isto the line K Gyas the Ine GD istothe line’ D B. Burthe lines G D and D B arecommenfurablein power onely.. Wherefore alfa the lines ZK and) are come neniurable in power onely, by the 10.0f this booke . Farther, forafmuch as theline TZ isto theline ZG, as thé lineZ K is to the line K G, therefore by the 12.0f the fifth, all che antecedentes, namely,the whole line T K are to all the confequentes,snamely, to the line K'Zjas one of the antecedentes,nam ely; the line Z K 1s to one of the confequentes,namelyjte the line K G.Wherefore the line 2K isthe meane proportional betwene the lines T K and K Gy And therefore (by,the Corollary.of the z0.of the fixth) B as the firitnamcly, the lide T K, is to the third.namely,to the line K G : fois the fquare of the line T K q. Here are the ower partes of the propoft- t20 more order byhadled 2, then in the former de- 3, wioftration. to the {quare.of the fecoud,namely, of the line K Z. And forafmuch as the parallelograme BI ( which ig équallto the fqtare’of the rational line A )‘is applied vpon the rational! liné DB; it maketh the breadth D-Lrationall and commedifurable in length ynto the line DB, by the20.of the tenth. And. ther~ fore the line G T (which is equall ynto the ine D1) is commenfurable in length to the fante line D B. And for tharas the line G Dis to theliné DB, fo t$ the line KZ to the line K G, butas the line K Z is té the line K G, fo is the line T K to theline X Z, therefore (by the 11. ofthe fifth) asthe line GD isto the line DB, fo is thelitte If Ko the line X.Z:.. Wherefore ( by the 22. of the fixtlt} as the {quare ofthe line G Dis to the fquare of the line DB, fo is the fquare of the line T X.to the {quare of the line X Z.But the fquare of the line G D is commenfurable to the {quare of the line D B (for the names G Dand DB of the binomial} line G B are commenfurable in power ) , Wherefore the {quare of the line TX. fhall be commmentfurable'to the fquare of the line KZ; by the 10:0 this booke . But as the {quare of the line T Risto the iquare of the ine X:Z,fo is. it prouedtharthe nighthneT Xuis to the right line K G.Wher fore the right line T Kis commenfurab!e in length ro the right line K.G..Whereforeitisalfocommen: furable in length to the line T G (by the 15.of the renth) . Whichline T G is (asit hath bene proued} asationall line, and equall to theline DI. Wherefore thelines T X and.X G arerationall commen{u- rable ih length . And forafinuch as it hach bene’proued, that the line Z Kis commenfarable in power nely yvntothe rationall line x G ,-cherefore the lines’ Z X and-K Gare rationall‘commenfurable in power onely . Wherefore the line G Z isa refiduall line. And forafinuch asthe rationallline T G is commenfurable in length to either of thefelines,D Bund X,G . Wherefore the lines DB & X G thalk be commenfurable in length, by the 12.0fthe tenth. But the line ZX isto theline K G,as the line G D isto theline DB. Wherefore alternately, by the 16.0f the fifth, the line K Z isto the lineG D,as the fine K G isto the line DB. Wherefore theline Z Ais commenfirable in length vito theliné G D. Wherefore the lines ZK and KG (the dames: of the refiduall dine’G Z ) are icommenfurable in length to the lines GD and DB, which aseithe: names‘of: the\ binomial! line,GB-: and the line Z Kis to the line KG in the fame proportion, that the line GD is to the line DB. Wherefore if the whole line -2 X be-in. powet more then .the. line conueniently ioyned XG, by thé Iquare ef a line commenfurable ih length to the lihe ZX, then the grearér name GD thall be in power more then the lefle name DB, by the {quare of aline commenfurable in length to the line GD, by the 14.0f the tenth. And if the line ZK be in power more then theline X G, by the fquare ofa line incomment{urable injéngrh-to.thaline Z-X,, the line alfo.G D.fhallbein power more then the line DB, by the fquare of a line incommienfurable in length vnto the linéG D (by the fame Propofiti- on) . And ifthe greater or leffe name of the one be commenfurable in length to the rationall line puts the greater ordeiic name alfo of the otherhall be commenfurablein length to the fame rationall line, > (by the 12.0f this booke )..But if neither nante of the one be commenturable in length to the rational Iine put, neither same of the other alfo fhall be commenfurable in length to the fame rational! line put by the 13.0f the fame}, Wherefore the refiduallline GZ fhall be in the felfe {ame order of refidu~ all lines, that the binomiallline G B is of binomial lines ¢ by the definitions of refiduall and binomialk y. Srvc. e wu UAE Seem Ss ae see are “ , —_ of Euclitles Elements, Fol.307. liries.. {quare therefore of a rati6nall line applied to a binorniall line. &¢ : which was required’ be proued, q Lhe do. T heoreme. TE hers 3. Propofttion. The {quare of avational line applied vntovwrefiduall,maketh the breadto or other fide a binomial line whofe nates are commenfurable to the name: - of the refiduall line, and in the felfe Jaine proportion : and moreouer tha’ binomiall line is in the felfe fame order of binomiall lynes that the refiduad line is of refiduall lynes. | Px 7 | V ppofe that A be a rational linesand B.D refiduallline.Aud unto the fauar WX irns-4 +7 : o3 : ; F nae OSA of the line A let that which is contained onder thelinesBD.and KH be equal 57) 4! Wherfore the {quare of therationall line dappled vutothe-refiduall line BD ISSAOS maketh the breadth or other fideK HE hen kay thatthe line K His abinomi. all linewhofle nanzes are. commenfurableto. A } —— Sy the names of the refiduall line B.D, and in . . D 4 the felfe [ame proportion, and that the line yah 3S on K isin the felfe fameorder. of binoeiall A. rc BO age lines,that the live B.D 1s of refiduall fipese¢ ~ KutothelineB D let the. line conucntently goyned be D.C. Wherfore the lines BC and D.C are rationall commenfurable im power onely. And onto the {quare.of the line A let-the parallelogramme contained under the lines B Cand G be equall.But the{quareof the line A és rational Wherfore the parallelogranune contained vider the lines BC and Gis‘alfora» tionall.Wherfore alfothe line Gis nationalland commenfurable inlength tothe line BC (by the 20.0f thetenth). Naw foraftianch as the parallélognamme contained under the lines.B Cand Gisequalltothat whichis contdrwed under thelinesB Dand K H, therfore (by the 16.of the fixt)as the line B Cis ta the line BD fassiheline K HtothelineG. But the line BGs ereaten then the line BD. Wherfore al{o thedine KH is greater then the line G. Vnta the line G let the bne K E beequall. Wher fore the iaeK Bis ¢ationalland comenfurablein length to the line. B-G3as alfothe live G.2was (by thex2.of the tenth )dnd for that as B.Cis ta BD fois KH to K E.Wherfore by couerfion of propertio( hy the corollary of y 1.9.0f the fift). aB Crete D C, fois KA ;to B.Hivds A ss to.£ Ai fa letrth@liine FH be te the line EF (how this 18 > tobe done,wemill decareattheende ofthis demanfiration).~ Wherfore thérefidue KF 4s to. the vefidue FP Hyasthe whole Kitsstathewhole Hh £(by the 19 of the fft) that is,as,the line. BG1s tatholine ©.D.But the lines B. Gand C.D.are commenfurableinpower anely. Where~ fore alfo the lines K F and F H are commenfurableiapower only; And for that as K i iste H E,fois K F to F H,butas K HistoH E,fois alfoH F to F E, therforeas K F isto F H, fois F H to F E. Wherfore(by theconallary of the 1.9 sof the fixt) as the firft is to. the third, foisthe [quare of the firft,to the [quare of the fecond. Wherefore as K F isto F E, fois the Square ofthe line K F tothe fquare of the line F H, but thefe [quares are commmenfutable, s? as cpurinen farableiulength But th line KE isrationall and commenfurablein length tothe. hwe B G:whereforerhe line KF is alfo rationall and tenimen{urablein leeth tothe line BE: And for that as the line BC is ta thelineC D fois K F.t0F i ;therfore altermately (by the: 16.0f the fift) as B Cis to K F,foisC D to F H.But the line B Cis comrmen{urable in length Q. D.jii). to Om es f com forghe lines K F and F H are commenfurablein power. Wherfore thelines KF and FE. are. commen}urable in length. Wherfore(by the fecond part of the 15. of the tenth) the lines KE. and E F arecommenfurable in length. Wherfore (by the fame) the lines KE and FE\are. Construttion, Demonfira- tion} ~ An Affumpe, 0 a et aoe oo re. SE See _ .3 = - < Ww BP mw > 9 4s 5° 6 eS eee The tenth Booke tothe line K F Wherfore the line C.D 1s. commen{urablein length to the line'F H. But the line C D is rationall.Wherfore al{o the line F H is rationall. And the lines B CandC D are rationall commen|{urable in power onely Wherfore the lines K F and F H are rationall com- menfurable in power onely. Wherfore the line K H is a binomiall line, whofe names are com- menfurable to the names of the refiduall line,and in the [ame proportion.I fay moreouer that itis binomiatl of the {elfe fame order of binomial lines, that the line B Dis of vefidual lines. For if the line B C be in power more then the line C D by the quare of a line commenfurable in length to the line BC, the line K F is al{oin power more then the line F H by the [quare of a line commenfurable in length to the line K F (by the'14:of the tenth). And if 2 a ee a the line BC be commenfurable in length 3 aye | ieee eee in Uta Oe a to the rationall line put, the line KF is alfo eraae (by the 12. of the tenth) commen{arable Re 7 - a smlength to the-rationall line, and fo the — ¢ ES A ITE bjue BD isa firftrefiduall lyne, and the line K H isin like fort a firft binomial line. If the line C D be commmenf{urable in length to the rational line,the line FA is alfo commen: fuvable ia length tothe fame line,and [0 the line B Dts afecondrefiduallline, and the line KHa fecond binomiallline. And if neither of the lines BC nor CD be commenturable in » J 3 length to the rationall line,neither alfo of the lines K F nor F H is commen{urablein length to the ame,and [othe line BL is a third refiduall line, ana thelineK Ha third binowmiall line.But if the line BC be in power more then the line C D by the [quare of a line incommen furablein length tothe line B C,theline K F is in power morethe the line F H by the (quare of a line inconrmenfurable in length to the line K F ( by the 14.0f the tenth) And if the line B € be commenf{urablein leneth to therationall line put, the line K F is.al{o commenfura- ble in lenath tothe {ame line,and fothelineB D is a fourth refiduall line, and thelineK a fourth binomiall line.And if the line C D be comefurable in legth to the rational line, the line F His allo comefurable in leeth to the fame,e fo the line B Dts a fift refiduall line, G the line K H a fifi binomiall line. And if neither of the lines BC nor C D be commen|urable in leneth'to the rationall line, neither alfoof thelines K F nor F H ts commen{urable in length-to the fame,and fo the line B Dis afixt refiduall line,and the line K H is.a fixt bino- miall line Wherfore K H is a binomiall line,whofe names KF and F H are commenfurable to the names of the refiduall line B D,namely;toB Cand D, and in the felfe [ame propor- tion,and the binomiall line K H 1s in the felfe fame order of binomiall lines that the refidu- all B D5is of refidnall lines Wherefore the {quare of a-rationall line applied unto a refidualk line,maketh the breadth or other fide'a binomiall line) whofe names are commenfurable to the names of thevefiduall line andin the felfe {ame proportion, and moreouer the binomial lineisin the felfe {ame order of binomiall lincs, that the refiduall line is of refiduall lsmese which was required tobe demonftrated. ve | The Affumpt confirmed. Naw let vs ) declare how, ast. rf oF P K the line K H is to the line BH, fo } aba | to make the line H F to the line F E. Adde-vntothelineKH directly a line equall toH Ey and let the whole line be K L,and(by the tenth of thefixt let the line H E be deuided as the whole line K Lis denided in the point Hilet the line HE be fo deuided in the point F Wher- fore. Se of Euchdes Elementes. Fol.208. fore as theline Kilt is tothe line HL; thatis, to the-line LE > [ors the line H F to the ine FE. An other démonttration after Fluffas. Suuppofe that 2be a rationallline , and let 8 D bearefiduall line ..And vpon theline BD apply the parallelogramme D 7 equall to the {quare of the line -4(by the 45.of the firft)making.in breadth the that they areeche equalltothefquare of the line 4): therfore reciprokally (by the i4.0fthe i $3) back, se ites line B Gtotheline B D. Wherefore by con- uerfion of proportion( by the corrollary of the 19.0f the fifth )as the line 37’ isto the lineT E, foistheline 8G tothelineGD. Astheline BGistothelineGD, {etéttheline 7 Z be to the line ZZ by the corrollary of the 10. of the fixth. Wherefore by the 11. of the fifth the line BT isto theline 7 £, asthe line 7 Zisto the ee line Z £.For either of them are as the line BG isto theline GD. Wherefore the refidue BZ is co the refidue Z7,as the whole BT is to the whole 7 Eby the 19. of the fifth. Wherefore by the 11. of the fifth the line 8 Zis to the line ZT as theline ZT is to the line Z £.Wherfore T G the line 7 Z isthe meane proportionall. be- twené thelines B82 and Z£. Wherefore 'the {quare of the firft,namely,of the line BZ, is to the {quare of the fecond, namely, of the line 27',as the firft,namely,the line B Z,is to the third,namely,to theline z E(by the corollary of the 20.of the fixth). And for that as the line B G 1s to the line GD, fois the line 7 Z tothe line Z E: butastheline7 Zisto the line Z-£,fois the line B Z tothe line 77. Wherefore as the line B Gis to the line G D , {o is the line BZ to theline ZT (by the 11.0f the fifth ),. Wherfore the lines B Zand ZT are commenfurable in pow- er onely,as alfo are the lines B Gand G D(which are the names of the refiduall line B D ) by the 10. of this booke. Wherfore the right lines B Zand Z Eare comentfurable in length, for we haue proued that they are inthe fame proportion that the fquares of the lines.2 Zand 2 Tare. And therefore (by the co- tollary of the 15,0f this booke)the refidue B E ( which is arationall line). iscommentfurable in length vnto the fameline 2.2, Whereforealfo the line & G(whichis commenfurable in length vnto the line 8 E)fhall alfo be commenfurable in length ynto the fame line B 2 ( by the rz. ofthe tenth ). Anditis pronedthattheline BZ is to the line 277 commenfurable in power oncly .. Wherefore the right lines B Zand 27 are rationall commenfurable in power onely.. Wherefore the whole line B T isa binomiall line( by the 36. of this booke) s And for that as theline BG isto thelineGD., fois theline BZ to the line 27: therefore alternately (by the 16.ofthe fifth)the line B G isto the line B Z, as the line GD is to the line Z 7'.But the line 8 G 1s commenfurable in length ynto the line Bz : Wherefore ( by the 10. of this booke)theline@ D iscommenturable inlengthvato the line ZT . Wherefore'the names 2 Gand GD of the refiduall line B.D are commenfurable in length vnto the names B Zand ZT of the binomi- alline BT :and theline BZ isto the line ZT in the fame proportion that the line B Gis to theline GD as before it was more manifeft-And tharthey are of oneand the felfe fame order is thus proved. Ifthe greater or lefle name of the refiduall line namely,the right lines B G or G D.bé.cémenfurable in length to any rationall line put: the greater name alfo or lefle,namely,B Z or ZT fhalbe commenfurable in length to the {ame rational! line put by the 12.0f this booke. And ifneither of the names of thé refidu- allline be comment{urable in length ynto the rationall line put,neither of the names of the binomial line thalbecommenfurable in length vnto the fame rational] line put ( by the 13. of the tenth ). And if the greatername BG be in power more then the lefle name by the fquare of a line commenfurable in length vnto theline 8 G,the greater name alfo B Zfhalbein power more then the leffe by the fquare of a lineccommenfurable in length ynto the line B Z..Andif the one bein power more by the {quare of a line incommenturable in length,the other alfo fhalbe¢ in power more by the {quare of a line incommen- {urable inlength by the 14.0f this booke. The {quare therefore of a rational! Jine.S:¢. which was requi- red.to be proved. The An other dew mon ftratio af ter Fiuffas. Construction. D emonfira- 110M. OG Lhe tenth Booke | q Ube 90. I beoreme. Ihe 114. Propofition. , . bis i$ 3 @ 2 war " | 2 aot eg Tf a parallelograme be-cotained ‘pnder a refiduall line <> a binomiall hne, : nerfe of both ‘whofe names are commenfurable to the names of the refiduall line, and in | alate ee! st the felfe fame proportion: the hne which contayneth in power that fupere | é . gf : ‘ Pies x: ficies is rationall. DN V ppofe that a parallelogramme be contained under arefiduall line AB and a : KS binemiallline C D,and let the greater name of the binomial line be CE, and : KX \ the leffe name be E D,and let the names of the binomiall line, namely, C E and [SX E D be commenfurable to the names of the refiduall line,namely,to AF and F Bad inthe felfe [ame proportion. Andletthe 4 c = =a = ; — wane a : —_ So he ee > —— > - ~—— (ae -——-- —— Saas “ ; a —— ‘ ==Tsa — a : x z es = = — a — —- ee eee 3 =e = - FSS SS = — <= — > = Linwwhich containeth in power that parallelo- - - grane beG.Thé I fay that the line G isratio. ¢ as 9 nalT ake a rational line, namely,H. And unto 2 Conftruction. the line C D apply aparallelogrameequaltoy = + jf quare of the line H, and making in breadth = x3 i theline K L. Wherefore (bythe 112. of the Wi Pemonfira- temh) K Lisare iduall line,whofenameslet Te A Aa | Si0t bek M and M L,which are (by the fame) co- | Alt il i menfurableto the names of the binomiall line,thatistoC E andE D, and are in the felfe ae ane proportio. But by pofition the lines C E and ED are comenfurable to the lines A F and | F 3,and are in the felfe {ame proportion Wherfore (by the 12.0f the tenth) as the line A F is to the line F B,fois the lineK M tothe line M L.Wherfore alternately (by the 16.of the fift) as ihe line A F isto the line K M,fo is the line B F tothe line LM.Wherfore the refidue A B | is io the refidue K‘L,as the whole A F isto the whole K M.But the line A F 1s commenfura- bletothe line K Msfor either of the lines AF and K Mis commenfurabletothe line C E. Woerfore alfo the line A B is commenfurableto the line K L. And asthe line A Bis to the lime K L,fo (by the fir of the fixct) is the parallelogramme contained under the lines C D and AB tothe parallelogramme contained under the lines C D and K L.Wherfore the pa- ralelogramme contained under the linesC D and AB is commenfurable to the parallelo- grimme contained under the lines C D and K L.But the parallelogramme contained under thilinesC D and K L is equall to the [quare of the line H .Wherfore the parallelograme co- tamed under the lines C D & AB is comen{urable tothe {quare of the line H.But the paral- leligrame contained under the linesC D and A Bis equall to the {quare of the line G. Wher- fore the (quare of the line H is commenfurable to the [quare of the lineG. But the {quare of thiline His vationall Wherfore the [quare of the line Gis alforationall. Wherfore alfo the line G is rational,and it containeth in power the parallelogramme contained under the lines AB andC D.1f therfore a parallelogramme be contained under a refiduall line and a bino- small line whofe names are commenfurable to the names of the refiduall line,and in the felfe fane proportion, the line which containeth in power that [uperficies,is rational: which was resuired to be proued. etl - ag bt , ) 4 A 4 | ‘y He c is | Ral at { i] it A en’ " Ae ip i ‘i ue oft i bi ih (Gg! i (c a ihe FE Hi I! t t q Corollary. / Hereby itis mansfe/t, that arationall parallelogramme may be contained ander irrationall lines. An: of Euchdes Elementess Fol.309. ¢ An otherdemontftration after. F/uffas. Suppofechat the fuperficies pie becontayned vnder:a: refidualldine.a #),!and a binomiall line adi whofe names a8 and x p let be commenfurable in length yato the names oftherefiduall line ag, which letbe ar and rs. Andletthe line az, beto theline s p, in the fame proportion that the lihe!A F istothéline ne. AndJetthe rightline 1 contaynein power the fiperficies ps . Then Lay, that theline 1-isrationall line. Take.arationallline, whichletbe. ¢. And vpon the line aD ; defcribe. (by the.4s ofthe firft) aparallelogrammeequall to, the {quare of the Jine-c, and making in Confiruttions breadth the line pc . Wherefore (by the 112. of this booke ).c p is a refiduall line, whofe names éwhichilet bec o and op )fhall be c6menfurable in Jégth vnto the names az anda pj,andthelmeco py fly fhall be vnto the line o p,in the fame pro- aytiet a portion that the line az istotheline zp. 520%. 3utas the line az isto the line £p, fo by ‘ {uppofition, is the line ar tothe line Fs. ae Wherfore as the line c oistotheline op, | foistheline ar to theline rs . Where- forethelines co and 6 p are commentu- rable with the lines.a r.and Fz (by.thez2. ofthis boke). Wherfore the refidue,name- poe ly, the line cp isto therefidue,namely,to | | the line az,astheline c oistothelinear (by the r5.of the fifth) Buitis proued,that 1 + ARY the Line co iscémenfurable vnto theline L ° a. Wherefore theline cD, 1s commen- furable vnto theline as .Wherefore (by 2 the firft of the fixth) the parallelogramme c ais commenfurable to thé parallelogramme p 8. But the paraliclogramme cwis (by conttruétion) rational} (foritisequallto the {quare of the rationall line o). Wherefore the parallelogramme 8. pis afd rationall. Wherefore the line 1, which by {uppofition c6- tayneth in power the fuperficies a. p, is alfo rationall .. If therforea parallelograme be contayned.é&c: which was required to be proued. , r a ae he c . oT he'gt. I'beoreme. The rrs. Propofttion. | oe Qf amediall line are produced infinite irationall lines , of which none is of the felfe fame kinde “with any of thofe that were before. V ppofethat Abeamediall line. Thenl{ay, that of the line A may be produced fr infinite trrationall lines, of which none hall be of the felfe fame kinde with any “NYS | of thofe that were before. T ake a rationall line B .. And upto that which is con- Aneel + ined under the lines A and B let the [quare of the line C be equall (by the ry. of the fecond’S: Wherefore the tine C is irrationall . For a [uperficies — / A yy (FbORe contained onder arationall line and an irrationall line, 7s (by the Af- ——— fumpt following the-28 of the tenth ) irrational > and the line which containcth in power an irrationall {uperficies,is (by the Afjumpt going before the 21.0f the tenth) irrational. And itis not one and thefelfe ~ fame with any of thofe thirtene that were before. For none of the lines that were before applied to a rational line maketh the breadth mediall. =‘ Againe unio that which is contained under the lines B and C, let the {quare of D'be equal .\Wherefore the [quare of D is irrational . Wherefore al{o the line D | is itvatjonall and not of the {clf [ame kinde with any of thofe that were before. F or the (quare . of none of the lines which were before, applied to arationall line,maketh the breadth the line C . dn like fort alfo fhallit (0 followe, if a man praceedeinfinitely . Wherefore it is manifeit, that of a mediallline de produced infinite irrational lines,of which none is of the felfe fame kinde with ang of thofe “if were before : which was required tobe proued. An SS Thetenth Booke An other demonftration: > - ‘ Ave other de~ °° “Suppofe that A.C bea mediall line Then t fay ,that ofthe line AC may be produced in- monTretion, feitetrrationall lines, of which nome [hall be of the ‘[elfe fame kinde with any of thofe irra tuonall lines before named .¥-nta the line.A Cand from the patut A, draw ( by the 11..0f the Sirft}a perpedicularline AB, andlet.AB be arationall line,and make perfette the parallela- gramme BC.Wherefore BC istrrationall, by that which was declared and proned(in maner s of an Afsumpt) in the end of the demonftration of the 32 ; andthe line that containeth it in i power 15 alfo irrationall . Let thelineC D con- -o1g | : taine in power the fuperficies BC. Wherefore 222. 5 20 Coie np CD is irvationall & not of the felfe fame kind | : with any of thofe that were before » for the | fquare of the line C D applied to a rationall | line, namely, A B,wsaketh the breadth a mediall line,wamely, AC. But the {quare of none of the fore{aid lines applied to a rationall line maketh the breadth a mediall line . A caine, make perfecte the paratlelocramome ED. Wherefore the parallelogramme ED is alfo irrationall (by the fayd Affumpt in the end of the 38. his des monfiration briefly proued ) and the line which contatneth it in power 7s zrrationall: lerthe line which containeth 1 in power. be, DoF . Wherefore D F is irrationall and not of the felfe Jame kinde with anyof the forcfaid irrational lines: For the Jquare of none of theforefayd srrationall lines applied unto arationallline, maketh the breadth the line CD. Where ore of a metliall line are produced infinite irrational lines: of which none. is of the felfe fame kinde with any of thofe that were before : which was required to be demonftrated. ado — ee ee eee rae se ae j.¢ he. 92.'t heareme, Lhe..116.Propofition, IN ow tet bs prove that in [quare figures, the diameter is incommenfurable in length to.the fide. Demonftratio ta av, S |ct be pofsible,let it he comenfurable in legth I fay that thé this will follow, that leadingtoan Jena one andthe felfe fame nuber fall bebath an euen uumber & an odde number. smapolyibilirsee ¢ first) that the {quare of the line.AC is doubleto the [quare i U i] Ad, | iy } : : ‘ ; i ) : el | 4 a | i 4 i i fy Wb ' i) " D i | l, | ‘ ee) ie | \ ‘ oy 4 a f antsy Bei fae } ty | \, - ( Mt | a rig y Wn J HEB 8 ie! " ' 1 | i nn b % nh } yOMDS BL ALR iy nM aN a | ! i RG y AG i : ! i i { , > - i f 7 ¥ ppd ya Bd t + i ne > hegny |} a r { iP 2 is Y eC TPH TL } “a ie it ‘ho Bd! A Maen ie i 1m : q bat , bd: att , i H heen | ERA : wy ' 4 { Gq ui Brat : va it) att ayy) { im 4 5 tam ta 4 q Sg ae al , n fi pa | ’ a)! Peay ; , Th 4 - bh iD 5 ’ ‘hh 4 Dt tae tie ls ‘tel hike Ti att ee " Hy paeay Ay ty mein Pe in all ih J ) it a tH a ein bi ah hn Sint PP) “A TE Baya wT } me Pt iy Dac avi ; t tk Be Le tat hey a De i as " it I r . Hiioet) |: bT nee 5 1 Maa) i at 1 a 4d F “yt i) oh ae ala . ap I 4) Miah H { 1) f | OB H ie ; rd tian Be hia : Wy). i : i My \ , fi Hie . 4.5 a Li ell ‘ t \ pofition), therfore the lune AC hathuntothe line AB that proportion that a number hath toa eumber (by the | s.of the tenth). Let the ne.AC haue unto the line AB vel that proportion that thenumber E F. hath to the num- \ ber G.And let EF and G-be the leaft numbers that haue one ana the [ame proportion with them.WherforeE F is a 20 weahie. For if EF be vmtie,and it-hath to the num- ber G that ANG that theline AC hath tothe lyne aE (2 Band theline A Cis greater then the tyne A B.Wher fore veitie E F isgreater then thenumber G, which ts ow nny “7 : tos > ** 1 S 1 impofvole. Wherfore F E is wot uiitie, wherforestisa@ ee Gao ss = —— aan a — = —— * F 7 = ZG 4 4 - = = > . . a ~? > at z = : ee 7 a4 as 3 = > — — ++ =. ~ ~ . - — q > ee - —— _ ><) 2 eee “x ’ SS > See > ‘ = - ~ < -—- = 2a a a a ee pee soe tr a Ba — : = —- - =, 7 = ee ect 7 i eee 2 SSS Ee — = — = es — . —- 7 - — : ~ ——* = - : a . = —— : : ——== Sa —- =" : - = = ———2 " _ = - = —- > - i - = aaa — = acne = = = = = ~ — as i ae ee, SE ee =< = aed — <= ten ee = ee joe ~ - i = — Ss. - eee —_——_— - +s - = " a= ~ — ~~ es: : exits at - n — Tee rs — Gas *S5 5s aie = = or = — - BS eee SS F*= - = = - - - _—_— ~ = Zz - = o SSS ——_—__— -____—} —— =—= SA Spa = ae 5 tte = e = = Set a ae =e — —s=s > = “= = Lee == Se = sae Si= z = ; - _ = < — — ———__———_——— <== RS Se Sn - "er - ee = ee ~ - = 7. . et, OP aes: x : = — 3 , = = — SS EEE TTL : - = ~ — The tenth Booke i other :which is impoffible.Wi herefore the diameter Nis not commnenfurable tn leneth tothe fide BWherefore it is incommenfurable:which was required tobe demonitrateds An‘othér démonttration after Flu (/as. Suppofe that vppon the line AB be deftribed a f{quare whofe diameter let bethe line A C .: Then Lfay ‘that the fide: A.B is incommenturablé indength vntothe diameter.A C-For-.. afmuch as thelines AB and BC areequall, therefore the fyuare of the line A € is double to the fquare of the line A B by °~ the 47.0f the firlt-Take by the 2\of the eight nabershow many >* {oeuer in,continuall proportion fré ynitie,and inthe propotti-.). 4 on.of thé fquares of the lines A BandAC. .. Which letbe the .. _ numbers D,E,F,G.And forafmuch as the firlt from vnitie name °° ly Eis no fquare number, forthat itis a prime Atiny ber . neither” * is, alfo any other of the fayd fuimbers.af{quake bum ber.except.\y the third from vnitieand fo.all the reft leuing one betwene,by , the ro.of the ninth. Wherefore Dis to E,orEtoFj,orFto G, ” in that*proportion that'a fqitare number isto Whumber ist® | fquare: Wherefore bythecorrollary of thezs oftheeight,they are not in. that proportion .the.one. to. the other-that Fx Fe i nutaber isto a iguarénumber. Wherefore neither alfo haue the fquares'of the lines AB andA CCwhich are in the fame pro-- 3 , portion). that porportion thata fquare.nvmberharhitoa fquare! auwber, Wherefore by'the9.of this. booke theirfides;namely, Go. the fide ABand the diameter AC are inecommenfurablein _ lerigth the’one to the other whith wasrequired'to be proned* This demontttration I thoughteoodto adde; forthat the former demonttrations femenotiofalland they are thought of fomic to be noneof T heons,as alfothepropofi tion to be none of Exuckides, | | molinaice a9 Here followeth an inftruction by fome ftudious and fkilfull Grecian (perchance 7. beon) which teacheth vs of farther vfe and. frilite of thefe irrationall lines... - Seing that there are founde out richt lines incommen{urable in leneth the oneto the o- ther,as the lines A and B , there may alfobe founde out many other magnitudes hauing léeth and breadth ( {uch as ave playne [uperficieces) whith fhalbe incomme|urable the oneto the o- ther -F or if (by the 13:0f the fixth) betwene the lines & and B there be taken the mane pros portionall line, namely C, then (by the [econd corrollary of the 20. of the fixth) asthe line A is tothe line B,fots the ficure defcribed feet the line A tothe ficure de[cribed vponthe line Cy being both like and in like fort defcrobed, that s, whether hey be[quares (which are dl- wayes like the one to the other) ,or whether they be any other like rettiline ficures, or whether they be’circles aboute the diameters A-dnd C. For cir- cles haue that proportion the oneto the other, that the Squares of their diameters haue(by the 2.of the see Pe Nn Wherfore (by the fecond part of the to. of the tenth) the. , figures defcribed upon the lines Nand C being like and 3 in Vike [ort de{cribed are incommen|urable the one to the é } other Wherfore by this meanes there aré foande out ‘[uperficieces incommen|urable the one —: + of Euchdes Elementes. Fol.3it, be ertéted folides of eqaall altitude, whether thofe folides be compofed of equidistane {uperfi- cieces,or whether they be pir amids,or primes, thefefolides[@ erected fhalbein that proportia the one to the other that theyr bafes are( by the 3 2.0, the eleuenth and s.atid 6 Of the welfth) Howbeit there is no [uch propofition concerning prifmes And {oif the bafes of the fo lide : be commenf[urable the one to the other, the folides alfo [halt be commenfurable the one to the other,and if the bafes be incommenfurable the one to the other,the folides alfo [hall be incon- menfurable the one to.the other (by the'r0.0f the tenth). And if there be two cirtles Aund B- and upon ech of the circles be erected Cones or Cilinders of equal altitude, thofe Cones es Ci linders [hall be in that proportion the one to the other that the circles are,which ave their ba: fes (by the 11.0f the twelfth) :and fo of the circles be commenfurable the one to the others the ‘Cone: and Cilinders alfo [hall be commenfurable the one to the other . But if the circies be in comen{urable the one to the other , the Cones alfo. and Cilinders fhalbe incomenfurable the one to the other, (by the ro:of the tenth) . Wherefore it is manifeft that not onely inlines and [uperficieces but alfoin folides or bodyes is found commmen|urabilitie or incommen urability. Anaduertifement by Johu Dee. a Although this propofition were by Excide to this booke alotted,(as by the auncient erecian pub- lifhed ynder the name of riforeles epi aroyay yea umav,it would feme to be,and alfo the property of the fame,agreable to the matter of this booke,and the propofition it felfe, fo famous in Philofophy and Logicke,as it was,wouldin maner craue his elemétal place, in this téth boke)yet the dignitie & perfec- tion,of Mathematicall Method can not allow it here:as in due order following: But moft aptly after the 9. propofitio of this booke,as a Corrollary of the laft part thereof. And vndoubtedly the propofitio hath for this 2000.yeares bene notably regarded among the greke Philofophers : and before ariferlestime was concluded with the very fame inconuenience to the gaynefayer , that the firft demonftration here induceth namely ,Odde number to be equall to ewen:as may appeare in Ariftettes worke, named Auxalitica prima, the firit booke and-40.chapter . But els in very many places of his workes he maketh mention of the propofition.Euident alfo it is that Eaclide was about 4rsfotles time , and in that age the moft excel- lent Geometrician among the Grekes. Wherefore, feing it was fo publike in his time, fo faméus,and fo appertayning to the property of this booke:it is moft likely, both to be knowne to Exchde , and alfo to haue bene by him in apt order placed.But of the difordring of it, can remayne no doubt, if ye confider in Zamberts tranflation,two other propofitions going next before it,fo farre mifplaced, that where they are, word for word, before duely placed, being the 105 and 106.yet here(after the booke ended) , they are repeated with the numbers of 116.and.117.propofition . Zembert therein was more faythfull to fol- lowas he found in his greke example,than he was [kilfull or carefull to doe what was neceflary. Yea and fome greke written auncient copyes haue them not fo : Though in decile they be well demonitrated, yet truth diforded,is halfe difgraced:efpecially where the patterne of good order, by profeifion is auouched to be.But through ignoraunce,arrogan- cy and temeritie of vnfkilfull Methode Mafters,many thinges remayne yet , in thefe Geometricall Elementes, vinduely tumbled in: though true , yet with difgrace : which by helpe of fo many wittes and habilitie of fuch,as now.may f) 7 A Parke RR.ij. ' ih) t ‘ A} i y | i 5 4 ! } : : > 4 ‘ Pile) kee f ee { i) j a] vt ’ oul’ ; Pi wh : + ie ® Wl 4 } 4 tS \ yet : 1 { q ai /’ hes 4 ‘ J Tey pe j 4 | 7 ide { me i}? lt } 4 | - { ia b ab 4 ‘ j ; ’ | 1 vi f —— — _— are 7s = ox - = ee ee The argument of the elewenth booke. . re AA pont fhe be- ans ae Soorrsve ao O Z LFentTe $s of a i GHAREIS IE Cle tintal. 5 A ~ fo 2 sept a OY 4 e ME FFIEL FUELS Cledby Exuclide —_ - . 373 the £O%2 | 6% pee 72ET COORES « cy f Firs? f- fit La EG 7%+ YUUNES Second vooke. ger * 5 ¢ T bird be0ve aA é Ma Vee se a ET BE) Relons FOUTS 12 COOKE « p Sewenth booke Esckt booke. & “ Ninth booke, Tenth booke. What is entrea ped of sw the pice bookes fol- bOT LEP &» | as Fise rerniar be- oe ; 7? Pf ” - fy v7 Fi dics tee fizais ; j >F a oS ae at Jf 2S) 08; f 073: ) EOE OF » a Pe Euclides eo- ta ot Pm pei Vicia 08 prrerres. é iz Comtart) or oF é - - t he f r fF és 20 ‘ e . . Dee y Peas Deh ave f 22S£S LiCMEI @q the eleuenth booke of Fue clides Elementes. 5 4 hardnes conceaue whata prifme is; name- ST ly,ifye imagine the fuperficies ABDC to be the ground & bafe of the folide,.and the two fuperficteces, namely; the fuperficies AEB, and the fuperficies CEF D to be erected vpon the fides of the bafe,the one on the-One- fide, namely, on the-line AB, and the other-on the other fide, namely, on the line D-C)not perpendicularly ; butin- cliningand bendingtheonetotheother,tull D Cc they meete ia the toppe,namely, on the line EF. Forfo ye fee that this folide figureis contained ynder many plaine fuperficieces, of whichtwo namely, the fuperficies AEC , and the fu- perficies BED which are the endes of the folide,arid oppofite the one to the other, are equall like and parallels,and all the other {uperficieces ,namely,the'bafe A BC D,& the two erected fuperficieces, thatis, the fuperficies AEF B, andthe fuperficies CEF.D are parallelogrammes . Yet notwithftan- ding,to make the thing more clere ynto the reader, I haue here {eta Prifme which will appeare bodi- : like,ifyou erecie bending wife the papers wherein are drawen the pa- . 4 rallelogrames ABEF, 8: CDEF, { that they may concurre in the line \ E F in the toppe.and fo erect the pa- pers wherein are drawen the trian- eles A CE and BDF, that the fide AE of the one triangle may exactly agree with thefide AE of the one par the fide C Eof the fame triangle, with the fide oa ESS with the fide D F of the parallelogramme C DEF : and finally, the fide BF of the fame triangle, with the fide BF of the parallelo- gramme ABEF. And fo thall you moft eafilie fee the forme ofa Prifme : that it confifteth of two equall,like,and parallell triangular fuperficieces,and of three parallelogrammes: wherof the one is the bafe, and the other two are erefted bending wife. Here alfo be- holde the forme thereof asitis by arte defcribed in a plaine toap- peare bodilike, Fluffashere:noteth that Theor and Campane difagree in defi- ninga Prifme ,and:he preferreth the definition geuen of Campane before the definition geuen of Ewclsde ( which becanfe hemay feme with out leffe offence to reiect,he calleth it Theons definition Jand following Campane he geueth an other definition, which is this. A Prifme 1s a folide figure whsch 4s contayncd Gnder fine playne fuperficseces , of which twoare triangles, like equall,and parallels and the reft are parallelogrammes. it . The example before fet agreeth likewife with this definition , and manifeftly declareth the fame. For in it were fine fuperficieces,the bafe,the two erected fuperficieces,and the two endes:of which the two endesare triangles like,equall and parallels,and all the other are parallelogrammes as this definiti- on requireth . The caufe why he preferreth the definion of Campane before the definition of Theor (as he calleth it,but in very deede it is Euclides definition, as certainely , as are all thofe which are geuer of him in the former bookes,neither is there any caufe at all, why it fhould be doubted in this one defini- tion-mote then in any of the other) as he him felfe alledgeth,is,for that itis(as he fayth) to large, and comprehendeth many mo kindes of folide figures befides Prifmes , as Columnes hauing fides, and all Parallelipipedons, which a definition fhould not doo: but fhould be conuertible with the thing defi- ned,and.declare the nature of it onely,and ftretch no farther, ge Me thinketh Flafzs ought not to haue made fo mucha dooin this matter, nor to haue bene fo fharpe in fightand fo quicke as to fee and efpy out fuch faultes,which can of no man that will fee right- ly without affection be efpyed for fuch great faultes . Forit may well be aunfwered that thefe faultes which he noteth (ifyet they be faultes)are not to be found in this definion.It may be faydthatitexten- deth it felfe not farther then it fhould, but declareth onely the thing defined,namely, a es sail | 0 gramme C D EF: and moreouer, the fide B F of the other triangle, . ‘ Se Se of Eucliles Flementes.: Fol.315. doth it agree(as F/u/ze cauilleth with all Parallelipipedons.and-Golumnes hauing fides . All Paralleli- pipedons what fo ever right angled,or not right angled which aredeferibed of equidiftant fides or {us perficieces haue theif fides oppofit.So that in any of them there is. no.one fide, butithatha fide oppo- fit ynto it.So likewife is it of eué fided Columnes, eche hath his oppolite fide direéily agaynit it, which agreeth not with this definition of Evelide.Hereit is euigently favd, that of all the fuperficieces,the two which are oppofite are equall,like,and parallels, meaning yndoubtedly onely two & nomoe. Which is manifeftby that which followeth. The other(faysh-he )are paralelogrammes, fignifiing moft euidently that none of the reft befides the two aforefayd ; which are equall, like, and parailels, are oppofite? bur two of neceffitie-are-rayfed vp ; and-coneurre in one common line, and the other is the bafe.So that it contayneth not vnder it the figures aforefayd,thatis fided Columnes,& al Parallelipipedons,as Flu/fas hath not fo aduifedly noted. Agayne whete,Flufas{erreth inhis definition, as aneffentiall part thereof , that of the fue fuperfi- cieces,of which a Prifme is contayned.jtwojof them muftbe triangles, that vndoubtedly is not of ne- ceflitie,they may.be.offome other figure . Suppofe that.in the figure before geuen that in the place of the two oppofite figures, which there were two triangles, were placed.two pentagés: yet fhould the fi- gure remayne'a Prifme fiill,and agree with the definition of Ewchde and falleth nos vnder the definiti- on. of Flafzs. So that his.definitiéfemeth to be to. narrowand firetcheth not fo farre asic oughtto do, nor declareth the whole nature of the thing defined. Wherefore itis not to be preferrd before Euchdes definition,as he woulde haue it. This figure of Exclide called:a Prifme,is called of Gem pane and cettayne others Figura Serratélss,for that it repreféteth in fome maner the forme of a Sawe.And of fomeothers.it This bedie 1s Called Cumews,thatis,a Wedge,becaufe it beareth the figure of a wedge. called Figure Moreouer although it were fo;that the definitié of a Prifme fhould be fo large, that it fhould cétaine — all thefe figures noted-of F/afses.as fided Columnes, & all Parallelipipedons: yet fhould not Flufsas have fo.greata caufe-to finde fo notably a fault, fo veterly to reieétir. Iris no-rate thing in all learninges, chiefely in the. Mathematicalls,tohaue one thing moregenerall then an other Is it not true that euery Ifofceles isa triangle;but not euery, triangle is an Ifofceles?And why may notlikewife a Prifme be more a) generall,thena Parallelepipedon, or a Columne hauing fides€andicontayne them ynderitas a triangle cétayneth ynder it an Lofceles and other kinds of triangles).So that euery Prallelipipedon, or euery fi- ded Columne bea Prifme,but not euery Prifmea Parallelipiped6 ora fided: Columne. This ought noe to’be fo much offenfiue, And indeede it femeth manifeftly. of many, yea’ & of the learned fo to-be také, as clearely appeareth by the wordes of #/eHus in his Epitome of Geometrie,where he entreateth of the production and conilitution of thefe bodyes.His wordes are thefle . aH restiline fieures being erededGpom Pfellus. thesr playnes or bafesby right angles make Prifmes. Who perceaveth not but thata Pen tagon ereéted vpé his bafe of fiue fides maketh by his-motion a fided Columne of fiue fides?Likewifean Hexagon. erected at right angles.produceth a Columne hauing fixe fides; and {o.of all other re@tiline figures . All which folides or bodyes fo produced,whether they be fidedGolumnes or Parallelipipedons, be herein moft plaine words (of thisexcellétand auncient Greke author P/elus,)called Prifmes. Wherfore if the defini- ud of a Prifmegeué of Euchdefhould. extend it felfe folargely as Flaffasim agineth, and dhould enclude fuch figures‘orbodyes,as henoted:he ought notyet forallrhatfo much to be offended s and fona- rowly to haue fought faultes.For Exelde in fo neti mought haue that m eaning & fenie ofa Prifme which 1’/ellws had.So ye fee that Evclide may be defended either of thefe two.wayes,either by that thae the definition extendeth notto thefefigures,and fo not-+to be.ouer generall nor ftretch farther.then it ought: or ells by that that if it fhould.fretch fo faritis not fo haynous.For thatas ye fe many haue.také #tin'that fenfe,In deede.cémonly.a Prifine is taken in that fignificatié and meaning in which Cempanus Finffas and others take it.1n which fenfe it femeth alfo that in diuers propofitions in thefe bookes fol- lowing itought of neceflitie to be taken. 12... ASphere is a figure which 1s made, when the diameter ofa femicircle Twelueth dif Serrarzzs. ; abiding fixed, the femicircle ts turnedround about; vntill it returne'dnte frtion, 1 ss the felfe fame place from whence it began to be moued. * 9 : To the end we may fully and perfeétly vnderftand this. definiti- ; LE : ‘ on, how a Sphere is produced of the motion ofa femicircle, it thall Ce” aE | ; be expedient to cofider how quantities Mathematically are byima- i See i gination conceaued to be produced, by flowing and motion,as was bie yj Fas i fomewWhat touiched in the beginning of the firft booke . Euer the tf Wey, Yi; == i lefle quantitie by his motion bringeth forth the quatitie nextaboue beste Uf, = 5 \ it. Asa point mouing, flowing, or gliding, bringeth forth aline, BE Bat Ly, = 3 } AA which 1s thefirft quantitie; and nexttoa point. Aline mouing pro- Yew = m f } duceth a fuperficies, which is the fecond quantitie, and next vnto a = Bray Tine . And laft ofall, a fuperficies mouing bringeth forth a folide or ; =< ¢ body, which is the third & lait quantitie . Thefe thinges well mar- SS ey Red; it fhall not be very hard to attaine to the right ynderftanding SHAAN ofthis definition . Vpon the line A B being the diameter,delcribe 2 : , femicircle oni as ————— 0 es a a aT femicircle A'C B, whofe: | a centrelet be D : the dia~ © Lge a meter AB being fixed Fetes on his endes or’ pointes, ee m,-’ imagine the whole fu- See 1 BS --perficies of the femicir- cle to moue round from fome one point affigned, tillit returne to the fame point againe. Se fhall it produce a perfeét Sphere . JL of Globe, the forme whereof you {ee in a ball or. bowle . And it is fully round and folide, for tharit is‘defcribed ofa femicitele which is per- har isto be feétly round, ‘as our countrey man Yohunnes de Sarvo Bafco in his booke 7 of the Spheré,of this défnitionwhich he taketh ‘out of Ewelde, doth Vaven beede of wellcolleGte . But iis to be noted and taken heede of, that none be deceaued by the definition of a an the dzffins - Sphere geuen by Johannes de Sacro Bufco-: A Sphere faythhe) ss the pafsage or mouing of the circumference tion of @ {phere ofa femscircle, till st retutne Gato the place where st beganne, Which'agreeth not with Exchde. Enelide plain geven by lo- __ ly faytli;thata Sphereisthe paflage or motion ofa femicitcle; and not the paflage or motion of the cir- bannes de Sa- éuniference of a femmicircles neither ¢an it bé true that the circumference ofa femicircle;which is 4 line? eré Bufcos fhould defcribe a body .It was before noted that every quantitie moued, defcribeth and produceth thes Gtiantitie next vnto it . Whercfore aline nioued can not bring forth a body, buta fuperficies enelyAs if ye imagine a rightline faltened at one of his endes to moue about from fome one point uilkit returne to the fame againe, it fhall defcribea plaine fuperficies nanely,acircle .S6alfo if ye likewife conéedire of aicrooked liné,fuch as is the circumferetice of a femiciicle, that his diameter faftened on both the endes it fhould moue from a point afligned tilliereturne to the fame againe, it fhould defcribe & pro- ducea round fiperficies onely, whichis the fuperficies and limite ofthe Sphere, and fhould not pro- disce the body and foliditie of the Sphere . But the whole femicircle,which isa fuperficies by his mote On, as is before {aid; produceth a body that is, a perfect Sphere . Sofee you the errour of this defniti2 on of the author of the Sphere : which whether ithappered by the author him felfe, which'I thinke fot : or that that particle was thruit in byfome one after him, which is more likely, it it not certaine: Butitis certaine, thatitis vnaptly put in, and maketh an wtrue definition : which thing is not here {poken, any thing to-derogate the author of the booke, which afluredly was a man of excellent know= fedge* neither-to the hindrance or diminifhing' of the woithines of the booke, which vadoubtedly is z very necéflary booke, then which I know noné more mecte'to'be taught and red in feholes touching: the groundes and ‘principles of Aftronomie and Geographic : but onely toadmonifhe the young and Theodofius V2 {kilfull reader of not falling into errour. Theedofius in tis booke De Sphericis ( a booke very necefla- “facie fy forall thofe which will{ee the groundes and principles of Geometrie and Afttonomie, which alfo E difinition of 4 haue tranflated into our vulgare tounge,ready to the preffe) defineth a Sphere: after: thys maner: #i p heree A Sphere +8.a folide or body contained Guder one [eperficies, in the midle wherof there s3 a point, fra which all lines drawen torhe circumference are equall. This definition of Theodefins 1s more effentiall and naturall,then is the othergenen by Ewchde. The other did not fo much declare-the inward nature and fubftance of 2 Sphere,as it fhewed the induftry and knowledge of the producing-of a Sphere,and therfore is a caufall definition geuen by the caufe efficient, or rather’ defcripcion then adefinition . But this definition’is very effentiall, declaring the nature and fubitance of a Sphere « As ifacircle fhould be thus defined, as it well may: 4 csrcle ss the pafSage or mowing of a line from 4 post tillirveturne to the fame point againes itis a caufall definition, fhewing the efficient caufe wherofa circle is produced, namely, of the motion ofa line. And it isa very good defcription filly fhewing what a-citde is. Such like defcription is the definition.ofa Sphere geuen of Exdisde\ by the mation.of < femicirtle,. But when a circle is defined to bea plaine {uperficies, in the middeft wherofis a point, fom which_all- lines ‘drawen to the circumfe- rence therof,are equall : chis déefiiitiod is effentiall arid frmall ;and declareth the very nature ofa cir- cle . And vnto this definition ofa circle, is correfpondent the definition ofa Sphere geué by Theodofins, faying : thatitis a folide or body, in the middeft. wherecf there is a point, from which all the lines drawen'to the circumference are equall. So fee you the afinitie betwene a circle anda uae whae a circle is in a plaine,thatis aSphereina Solide.. The fulnesand content of acircleis deferibed by the motion ofa line moued about ; but the circumference tkerof, which is the limite and border thereof, isdefertbed of the end and point of the famé line moued about. So the fulnes,content, and body ofa Sphere or Globe is defcribed of a femicitcle’‘moued abort. But the Sphericall fuperficies, which is the The circamfe- limite and border of a Sphere, is deferibed of the citcum‘erence of the fame femicircle moued about. A Do t & ELAM MAE ¢ z T be eleuenth Booke Pas yonce of & And this is the fuperficies ment in the definition, when it is fayd, that it is contained vnder one fuper- fy heve fictes, Which fuperticies is called of Lohannes de Sacro Bufer & others, the circumference of the Sphere, e me ee diffi Galene in his booke de Asffinitsonsbas medicis,ceueth ye: an other definitid of a Sphere ,by his proper= kK GEL tje~ _ : tie or comon accidéce of mouing,which is thus. 4 Spheress a figure mof? apr to all mution,as hauing wo ba RILIORN & whereon tofiay.1 his is avery plaine and witty defitition, declaring the dignitie thereofaboue all figures {phere _, enerally.All other bodyes or folides,as Cubes,Pyramid:, and ether s hane fides, bafes,and angles, all The diguitie Whichare ftayes to reft vpon , orimpedimentes and lets:o motion . But the Spheré hauing no fide or J of 4 [pberee peste: bale: of Euclider Elementes. Fol.316. bafe to ftay one,nor angle to let the courfe hereof, but onely in a poynt touching the playne wherein it ftandeth,moueth freely and fully with out bt.And forthe dignity and worthines thereof , this circular and Sphericall motion is attributed to the leauens , which are the mof} worthy bodyes .. Wherefore there is afcribed ynto them this chiefe kind: of motion. This folide or bodely figure is alfo commonly called a Globe. A Sphere cal. leda Globe. 13° The axe of a Sphere isthat right line which abideth fixed about Thirtenth dif- which the femicircle was moved, finition. As in the example before geuen in the jefinition of a Sphere,the line A B, about which his endes being fixed, the femicircle was moued ( whih line alfo yet remayneth after the motion ended ) is the axe of the Sphere defcribed of that femicirck , Theedofius detineth the axe ofa Sphere after thismaner. __ Dheaxe of aSphere is a certayne right line draven by the centre, ending on either ‘fide 12 the [uperficves of the Theodofius Sphere,about which being fixed the Sphere is turmd.As the line AB in the former exam ple. Therenedeth to diffinition of this definition no other declaration , but orely to confider , that the whole Sphere turneth vpon that rhe gxe of « line AB,which paffeth by the centre D,and is extended one either fide to the {uperficies of the Sphere, Sphere. wherefore by this definition of Theedofius its the axe of the Sphere. 14. I'he centre of aSpheres that poynt which is alfo the centre of the fee Koyonh micircle. diffivitions This definition of the centre ofa Sphere is geuen as was the other definitionof the axe » Namely, hauing a relation to the definition of a Spher: here geuen of Ewelde: whereit was fayd that a Sphere is made by the revolution of a femicircle,whof: diameter abideth fixed. The diameter of a circleand of a femicrcle is all one. And in the diameter either of a circle or ‘of a femicirdleis contayned the center of either of them,for that the diameter of eche :uer paffeth by the centre . Now (fayth Ewclide ) the poynt which is the center of the femicircle,by whofe motion the Sphere was defcribed , is alfo thecentre of the Sphere, As in the example there geuen, the poynt D is the centre both of the femicircle & alfoof the Sphere . Theedofivs geueth an other definitioi of the centreof a Sphere which is thus . The centre of « Theodohus Sphere 1s a poynt with in the Sphere from which ill lines drawen to the [uperfictes of the Sphere are equall . Asin diffinition of a circle being a playne figure there is a poynt n the middeft,from Which all lines drawen to the circum- sho center of a frence are equall , which is the centre of the tircle : fo in like maner within a Sphere which isa folide Ipbere and bodely fgure,there muft be conceaued apoynt in the middett thereof,from which all lines drawen Puerto “ro the fuperficies’ thereof are equall / And ths poyat’is the centre of the Sphere by this definition of *Theodofias.Flufasin defining the centre ofa Sohere comprehendeth both thofé definitions in one, after | this fort: The centre of a Sphere itt poynt affigied in a Sphere from which all thé lines drawen to the fuperfi- : ‘ies are equall and st 4s the Jame which was alfo the centre Sie femicircle which defersbed the Sphere.This defi- F, sufsas diffi-~ ; ‘nitionis fuperfluous and contaynéth more thé nedeth -For either part thereofisa full and fufficient dif- métion of the finition,as petore hath bene fhewed, Orells lad Exclide bene infuficient for leauing out the one part, center of 4 or Theodofiws for leauing out the other . Paradienture Fé/fasdid it for the more explication of either, Iphere. “that the one part mught open the other. 15° Thediameter of a Sphere is acertayne right line drawen by the cétre, oan dif ; |} . and one eche fide endin 1g at the [uperficies of the fame S phere. — 2» bas This definitié alfo.isnot hard,but may eafdy be couceaued by the definitid of the diameter of a cir- cle.Foras the diameter of a circle is a right lire drawne fr6 one fide of the circiifrence ofa cirtle to the other,pafling by the centre of the circle: fo inagine youa rightline to be drawen from one fide of the fuperficies ofa Sphere to the other,paffing bythe center of the Sphere, and that line is the diameter of Difference be- the Sphere. So itis notall one to fay,the axe of a Sphere , and the diameter of a Sphere. Anyline ina pone phe di- Sphere drawen from fide to fide by the centreis a diameter.But not euery line fo drawen by the centre ameier Gare is the axe of the Sphere , but onely one right ine about which the Sphere is imagnined to be moued, : p So that the name of a diameter ofa Sphere is nore general,then is the name of an axe.For euery axe in of 4 [phere, a Sphere is adiameterof the fame: bit not eurry diameter of a Sphere isan axe of the fame.And there- fore F/u/fzsfetcech-a-diameter in the definitior of an axe€ asa more generall word in this maner.The axe | SS.4, of T heeleuenth Booke of « Sphere,ss that fixed diameter about which the Sphere is mowed, A Sphere (asalfoa circle)may haue infi- nite diameters, but it can haue but onely one axe. Siaivid 16 Alconeis a folide or bodely figure which is made when one of the fides diffinitions of a rectangle triangle, namely , one of the fides which contayne the right angle abiding fixed the triangle is moned about ,ontill it returne nto the felfe fame place from whence it began first tobe moued. Now if the right line which abideth fixed be equall to the other fide which is moued about and containeth the right an cle:then the cone is a rectangle cone.But if it be leffe then is it.an obtufe angle cone. And if it be greater , thé is it ana cutea angle cone, This definition ofa Cone is of the nature and condition that the definition of a Sphere was, for either is geuen by the motion of a fuperficies. There,as to the production of a Sphere was imagined a femicircle te moue round, from fome one point till it returned to the fame point againe : fo here mutt yeimagine a reQtangle triangle to moue about till it come againe to the place where it beganne . Let asc be areétangle triangle , hauing the angle asc a right angle, which let A be contained ynder thelines azands c. Now: {uppofe the fide az , namely, one of the lines which cotaine the right angle ag c to be fattened, and abour it fuppofe the triangle arc to be moued from fome one poynt afligned till it re- turne to the fame agayne (as vppon the diameter in the definition of a Sphere ye imagined a fe- micircle to moue about): fo fhall the folide or body thus defcribed bea perfec Cone. Asyou may imagine by this figure here fet. And the forme ofa Cone you may fufiiciently con@eaue by the figure fet inthe margent. There are of Cones three kindes, namely, a reGtangle Cone, an obtufeangle Cone, and an acute angle Cone,all which were beforein the former definitio defined: Name~ ly the firft kindeiafter this maner. If the right line which abideth fixed, be equall tothe other fide which moneth round about, and containeth the right angle,then the Cone ss arédtangle Cone. Feri kinde As fuppofe in the former example, that the line a 8 which is fixed,and about which the triangle of Cones. was moued, and after the motion yet remayneth, be equall to the line 8 c,which is the other line con- BA OPN Ts: tayning the right angle,which alfo is moued about together withthe whole ray si then is the Cone defcribed,as the Cone ap ¢ in this example,a right angled Cone : fo called for that the angle at the toppe of the Cone isarightangle. For forafmuch as the lines a Band 2c of the triangle a Bc are e- quall, the angle 8 a c is equallto theangle Bc a ( by the s.of the firft) . And eche of them is the halfe of the right angle a8 c (by the 32. ofthe firit) . In like fort may it be fhewed in the triangle ABD, thatthe angle s p a is equall to the angle 8 aD, and that eche of them is the halfe of a right angle. Wherefore the whole angle c ap, which ts compofed of the two halfe right augles,namely ,p ABand ¢ AB is arightangle . And fo haue ye whatisa right angled Cok Cone. | | But sf st be leffe,them is st an obrufeangle Cone . As in this ex- ample, theline 4 8 fixed is leffe thenthe line 8 c moueda- bout. Wherefore the Conedefcribed of the cireumuoluti- on of the triangle as ¢ abouttheline 48, isah obtufean- ele Cone, for that the angle at the toppe D4 C is greater thena right angle. Wherefore it is an obtufeangle. And therefore the Cone is called an obtufe angle Cone. E And Ss of Euclides Elemente’: 317, And if it be preater,then ss it an acuteangle Come.As itt this figure, the line AB faftened, is greater then the line BC moued about . Wherefore the Cone de- fcribed by the motion and turning of the triangle ABC about AB isan acuteangle Cone, hauing the . angle atthe toppe BA C an acute angle .fOfwhome the Cone is called an acuteangle,Cone. For the ea fier fight 8 cofideration ofall thefe kindes of Cones, and alfo for the plainer demonftration of the varie» ties oftheirangles in their toppes, I haue defcribed \ them all three in one playne figure , of which the Pee ee Cone ACB isaright angled Cone, hauyng his fix- = Sey oa ones ed fide C F equall to the line FB, and hys angle nil Sieetalie noise ACB arightangle: the Cone AEB is an obtufe angle Cone, and ADB an acuteangle Cone, By which figure ye may eafily demontftrate D (by thezr. of the firlt) that the angdleADB | of the Cone ADB, whofe fixed line DE is greater then the fide FB, is leffe-then the rightangle A CB, and fo is an acute angle, C And alfo (by the fame 2t. of the firlt) ye \ fhall with like facilitie perceaue how the 2 angle AEB of the Cone AEB whofe fix« ‘ edline E Fis leffe then the fide F B,is grea- ter then the right angle ACB : and there. 3 fore isan obtufe angle. This figure of a Cone is of Campane, of / Vitellio, and of others which haue written in bi" 5 7 . Al Cone called thefe latter times, called a round ‘Pyramis, ANS = of Campane a which isnot fo aptly . Fora Pyramis, anda ee roundé Pize- Cone, are farre diftant, & of fundry natures, | wie, A Coneisaregular body produced of one circumuolution of a rectangle triangle , and limited and bordered with one onely round fuper- ficies . But a Pyramis is terminated and bordered with divers fuperficieces . Therefore can not a Cone by any iutt reafon beare the name of a Pyramis. This folide of many 1s called Twrbo,which to our purpofe may be Englifhed a Top or Ghyg: and mor eouer, peculiarly Campane calleth a Cone the Py- ramis of a round Columne,namely,of that Columne which is produced of the motion of 2 parallelo- gramme (contained of the lines A B and B C) moued about,the line A B being fixed. Of which Co- lumnes fhall be fhewed hereafter, : | 17 The axe of aCone is that line, which abideth fixed, about which the triangle 1s moued. And the bafe of the Cone is the circle *phich is defcribed Senententh » by the right line whichis moued about. ) diffinition. Asin the #3 the line 4 & is fup- pofed to be the liné ‘about which the | A right angled triangle 4B C (to the pro- duction of the Cone ) was moued : and that line is here of Exchde called the axe ofthe Cone deferibed . The bale of rhe Cone ssthe circle which is.defcribea by the right line which is moued about. As the line 4B was fixed and ftayed , fo was the line BC (together with the whole triangle 4BC) moued and turned a- G bout .. A line moued) as hath bene fayd : before,produceth a fuperficies : and be- caufe the line BC is Moued ‘about’a = point; namely, the point. 8, being the | | end of the axe of the Cone 4B, itproduceth by his motion, and revolution a circle, which circle is the bafe of the Cone :'as in this exam ple, the circle cz. | —— ~~ SS, ij. The Se T be eleuenth Booke The line which produceth the bafe of the Cone, is the line of the triangle which together with che axe of the Cone contayneth the right angle. The other fide alfo of the triangle,namely,the line 4 C, is moved about alfo with the motion of the triangle,which with his reuolution deferibeth alfo a.fuper- eA conicall [u- ficies, which is a round fuperficies, 8 is erected vpon the bafe of the Cone, & endeth ina point,name- perficies. ly, in the higher part or toppe of the Cone. And itis commonly called a Conicall fuperficies. Eigheenth 18 A cylinder isa folide or bodely figure which is made, ‘when one of the diffinttion. fides of a reétangle parallelogramme, abiding fixed , the parallelogramme ' is mowed about, vntillit returne to the felfe fame place from whence it bee gan to be moued. This definition alfo is of the fame fort and condition,that the two definitions before geué were, namely , the definition ofa Sphere and the definition ofa Cone . Forallare geuen by mouing ofa fu- perficiesabouta right line fixed,the one ofa femicircle about his diameter, the other of a rectangle tri~ angle about one of his fides. And this folide or body here defined is caufed of the motion ofa rectangle parallelograme hauing one of his fides contayning the right angle fixed from fome one poynt till itre- turne to the fame agayne where it began. As fuppofe ABC Dtobea rectangle parallelogramme , hauing his fide A B faftned, about which imagine the whole ‘Ma LA parallelogramme to be turned , tillit returneto the S 444 poynt where it began, then is that folide or body, by N 34 this motion defcribed,a Cylinder: which becaufe of S 74 his roundnes can notat full be defcribed ina playne & Figs fuperficies,yet haue you for an example thereof a fut- XS = ficient defignation therof inthe margent fuch-asina S =m — pilaine may be.If you wil perfectly behold the forme S =A; of a cilinder.Confider a round piller that is perfedt- 4 Sas ly round. 4 Sen EM | PRIYA UE De AN Ll VTE iia INSET ATA, UPR ELA oi Mii Hi ie | : Ninetenth 19 Ihe axe of acilinder is that right line which abydeth fixed, about diffinition. which the parallelogramme is moued. And the bafes of the cilinder are the circles defcribed of the two oppofite fides which are moued about. Euen as in the defcription of a Sphere the: line faftened was the axe of the Sphere produced : and in the defcription ofa cone,the line fiftened was the axe of the cone brought forth: fo in this defcripti- on of a cilinder the line abiding, which was fixed,about which the reétangle parallelogramme was mo- ued is the axe of that cilinder.As in this example is the line .4 B. The bafes of the eslinder €§c.In the reuo- lution ofa feces nemo onely one fide is fixed , therefore the three other fides are moved about: of which the two fides which with the axe make right angles, and which alfo are oppofite fides,in their motion defcribe eche of them a circle which two circles are called the bafes of the cilinder.Asye fee in the figure before put two circles defcribed of the motié of the two oppofitlines ADandBC,which are the bafes of the Cilinder. ok A cillindvicall The other line of the rectangle parallelogramme moued , by his motion deferibeth the round fu- piss perficies about the Cilinder.As the third line or fide ofa retangle triangle by his metion defcribed the JuperficteS- sound Conical fuperficies about the Cone.And as the circtiferéce of the femucircle defcribed the round {phericall fuperficies about the Sphere. In this example it is the fuperficies defcnibed of the line DC. Corollary By this definition it isplayne that the two circles , or bafesofa cilinder are. euer equall.and paral- * els: for that the lines moued which produced them remayned alwayes equall and parallels . Alfo the axe of a cilinder is euer an eretted line vnto either of the bafes. For with all the lines defcribedin the bafes, and touching it,it maketh right angles, A rounds Co- Cumpane,V iteléo,with other later, writers,call this folide or body a round Columne or piller, And dumne or Campane addeth vnto this definition this,as a corollary. That of a round Columne, of@ Sphere, Aad ? parmages Jpbera. Seas re of Euclides Elementes. Fol.318. ofa circle the cétre is one and the felfe fame. That is ( as he him felfe declareth it & proucth thefame) > 4 Corollary here the Columne,the Sphere,and the circle haue one diameter, a eee added by Cam ane. a ee 20 Like cones and cilinders are thofe, whofe axes and diameters of their “Twenty diffie bafes are proportional Nitiote The fimilicude of cones and cilin- {\ ders ftandeth in the proportion of thofe | right lines,of which they haue their ori- ginalland {pring . For by the diameters of their bafes 1s had their length and breadth , and by their axe is had. their f heigth or deepenes, Wherefore to fe / SS Se x whether they be like or ynlike , ye mutt Lo | SPe aa | compare theiraxes together , which is CS Se SE Sere SONY ae AY their depth, and alfo their diameters to- Secs Se eg ut gether,which is thierlength & breadth. As if the axe B G of the cone 4B Cbs to to the axe EZofthecene DEF ,asthe as diameter .4C ofthecone 428 Cistothe a Rim. diameter D F of the cone D £ F,then are SS Wee: the cones 48C and DEF like cones. Likewife in the cilinders. Ifthe axe Z NW of the cilinder £ A .7N haue that pro- portion tothe axeO 2 of the cilinderR OP 2 which the diameter H hath to the diameter R 7: then are the cilinders HLMNand ROP @ like cilinders,and fo of all others. 21 A (ubeisa folide or bodely ) | figure contayned vnder fixe e Sinem quall [quares. - As is a dye which hath fixe fides, and eche of them isa full and perfe@ fquare, as limites or bor= ders vnder which itis contayned. And asye may concetue in a piece of timber*contayning a foote {quare euery way, orinany fuch like. So that a Cube is fuch a folide whofe three dimenfionsare equall;the length is equall to the breadth thereof, and eche of them equall to the depth.Here is as it may bein aplayne fuperficies fer an image cherof, in thefe two figures wherof the firitis as itis com- monly decribed in a playne,the fecond (which is inthe beginning ofthe other fide of this leafe) is drawn as itis defcribed by arte vp6a playne fu- perficies to thew fomwhat bodilike.And in deede the latter defcriptiéis for the fight better thé the firft.But the firft for the dem6ftrations of Euclides propofitions in the ‘fiue -bookes following is of more vfe,for that in it may be confidered and fene SS. iit. all T he eleuenth Booke a)| the fxefides ofthe Cube. And fo any lines or fections drawen in any one of the fixe fides. Which can not be fo wel fene in the o- ther figure defcribed vpona playne}. Andas touching the firft figure ¢ which isfetat the ende of the other fide of this leafe) ye fee that there are fixe parallelogrammes which ye mutt conceyue to be both equilater and reétangle, although in dede there can be jn this defcription onely two of them reétangle,they may in dede be defcribed al equilater.Now if ye imagine one of the fixe paral- lelogrammes,as in this example,the parallelogramme AB C Dto 7 _ be the bafe lieng vpon a ground playne fuperficies. And fo con- ceiue che parallelogramme E F G H to bein the toppe ouer it, in fuch fortr,that the lines A E,C G,DH, & BF may be erected per- sendicularly from the pointes A,C,B,D,to the ground playne fu- perficies or fquare AB CD. For by this imagination this figure wil fhew vnto you bodilike. And this imagination perfectly had, wil make many of the propofitions in thefe fiue bookes following, in which are required to be defcri- bed fuch like folides(although notall cubes) to be more plain ly and eafily concrined. In many examples of the Greeke and alfo of the Latin, there is in this place fet the diffinition of a Te- trahedron,which is thus. “ ~ 96 beige Twenty tive 22. AT etrabedrowis a fokde whichis contained Yuder fower triangles ll and equilater. d i < A forme of this folide ye may fee in thefe two examples here fet, whereof one is asitis commonly defcribed in a playne . Neither is it hard to conceaue. For (as we before taught in-a Pyramis ) ifye imagine the triamegle BCD to lie vpona ground plame fuperficies, and the point Asgo be pulled vp together with the lines A B,A C,and AD, ye thall paiauc the forme of the Tetrahedron to be contayned vnder 4.trianglesawhich ye mutt imagine to be al fower equilaterand equiangle, thougl they can not fo be drawen ina plaine . And aTe- trahedron thus dé{cribed, is of more vfe in thefe fine bookes follow- ing, then is the other, although the other appeare in forme to the eye ATetrahee tnorebodilike. drou one of _ Why this definition is here left out both of Campane and of Fluffas, the tine yeau~ can not but maruell, confidering that a Tetrahedron,is ofall Philo- 5 ee fophers counted one of the fiue chiefe folides which are here de- ——_* fined of Euclde, which are called cémonly regular bodies, with- out mencion of which,the entreatie of thefe fhould feeme much maimed : vnlefle they thought it fufficiently defined vnder the definition of a Pyramis, which plainly and generally taken ,inclu- deth in deede a Tetrahedron, although a Tetrahedron properly much differeth from a Pyramis,as a thing fpeciall ora particular, from a more generall.For fo taking it,euery Tetrahedron 1s a Py- rarnis, but not euery Pyramis is a Tetrahedron . By the generall definition of a Pyramis, the fuperficieces of the fides may be as | LR many in numberasye lift, as 3.4.5.6.ormoe,according tothe /—- ? ee ae NS forme of the bafe, whereon itisfet, whereof before in the-defi- | nition of a Pyramis were examples geuen. But in a Tetrahedron the fuperficieces erected can be but three in number according to the bafe therof,which is euer a triangle . Againe, by the generall definition of: Pyramis,the fuperfi- cieces ereéted may afcend as high as ye lift, butina Tetrahedron they muft all be equall to the bafe. Wherefore a Pyramis may feeme to be more generall then a Tetrahedron, as be‘orea Prifme feemed to be more generail then a Parallelipipedon, or a fided Columne : fo that enery Parallelipipedon is 2 | Priftne, butnoteuery Prifme isa Parallelipipedon . And every axe in a Sphere isa diameter + but not Piellas calleth euery diameter of a Sphere is the axe therof .So alfo noting well the definition ofa Pyramis,euery Te- a Tetrahcdrgsy tahedron may be calleda Pyramis, but not euery Pyramisa Tetrahedron. And in dede P/e/xs in num- bring of thefe fiue folides or bodies, calleth a Tetrahedron a Pyramis in_manifelt wordes . This I fay might make F/u/fus & others(as I thinke it did) to omitte the definition of a Tetrahedron in this place, as {ufficiently comprehended within the definition ofa Pyramis geuen before . 3ut why then id he not count that definition of a Pyramis faultie,for that it extendeth it felfe to larg¢,and comprehendeth vnder ita Tetrahedron ( which differeth from a Pyramis by that it is contayned of equall triangles) as he not fo aduifedly did before the definition of a Prifme. Twenty three 23 .AnOétohedronis a folide or bodily fi eure cotained bnder eight equall definition. and equilater triangles. : | ) of d diffinition. equa aS SS = ¥ Sr e SA = — SSS >a ~~ a — - eee @ € tials. . » a “a —s - “ Ae = —~ << = ? == 2 ~ —s = SS ee --~ = = =- ——— —, — ——= As of Euchites Elementes. Fol.319 Asa Cubeis a folide figure contayned vnder fixe fuperficiall f- gures of foure fides orlquares which are equilater, equiangle,and 2 equallthe one to the other: fo is an O¢tohedron a folide. figure : contained vider eight triangles which are equilater and equal! the een one to the other. As ye may in thefe two figures here fet beholde. A / “5B Ww hereof the firit is drzwen according as thie {olide is commonly : — —— ee ee ee —— ‘ _ = a es a ——- ee ey, ee a a op = + eee Fe on = a P< ae : 4 ee — a - = ers - ers a —— ees =~ - on =u ie ee a LE SS YP — ee ee ee wasn wet T he eleuenth Booke So here you may behold twelue Pentagons,which if you imagine to be equall,equilater, & equiangle, and to be lifted vp,ye fhall (although fomewhatrudely)conceaue the bodily forme of a Pentagon. And fome light it will geue to the vnderitanding of certaine Propofitions of the flue bookes following con- cerning the fame. i Tventy fine 2s AnIcofahedronis a folide or bodily figure contained ynder twentie diffinttion, equall and equilater triangles. Asthefolides before lat mentioned are all defcribed by the number and forme of the fu- perficieces which containe them : fo this body likewife is defined by that that it is contayned of twenue triangles equall, equilater, and ¢ - quiangle . And aithough this folide alfo be ve- ry hard to conceane, as it iscommonly deirri- bed vpon a plaine (an example wherofyou haue in the firit figure here fet): yetts it of neceNitie that in that forme it be defcribed, if we will vndertland fuch defcriptions as are ft forth Exclide touching that body in the fiue bookes following. Howbeit you may byit (although fomewhat rudely ) fee the 20. triangles, which are imagined to be equall ,equilater,and equian- gle, ifyou coniider fine angles of fine triangles to concurre togetherat 2 point. And forafmuch as there arein this folide 20. triangles,and enery triangle hath three angles, the concurfe of the faid triangles will be in twelue pointes . Asin this example the pointes of the concurfe are A, B,C, D,E,F,G,H,K,L,M,& N.Where note that in this plaine the two poyntes Mand N are but one point, yer muft yeimagine one of thofe pointes to be erected vpward,and the ether down- ward . Now the fiue triangles which concurre in the point M, are theie, BMD,DMFE,FMH,HML,and LM? : the hue triangles which concurre in the point N, and are imagined to be erected downward, arée thele, ANC,CNE,ENG.G) K,andKNA: the other ten triangles which include this body,are thefe, ABC, BCD,CDE,DEF,EFG,FGH,GHK,HKL, KLA,;LAB. The fecond figure here appeareth more bodilike ynto the eye. Thefe fiue folides now Jaft defined namely,a Cube;a Tetrahedré,an OGohedron, a Dodecahedron Fine regular and an Icofahedré are called regular bodies.As in plain fuperficieces, thofe are called regular figures, bodies, whofe fides and angles are equal,as are equilater trianales .equilater pentagons, hexagons, & fuch lyke, ; fo in folides fuch only are counted and called regular,which are coprehéded vnder equal playne fuper= ficieces, which haue equal fides and equal angies,as all thefe fine forefayd haue, as manifeftly appeareth by their definitions, which were al] geuen by this proprictie of equalitie of their fuperficieces, which haue alfo their fides and angles equail, And in all the courfe of nature there are no other bodies of this condition and perfeétion, but onely thefe fine. W herfore they have ever of the auncient Philofophers bene had in great eftimationand adiniration.atid haue bene thorght worthy of much contemplacion, about which they haue bettowed mott diligent ftudy and éendeuour to fearche out the natures & pro- perties of them.They are as it were the ende and perfection of al} Geom etry, for whofe fake is written whatfoeuer is written in Geometry. They were (as men f. ay) firft inuented by the moft witty Pithaga~ The dignity of ras then afterward fet forth by the diuine Plere, and lait of all n erucloufly taughtand declared by the thefebodres, mottexcellent Philofopher Ewclide in thele bookes tollowing,and euer fince wonderfully embraced of ATetrabe- aillearned Philofophers.The knowledge of them containeth infinite fecretes of Nature. Psthagoras, Ti- drow aleribed Hise ee ts by them fearched out the copofition of the world,with the harm ony and preferuation aento the fore ah ri bes fiue pom to the fimple parres therof,the abe Tetrahedro they aferi- An oélohe- ee se the fre,forr hat it aicendeth vpward according to the figure of the Pyramis .To the ayre they an afcribed the O@ohedron,for that through the fubtlée moifture which it hath,it extendeth it felfe enery dron afcrives way to the one fide,and to the other,accordyng as thatfigure doth. Vato the water they affigned the pute the cyre, Tkofahedron of Euclides Elementes. 34.0, Ikofahedron,for that it is continually flowing and mouing,and asit were makyng angles on euery fide according to that figure.And to the earth they attributed a Cube,as toa thing ftable,firme and fure as the figure fignifieth.Laft ofall a Dodecahedron,for that it is made of Pentagés, whofeangles are more ample and large then the angles of the other bodies,and by that meanes draw more to roundnes,& to the forme and nature ofa {phere, they afligned to a fphere, namely,.to heauen.Wh o.fo willread lero in his Tnews,fhall -ead of thefe figures,and of their mutuall proportion, ftraunge matters, which here are not to be entreated of,this which is fayd,fhall be fufficient for the knowledge of them,andfor the declaration of their diffinitions. After all thefe diffinitions here fet of Ewclide,. Flufsashath added an other diffinition, which is ofa Parallelipipedon,which bicaufe it hath not hitherto of Eclde in any place bene defined, and becaufe: it is very good and neceflary to be had,I thought good not to omitte it, thus it is, A parallelipipedonis a folide figure comprehended vnder foure playne qua- drangle figures of which thofe which are oppofite are parallels. As in playne fuperficieces a parallelogramme is that which is contained A vnder foure fides beyng lines, and whofe oppofite fides are equidiftant and fy parallel lynes, fo in folide figures a Parallelipipedon is that folide which is $f, contayned vnder foure quadrangle fuperficieces, whofe oppofite fides are al- i what conceiue therof by the example in the margent. Ya two bafes are Poligonon figures, lyke,equall,equilater,an grames:ye may better coceiue the forme therof by the figure put vnder the figure of the parallelipipe- don, which apeareth more bodilike. There may of thefe be infinite formes according to the dinerfitie of their bafes. Becaufe thefe fiue regular bodies here defined are not by thefe figures here fet, fo fully and linely expreffed,that the ftudious beholder can through- ly according to their definitions conceyue them. I haue here geuen of them other defcriptions drawn in a playne, by which ye may eafily attayne to the knowledge of them. For ifye draw the like formes in matter that wil bow and geue place, as moft apt- ly ye may do in fine pafted paper, fuch as paftwiues make womés paftes of, 8 thé witha knife cut euc- ry line finely,not through, but halfe way only if thé ye bow and bende them accordingly, ye fhall moft...j plainly and manifettly fee the formes and shapes of . | thefe bodies,euen as their definitions fhew.And it fhall be very neceflary for you to have ftore of that pafted paper by you;for fo fhal you wpon itdeferibe the formes of other bodies, as Prifmes and Parallelipopedons;and {uch like.. , (et forth in thefe fiue bookes following, and fee the yery-formes of thofe. .:}., bodies there mécioned: which will make thefe bokes concerning bodies,as |; eafy vnto you as were the other bookes, whofe figures you might plainly fee vpon a playnefuperficies. | fo parallels,as itis eafily to be fene and conceaued in a cube or die, all whofe | oppofite fides are parallel fuperficieces, & fo of others like,ye may alfo fome- There is alfo in thefe bookes following, mencion made of folides, whofe d Pigivee and the YZ, fides fet vpon the bafes are parallelogrammes : which kynde of folides Cam- ty pane calleth fided Columnes(and which as was before noted,may be copre- # ded ynder the definition of a Prifme) a forme wherof although grofely, be- hold in this example, whofe bafes are two like equal, equilater, equiangle, and parallel hexagons, and the fides fet vppon thofe a are fixe parallelo- Ln | “4n tkofahedron afsigned Gnto the water, A cube afsegned Gre the tarth, A dodecahe= dron ufsigned to heauen. Diffinition of a para Helspspe~ dom. ) | fided Ce- lune. of Tetrsbe- drone Ae oftobe- tbls if ye draw this figure confifting as ye fee of fower equilater and equian- gle triangles vpo patted paper,or vp pon any other fuch like matter that will bowe and geue place, and then cut not through the paper, butas it were halfe the thicknes of the paper, the three lines contained within the fieure,and bowe & folde in the fower triangles accordingly: they will clofe cogether in fuch fort, that they will make the perfecte forme of a Tetra- hedron. This figure (confifting of fixe equall {quares) drawen vp- on palted paper, and the fiue lines contained within the fi- gure being cut finely halfe the thicknes of the paper, or not through, ifthen ye bowe and folde accordingly the fixe e- quail fquares, they will fo clofe together, that they will caufe the perfecte forme of a Cube. T he eleuenth Booke This figure (which confifteth ofeighres quilater and equiangle triangles) drawen vp= - ontheforefayd matter,and the feuen lines contained within the figure being cutas bes | fore was taught,and the triangles bowed and |, folded accordingly, they willclofe together - in fuch fore, thae they will make the perfecte forme of an O@ohedron. Delcribe of Euchides Elementes. — Fot.z4.t. ‘Deferibe this figure which confifteth of twelue equilater and equiangle Pentagens, vpon the’ fore- A Dodecabe- faid matter, and finely cutas before was taught the eleuen linescontained within thefigure, and bow dis : and folde the Pentagons accordingly . And they will fo clofe sogether, that they will make the very : forme ofa Dodecahedron. . | : T he elenenth Booke aa , .o0 Afye deferibe this figure which confifteth of twentie equilater and equiangle wiangles vpon the re Tcofabe forefaid matter, and finely cut as before was fhewed the ninetene lines which are em within duit the figure, and then bowe and folde them accordingly, they will in fuch fort clofe together, that there will be made a perfecte forme of an Icofahedron. eo aA Becaufein thefe fiue bookes there are fometimes required other bodies befides the forefaid fine regular bodies, as Pyramifes of diuers formes, Prifmes,and others, I haue here fet forth three figures of three fundry Pyramifes, one hauing'to his bafe a triangle, an other a quadrangle figure, the other a Pentagon : which ifye defcribe vpon the forefaid matter & finely cut as it was before taught the lines contained within ech figure, namely, in the firft,chree lines,in the fecond,fower lines,and in the third, fiue lines, and fo bend and folde them accordirigly, they will fo clofe together atthe toppes, that they will make Pyramids of that forme that their bafesare of. Andif ye conceaue well the defcribing of thefe, ye may moft eafily defcribe the body of a Pyramis of what forme fo ener ye will. A triangled Pyramis. : of Euctides Elementess\\ Fol.32,7. RL - 3 The forme of a quadrarng- ded Pyramis, eee Sr —— er * a J he formeof a fine angled Pyravtise ‘ TPs Likewile The forme of 4 prt[iie. T be forme of a paralletzpi- pedon T be eleuenth Booke Likewife if ye defcribe this figure vpon the forefaid matter, and finely cutte the fowerlines cétained within the figure, and bowe and folde them together accordingly, the three paral- lelogrammes and the two triangles will fo ‘clofe together, that they will caufe the perfede forme of a Prifme cotained vnderthree parallelogrames andtwo equediftant triangles. And conceauing this defcription well, it fhall not be ha:d to defcribe any o- ther Prifme of any other forme. Touching the defcrip- tion of Parallel pipedons I fhall not neede to {peake. For if ye conficer well the defcription of : Cube, it fhall not be tard to de- {cribe a Parallelipipedon of what form: ye will. Onely where as ina Cube all the parallelogrames in the defcription of that fi- gure are {quares, in the de- {cribing of a Parallelipipe- don , the fayd parallelo- ramme may ke of what 5 aa ye will . So thatye take heede that the oppo- fite parallelogrmmes be equal & equiangle. Which oppofite parallelogrames in the figure asi: lieth ina peste any tvo paralle- ogrames leauirg one pa- rallelogramme betwene them ..An example wher- of beholde in this figure. Becaufe thefe fiue bookes following are fomewhat hard for young beginners, by reafon they mutt in the figures de(cribed in.a plaine imagine lines and fuperficieces to be eleuated and ereéted, the one to the other,and alfo conceaue folides or bodies, which, for that they haue not hitherto bene acquain - ted with, willa: the firft fight be fomwhat ftraunge vnto thé,I haue for their more eafe,in this eleventh booke, at the ead of the demonttration ofeuery Propofition either fet new figures, if they concerne the elevating or erecting of lines or fuperficieces, or els if they concerne bodies, I haue fhewed how they fhall defcribe bodies to be compared with the conftructions and demonftrations of the Propofi- tions to them wots 5 And if they diligently weigh the maner obferued in this eleuenth booke tou- ching the defcription of new figures agreing with the figures deferibed in the plaine,rt hall not be hard for them of them felues to do the like in the other bookes following, when they come to a Propofiti- on which concerneth either the eleuating or ereéting of lines and fuperficieces,or any kindes of bodies to be imagined. q Ihe RS ee aa | Ct Ss i — at i el a 3 7 ad Ay of Euchides Elementes: 3236 gq Tbe 1.T heoreme. . The 1.Propofition. > T hat part of a right line fhould bein a round playne fuperficies, ¢x part elenated pward is impofsible, : =~. Or if it be poffible,let part of the right line ABC, name- Vr ly,the part AB bein a ground playne [uperficies, and the q Aj a I rd part therof,namely,B C be eleuated vpwarde. And ONY is) duce diret#l , i ( 9 produce directly . upo the ground | | 9 © playne fuperfi- gam cies the right “yd C rs line A B beyond Demonflratia on leading to an impo sbi littes This figure more plainly fetteth forth the forefaid de- monftratid,if ye eleuate the {uperficies wherin is drawn the line BC. = Anotherdemonftration after F/u/vas. Ifit be poffible let there be aright line AB G, | | whofe part A B let be in the ground playne fuper- An other dee ficies A E D,and let the reft rherofB G be a Bate : monstration on high,that is,without the playne AED. Then I 7 | fay slat AB G isnot one aH line.For forafmuch COeree: as A EDisaplaine fuperficies,produce direttly 8 equally vpon the fayd playne A ED theright lyne A B towardes D,which by the 4.definition of the firit fhall be a rightline.And from fome one point ofthe right line A BD,namely,from C,draw ynto " the point Garight tyne C G. Wherefore in the E triangle B CG the outward angle A BGs equall to the two inward and oppofite angles (by the3z. 3 i of the firit) and therfore it is lefle then two right angles (by the 17.0f the fame) Wherfore the lyne A B G forafmuch as it maketh an angle,isnot aright line. Whérefore that part ofaright line fhould be in aground playne fuperficies,and part elettated vpward is impoflible. : Ifye marke well the figure before added for the playner declaration of Euclides demonftration, it will not be hard for you to conceiue this figure which Fluflas putteth for his demenftration : wherein is no difference but onely the draught of the lyne GC, : qT he z. T'heoreme. T he 2. Propofition, If two right line cut the. one to the other, they are in one and the felfe fame plane fuperficies:ex enery triangle is in one ¢7 the felfe fame fuperficies. | ; TF @. Suppofe Pa ROS = a - ~ - FS o a ee Se er ae = = a SSS oe 2 _ —— igi a se VS Se oe =. tit ' i 1 H Li Bi ia Kit y 44 : tah 4 The eleventh Booke \: Al V ppofe that thefetwaricht ines A Bard&G.D doo <3 cutte the one the other in the pomt E.T hen fay that A )} thefe lines AB and CD sareiv oneand the felfe vs | fame [uperfictes, ana that euery triangle isinone ¢ Confiruction. felfe fame playne f uperficies.T ake in the linesE C andE B points at all auentuxes , and let the fame be FandG.,, and draw a right : line from the-poynt B to the point C, and an other from the point E ta the point G « And draw the lines Hard GK. First [a ‘ ¢ #1. KR i - pHi ais that the triangle E B C is in one and the fam: ground [uperfictes. 4% impo (ii _ For of part of the triangle EB C,namelythe triangle FE CH, or lite: the triangle GB be in the ground (uperficies , and the refidue be in an other , then alfo part of one of the right lines E CorEB fhallbe in the ground {uperficies , and partinan other . So alfo.if part of the triangle EB C , namely , the part EF G be in the ground fuperficies and the refidue betnarother;then alfo one part of echeof the right lines EC andEB hall bein the ero wind fuperficiegsc7 another part in another fuperfictes, which (by the firft of. the elencnt}) B - -univomsl zepraned to be snpoffi ble Wherfore the triangle E BC 43 4n one and the felfe [ame playne [i- SEAMS NS per ficzes Foren mbat [uperficies the triangle B CE7s iz the fame, al{o is either of the linés og EC and 8 Band inwhat [uperfictes either, of the lines E.C and E B1s,in the felfe fameal. eee he hues AB and CD. Wherfore the right lines lines A Band C.D arein one ep the felfe fame playne fuperfictes arid ewery triangle isin one cy the [elfe fame playne [uperficies: which was required to be proued In this figure here fet may ye more playnely conceaue the demon- ftration of the former propofitrionwhere Ye may: eleiatewhat'part of the ” triangle ECB ye will,namely the part F C H or the part GB K; or finally o ii : : the part F CGB as is required in the demontftration. oe a0 a: _s ~The 3. Fheoreme: Or es Propofition. | a If two playne fuperficieces cutte the one the other s:theiy common fettion is APL cc’ sis%'0.53 ody) signa a ne Gua . V ppofe what thee two [uperficieces AB ey BC do a i, cutte the one the other, and let their common Jeciz-.. de s Sa | ombe thedineD BuDhemh fay that D Bis a right Mec Ga line.F o7 if not,draw from the poynt D to'the point — eee D j oy jae B a richt lineD F Bin the playne Superficies A B, and likewife oe poli hi- from the fame pojntes draw an other richt ine DEB trthe eg playne [uperficies B C.Now therfore two right lines D EBand DE 'B fhalbhave the felfe famecndet,and therefore dovinclade| 4 [uperficies mhich ( by the laft commonfentence )isimpoffible: Wherefore the lines D EBand DF B are not right lines . In AT ae er or Fe Fe of Euchdes Elementes. Fol.32406 Like fort alfo may we prone that no other right lme cam be drawne from the poynt D tothe point B befides the line D Bwhich is the comunn fection of the twofuperficieces A Band BC. If therefore two playne fuperficieces cutte the one the other , their common fection is a right lime: which was required to be demonjtratid. This nour here fet , fheweth moft playnely nor onely this third propofition,but alfo the demonftra- tion thereof, if ye eleuate the fuperficies AB, and fo compare it with the demontitration. q Lhe 4. I heoreme. The 4. Propofition. If from two right lines cutting theone the other, at thei common [ektiton, aright line be perpendicularly eretred:the fame fhall alfo be perpendiculare ly eveéted from the playne fuperficies by the fayd two lines paffing. aa V ppofe that there betwo right lines AB 2 | and CD cutting the one the other in the a poynt E. And from the poynt Ext there be / LGN erected a right lineEF perpendicularly to . ¢ the [ayd two right lines AB and D:then I fry thatthe ~ right line EF , is alfo erected perpendicular t: the plaine | if 4 fuperficies which pafseth by thelines ABand CD. Let» ngs . thefelines AE,EB,EC, and E D be put eqrall the one I> to the other. And by the poynt E extend a right line at all anentures , and let the fame be GEH . And crawe thefe right lines AD, CB, FA, FG, FD,FC,FH, and & F B. And forafmuch as thefe two right lines AE @ED» | Demonstra- B are equallto the[e two lines CE and EB, and they com- - $20Me prehend equallangles(by the r5.0f the firft) :therefore (oy... Er | the 4.of the first the bafe A Dis. equall to the rafe CB, and the triangle A E.D is eqnall to the triangle C EB.Wherefore lle the angleD AE, is equall to the angle EB C.. But the angle AEG is equall- tothe angle BEH (by the 15 of the firft). Wherefore there are two triangles AG E., and BEH hauing two angles of the one equall to two angles of the other, eche ta his corre{pondent angle,and one fide ofthe one equall to af of the other , namely one of the fides which lye betmene the equall aneles,namely , the fide A ¥ is equall to the fide EB. Wherefore( by the 26 .of the fir/t) the fide: remayning are equallto the {ides remayming. Wherefore the fide G Eis equall to the fideEN , and the fide. AG to the fide B And for- afiuch as the line AE. is equall to the line EB , and the line F E is common to them both, TT iy. and Conflrnttions ne ee et — ~ - ——— = 7 —w - — = ss 4 fj W . nN Ree TheeleuenthB ooke snl maketh with them right angles, wherefore ( by the fourth of the firft) the bafeP Ais ts guallto the bafe FB. And (by the fame reafon) the bafeF C is equallto the bafeE D. And forafinuch as. theline A D is equallto the line B Cand the line Nis equal tothe line F B as it hath bene proued . Therefore thee twolinesF Aand AD are equall to: thefetwo lines F Bee BC, the one to the other,¢y the bafe F D is equall to the bafe FC, Wherfore alfo the angle¥ A Dis equalltothe angleF BC. And againe forafmuch as it hath bene proued, that the line A G is equall to the line BH, but the line F A ts equall to the line F B. Where- fore there are two lines F Aand AG equall to two lines F Band BH and itis proued that the angle ¥ A G is equall to the angle F BH wherefore ( by the 4.0f the firft ) the bafeF G is equal tothe bafe YH. . Agayne fora{much as tt hath bene proned that the line G E.is equal to the line EH,and the line & F ts common to them both: wherefore thefe two lines G Eand E F are equall to thefe two lines H EandE F , andthe bafe¥ H is equall to the bafe EG: wherefore the angle G EF ts equail to the angle AHF. Wherefore either of the angles GE F,and HEF is a right angle . Wherefore the line E F is erected; from thepoint E perpendi- cularly to the line G H1.In like fort may we prowe,that the fame line F E maketh right ancles with all theright lines which are drarnevpon the ground playne {uperficies and touch the point B.But aright line is then erected perpendicularly to a plaine {uperficies,when it maketh ight angles with all the lines which touch it , and are drawne vpon the ground playne [uper« ficies (by the 2.definition of the elewenth ) . Wherefore the right line F ¥ is eretted perpen id; cularly to the cround playne fuperficies .And the ground plaine [uperficies is that which paf- eth by thefe right lines A Band GC D.. Wherefore the right line F Eis erected perpendicu- larly to the playne [uperficies which pafveth by the right lines A B and CD. If therefore from.tworight lines cutting the one the other andat their common fection a right line be perpendicularly erected: 1t phallal{o be erected perpendicularly to the plaine fuverficies by the fayd two lines paffing: which was required to be promed. Ma cs oo ee ee a ae aE eo a ne EEA CR ee — Soa > a oe ee In this figure you may moft euidently conceaue the former” propofition and demonitration, if ye erect perpendicularly vnto the ground playne fuperficies AC B D the, triangle A F B.: and eleuate the triangles A F D,& C FB in fuch fort,thatthe line A Fofthe tri- ' angle AE B may ioyne & make one line with the line A F of thetri-_ angle AF D:and likewife that the line BF of the triangle A F B may, ~ ioyne & make one right line with the ine B F of the triangle B F.C...” q The 5. I heoreme. The.s:Propofttion. If onto three right lnes'-which tonch theone the other , beeretfed a pers “pendicalar line from the common point where thofe three lies touch:thofe three right lines are in one and the felfefame plaine [uperficies. =e phe that unto thele three right lines B CBD, and BE, touch ing the one th I 4 | wht | i" } q Fs wa eh Aa Pp w a | - ! ran} sp ' | ¥ : Pipe |, | ; 4 r ‘ 4) = % i - 5 e} H ’ pa ay - ‘ i a yi a , . § > ‘ q Ma? 3 oe) st f | Uf. mat ; uM i 1g] t f f i a “| in Wy a vind} , ay a 4 Hie! { \ re. i wee ath) ¥ Le . read iH ft ’ } 4 ¢ aD i Py a yf . { i a ‘ah mT. r 4 va Ad 1a 7 oe time | y i ‘i 3 A ap a ' bey WERE im { : : t eT ban | & ryt " a7 a ; ’ ya ‘ , vey 1) ‘1 y " i) il oy dh , | 4 1 _ _ an | oieit » A ie fi or ' aay i ft ; | ef aa RRS ie’ : i] 4 i if " , ae “Se hl te {| oh a SS ee Se = pereaaie=ante = SSIS =. — a at -— the other in thepoynt B,be erected perpendicularly from. the poynt B, the lize Se AB:Then I [ay that thofe thre right kines B.C,B D and BE arein one e& the [elfe fame plaine fuperficies,F or if not,then if ut be pofjible, tet esses oO See pe ee a LO ESE Oe ae a —— = — . =e 2 line Sin sean" © Be oe a ee ' ' ren Us e é - of Euclides Elementés.: Fol.324 BE bein the eround fuperficies , and let the line BC be €5 reited upward (now thelines AB and BC are in one and the fame playne fuperficies (by the 2. of the elenenth )for they touch | the one the other in the point B ). Extend the plaine fuperficies wherein the lines AB andB€C are, and it fhall make at the length a common. fection with the ground fuperficies , which common {ection [hall be a right line (by the 3 .of the elenenth): let that common {ection be the line B¥ .Wherefore the three right lines AB, BC, and B F are in one and the felfe [ame [u- perficies , namely , in the fuperfictes wherein the lines AB and BCare.And forafmuch as the right line AB is ere‘ted per- pendicularly to either of chefe ines BD and BE, therefore the © line A Bis alfo ( by the 4.0f theelenenth ) erected perpendicn- ) larly to the plaine fuperfictes, wherein the inesBD and BE are. But the fuperficies wherein the lines B D and B Eare is the ground fuperficies. Wherefore the line AB is trected per- pendicularly to the ground plaine {uperficies. Wherefore (by the 2. definition of the elewenth) the line AB maketh right angles with all the lines which are drawne vpon the gronnd (uper- cies and touch it.But the line B¥ which is in the eround fuperficies doth touch it Wherfore the angle A BF is aright angle . And itis {uppofed that the angle AB C is aright anele. Wherefore the angle AB Fis equall tothe angle ABC, and they are in one and the felfe fame plaine [uperficies which is tmpofible . Wherefore the right line B C is notin an higher [uperficies .W herefore the right lines B C,B.D, BE are in one and the felfe fame plaine [u- perficies . If therefore unto three right lines touching the. one the one the other ,be erected a perpendicular line from the common point where thofe three lines touch: thofe three right lines arein oneand the felfe fame plaine [uperficies : which was required to be demon- strated. | " This figure heré fet more playnely declareth the demonftration ofthe for- mer propofition, if ye erect perpendicu- larly vnto the ground fuperficies,the fu- perficies wherein is drawne the line AB | and fo compare it with the fayd demon- tem ug {tration. | . 3 hag Re <5 Pe ci ee a _ The 6. heoreme. T he 6.Propofition,. fF (>, fi MTG m SBS OS 2520979 ¥, rag SS SS a 25s 5 i puperficies:thofe right lines are parallels the one to the other. Lftworight lines be eretzed perpendicularly to one ¢- eee fame plaine TT ity. Suppofe Demon/ffrati- on leadsng to an impofsibir> [theese Siar Uae ASTER SiS ee - SF ES - = a = ee aa. el eg ee — Conftrutiien. ® An Affampt * as M,Dee proucth ite Dimonstra~ tEAM, T he eleuenth Booke wT) V ppofe that thefetwo right lincs AB and C D be eretted perpendicularly to a RY a ground plaine fuperficies.. Then I fay that the line AB isa parallel to the line ys C D.Let the pointes which thofe two right lines touch in the plaine {uperficies be Waid B and D. And draw aright line from the point B to the point D . And ( by the r1.0f the fir[t ) from the point D draw vatothe line B D in the ground [uperfictes a perpen- dicular line DE. And( by the z.of the firft) "put the line D Eequall tothe line AB. And draw thefe right lines BE, AE, and AD. And forafmuch asthe line ‘AB is erected perpendicularly to the ground {uperficies ,ther- i © fore (by the 2.acfinition of the eleuenth)the line A B maketh right angles with all the lines which are drawné upon the ground playne fuperficies and touch it. But either of thefe lines B.D and B-E which are in the ground {uperficies,touch the line AB , wherefore either of thefe.angles A B.D and - AB Eisaright angle:and by the {ame reafon alfo either of the angles C DBs CD Eis aright angle. And foralmuch as the line AB 1s equallto the ine D E.,and the line B Dis common to them both,therfore the[e two lines A BandB D, are equall to the{e two lines ED and D B,and they contayne right angles » wherefore (by the 4.0f the first the bafe A D.isequall to the bale B E . And fora{much as the line AB is equall to theline D.E., and the line A D tothedine BE , therefore thefe two lines A Band BE are equall to thefe two lines E D and D A, andithe line A Eis a common bafeto them both. Wherefore the angle: AB E 1s ( by the 8.of the firft)equal tothe angle E DA. But the angle AB Eisaright angle; whefure.alfotheangle ED Ais aright angle:wherfore the line E D is erected perpedicularly to the line D A.and it is alfo erected perpedicularly to either of thefe lines BD and DC, wherefore the line ED is unto thefe three right linesB D,DA,anad DC erected perpendicularly from the poynt where thefe three right lines touch theone the other : wherefore (by the .of the eleuenth)thefe three right lines B D,D A, and DCarein one and the felfe [ame [uperficies . _And in what {uperficies the lines B D and D A are,in the felfe fame alfozs the line B A : for euery triangle is (by the 2. of the elenenth ) in one and the felfe fame [uperfictes . Wherefore the[e right lines AB, B D,and DC are in one and the felfe [ame Las , and either of thefe angles ABD and BDC isaright angle ( by {up- pofition).Wherefore( by the 28. of the firft)the line A B is a parallel to the line C D. if there- foretwo right lines be erected perpendicularly to oneand the felfe fame playne fuperficies, thofe right lines are parallels the one to the other:which was required to be proned. - VE Here for the better vnderftanding of this 6. propofition I~ haue defcribed an other figure: as roe which if ye erect the {uperficies.4 BD perpendicularly to the {uperficies BD £, and imagine only 2 line to be drawne from the poynt 4 to the poynt E (if ye will ye may extend a thred from the faide poynt 4 to the poynt € ) and fo compare it with the demonftration, it will make both the propofition, and alfo the demontitration moft cleare ynto you. @y An other demonftration of the fixth propofition by 7%. D a \ cou tia Sa rt gen A> vote ae oma aK Vn tne Ge aS ; Bes a" Sint dys. MG. LY NY ty U Bhishvoses 6. 3% bk Seite to ay : Suppole that the two right fines 4 ied be perpendicularly erected to one & the fame playne fu- oF hSO FAL OL TY IAT WS NB Tes | perficies & . ; ' er AGE oe ae ee 7 Ee —_— a, , iets pom rrr orem) of Euclides. Elementes. 326. perficies namely the playtie fuperficies.o p.ThenT fay that ap arid D ate paralicls Let the end points of the righrlines as and'c » which touch the plainefuperticies © p be the poyntes & and p,fro 8 to B ler'a ftraight line be drawne(by the firlt petition and (by the fecond petition) let the ftraight line s pv be ‘extéded,as to the povats47 & x. Now forafmuch as the rightline a3, fromthe poynt B'produced; doth cutte the line m w ( by conttraction). There- fore (by the fecond propofition of this clenenth booke)the right lines ag & ™ N are in one plaine fuperficies. W hich let be cx, cutting thefuperfi- cies o.p in the right line m N.By the fanre meanes may we conclude the right line c p to bein one playne fuperficies with the right line mn. But the right line m n.( by fuppoiition) 1s 15 the plaine fa perficies ar : wherefore cD 1s 1n the plaine fu- perficies qr »And az the night line was proued to be in the fame plaine fuperficies a x.Therfore a Band c pare inone playne {uperficies, namely er .And forafmuchas the lines a8 and.cp (by {uppofition ) are perpendicular vpon the playne fuperficies op , therefore (by the fecond definition of this booke ) with all the right lines drawne in the fuperficies o p and touching a B andcenp,thefame perpédiculars a 8 and c p,do make right angles. But(by conitruction )M N, being drawne in the plaine fuperficies 0 P toucheth the perpendiculars a 8 and c p at the poyntes g and p. Lherefore the perpen- diculars a z and c p , make with the right line m w two right angles namely an N,andcp™m: anduN the right line is proued to be in the one and the fame playne fuperficies,with the right lines a 8 & « p: namely in the playnefiipenficies Qk. Wherefore by the fecdnd part of thie 28. propofition of the firit booke,the,right lines a 2 andc p are parallels , Ifthereforevtwo rightlines be erected perpendicularly to one and the felfe fame playne fuperficies thofe right lines are parallels the oneto the other: which wasrequired to be demonitrated. | See eae Pee A Corollary added by M.Dee. playne fuperficies sare alfo them falnes in One and the fame playne fuperficies which ts likewi/e perpen drcularly erected vo theyfame playuc fapecficres,unto which the two right lines are perpendicular. c p being in the playne fuperficies cx, are by fuppofitior pee Ge ed to the playne fuperficies 0 P; wherefore by the third definition of this booke Qn is perpen was required to beproued, lo-Dee his aduifevpon the Affumpt of theo, * Asconcerning the making of theline DE, equall to the rightline A B , verelythe fecond of the firft, without fome farther confideration is pot properly enough alledged And no Wonder itis , fos thar inthe former bookes , whatfociiah Rath OF Hines 94 Ss Sos “Soby | bene {pokeny the fame hath alwayésbenetmagided ro... \ be in, one onely playne {uperficies confidereckor execu- ted’. Bubhére the perpendicular line AB, is not in the fame playiie fuperficiesjthat therightlinéD\E Ys. Ther- fore fome other helpe muit be put into the handes of young beginners stow tobring this “probleme to exe- cution : which is this , moft playne and briefe . Vnder- ftand th@f'B D the right line , is the common feétion of the playne fuperficies, wherein the perpendiculars A B and C Dare, & of the other playne {uperficies,to which they are perpendiculars.The firft of thefe(in my former demonftration of the 6), I noted by the playne fuperfi- cies QR:and the other,I noted by the plaine fuperficies O P.Wherfore B D being a right linecommon to both the playne fuperticieces Q-R-8e-O P;therby the ponits B and Darecomon to the playnes Q Rand OP. Now from TI. 20 as nant | { t | t T be eleuenth Booke from B D(fufficiently extended) cuttea right line equall to A.B, ( which fuppofe to be B F) by the third of the firit,and orderly to B Fmake DE equall,by the 3.of the firft,if D E be greater then BF. ( Which alwayes you may caufe fo to be,by producing of D E fufhciently ). Now forafmuchas B F by conftruéti- On is cutte equallto A B,and DE alfo, by conttruction, put equall toB F , therefore by the 1. common fentence,D E is put equall to A B: which was required to be done. In like fort,if D E were a linegeuen to whome AB were to be cutte and made equall, firftout of the line D B(fuficicntly produced)cutting of D G,equall to DE by the third of the firft: and by the fame 3-cutting from B A({ufhciently produced)B A, equalltoD G : thenis it cuidét,that to the right line D E,the perpédicular line A.B is put equall. And though this be eafy to conceaue, yet I haue defigned the figure accordingly, wherby you may initruct your imagination. Many fuch helpes are int this booke requifite , as well to enforme the young ftu- dentes therewith , as alfo to mafter the froward gayne- fayer of our conclufion , or interrupter of our demon- itrations courfe, q Ube 7. Theoreme. Lhe 7. Propofition. If there be two parallel right lines, and in either of them be taken a point at all aduentures : a right line drawen by the Jard pointes is in the felf fame Juperficies with the parallel right lines, | ace ppofe that thefe two right lines.A B andC D be parallels, and in either of thé take OSS 2 point at all Aaduentures, namely, E and FE. Then! fay, thata right line drawen MSN from the point E tothe pomt F, is in the Jelfefame plaine {uperficies that the pa- vallel lines are. For if not, then if it be pofsible, 7 DemonStra- “et it Lein.an higher [uperficies,as the line. EGF... A E L tion leading to 13,and draw the fuperficies wherin the line EGF an smpofstbi= is,¢} éxtendit, and it hall make a common fetti- 9 Witte on with the ground {uperficies which fection fhall (by the 3.0f the eleuenth) bea right line : let that fection be the right line EF | Wherefore two right lines EG F and E F includea [uperfitiess 9 Co B which (by the last common fentence)is impo/(able. | Wherfore a right tine drawen from the point E to the point Fis notin.an higher fuperficies. Wherfore a right line drawen from the point E tothe point F, isin eh fame {uperficies _ wherein ave the parallelright lines AB andCD. If therefore there be two parallel _ _ “ines, and in either of them be taken a point at all aduentures, a right line drawen by thofe pointes is in the felfe fame plaine fuperficies with the parallel right lines : which wasrequi- red to be demonstrated. i = cl | | E Pax By this figure it is eafie to fee theeformer demontftration, if ye eleuate the {tiperficies wherin is drawen the line EG F, = | of Euclides Elementes. Fol.324. The ¢.Theoreme. The 8:Propofition. | ) | If there be two parallel right lines of hich one is ereéted perpendicularly to around playne fuperficies: the other alfo ts erected perpendicularly ta the felfe fame ground playne fuperficies. ‘by the fir ft peticton) draw aright line from the point B to the point D. And drawe Soxth es from the point D unto the line BD a per- Confiruttions ; ye fir it) put the line | ’ D Eequall tothe line A B,and draw a right line from the a . | point B to the point E and an other fromthe point A to the | point E,and an other from the point A tothe point D Ana Demonfira~ ; whe foraf auch as the line A Bis erested perpendicularly to the L46%. ground [uperficieces therfore (by the 2.definition of thee- BL. lenenth) the line AB is erected perpendicularly to all the right lines that are in the ground uperficies and touche tt. Wherfore either of thee angles A B D cy ABE isaricht ancle. And forafmuch as vpon thefe parallel lines AB and CD falleth a certaine right lineB D, therefore (by the 29. of the firft) the angles AB DandC DB are equal to two right angles But the angle A B D is aright angle, wherfore alfothe angle C D Bisa right angle. Wherfore the line C Dis €- rected perpendicularly to the line B D. And fora{much as the line A B is equall to the line D E,and the line B D is commen to them both, therfore the[e two lines AB and BD are equal -tathefetwo lines ED and D B,and the angle A B D isequall to the angle E D B for either of them is aright angle. Wher fore (by the 4. of the firit) the bale A D isequall to the bafe BE. And foraf{much as the line A Bis equall to the line D E, andthe line BE tothe line A D,therforethefe two lines AB and B E are equall to thefe two lines A D ey D E, the one to the other,and theline A E ts a common bafe ta them both. Wherfore( by the 8 of the first) the angle AB Eis equall to the angle ADE: but the angle AB Eisa right angle wherfore the angle ED A,isalfo aright angle. Wherefore the line E Dis eretted perpendicularly to the line A D,and it is alfo erected perpendicularly to the line P B.Wherfore the line E Dis erected perpendicularly to the plaine {uperfictes wherin the lines B D and B A are (bythe 4- of this booke) Wherfore (by the 2. definition of the elewenth the line ED ts erected perpen dicularly toll the right lines that touche it and are in the (uperficies wherein the lines B D be and AD are.But in what [uperficies the lines B Dand D A are,in the [elfe fame [uperficies | is the line D C.For the line A D being drawen from two pointes taken in the parallel lines A Band C D is by the former propofition im the felfe fame fuperficies with them. Now foraf- | much as the lines A Band B D arein the fuperficies wherin the lines BD and D A are, but . in what (uperficies the lines A B ey B D aren the fame is the line DC Wherfore the line E Dis eretted perpendicularly to the line DC Wherfore alfo the line C D ts erected perpendi- . cularly to the line D E. And the line C Dis eretted perpendicularly tothe line D B. For by ; the 29. of the firft,the angleC D B being equall to the angle A BD is aright angle Where- ; forethe line C D is from the point D erected perpendicularly to two right lines D E and D B cutting the one the other in the point D.W herfore by the 4.0f the eleuenth,the line C D is eretted perpendiculaaly to the plaine [uperficies, wherein are the lines D E and DB. But the eee Confiruction. Demounflya- bn, lines EB F.and A-B unto the line E F 4 perpendicu wherin ave thelinesE F and c D,raife diculer line G K. And forafinuch as the line £-F 1s erected perpendiculerly to either of the lines G Hand G K, therfore (by the 4.0f the elewenth) the line E F is erected perpendicu- larly to the fuperficies wherein the lines GH ana GK are, butthe line EF isa parallel line. tothe line AB.Wherfore( by the 8.of the eleuenth) the line AB is erected perpendicu- larly.to the plaine [uperficies,wherin are the lines GH and GK, And by the fame reafon al- [othe lineC Dis ereited perpendicularly to the plaine {uperficies r GK. Wherefore either. of thefe lines .AB andC D Superficies,wherinthe lines G H and GK are. B larly to.one and the Selfe fame plaine [uperficies thofe right lines are parallels the one to the ath right lines whith are parallels to one Superficies with itare alfo parallels the o ix | T be elenenth Booke she ground plaine fu perficiesis that wherin are the lines D Band DB, to which fuperficies alfa the line AB ts [uppofed to be erected perpendiculerly, Wherefore theline CD is evected perpengicularly to the ground plane {uperficies,wheruntothelline.A B is erected perpendicu- larly. If therfore there be two parallel right lines, of which one is erected perpendicularly to 4 groniid plaine uperficies the other alfo is erected perpendicularly to the felfe [ame ground plaine fuperficies : which was required to be densonftrated. This figure will more clearely fer forth the former das monttration, ifye ere& perpendicularly the fuperficies ABD to the fuperficies B D F,and imagine a lyne-to be drawen from the point A to the point D, in ftede wher. of, as in the 6. propofition ye may extendea threede, 7 Ihe 9. T heoreme. The 9. Propofition. ‘Right lines which are parallels to one and the Jelfe fame right line, and Aap bg if: te i M : . dre notin the felfe fame [uperficies that it is in: are alfo parallels the one to the other, <7 ppofe that either of thefe right lines ABandC D bea parallel tothe line EF <) ot being in the felfe fame Superficies with it. Then 1 [ay that theline AB is ie | parallel to the line G D.T ake in thelineE Fa point at all cduentures, and Ter tine fame be Gx Avid from the point G ratfe up in the faperficies wherin are the ler line GH, and againein the {uperficies Up from the fame point G tothe line EF a perpen vherin are the lines GB ¢ is erected perpendicularly to the plaine ut if two richt lines be erected perpendicu- tr (by the 6.0f the eleuenth) Wherfore the line AB isa parallel to the line C D. Wherfore the felfe {ame right line,and are not in the felf fame ne to the other «which was required to be proued. This : a — £E D ,E F be put equall the one tothe other : and draw thefe of Euclides Elementes. Fol,328. This figure more clearely manifefteth the formet propo- fition and demonttration,if ye eleuate the fuperficieces A B EF andC DEF that they may incline and concurre in the lyne EF. q The 10. I heoreme. The 10. Propofition. If two right lines touching the one the other be parallells to two other right lines touching the one the other, and not being im one and the felfe fame fuperficies with the two firft : thofe right lines cotaine equall angles. V ppofe that thefetwo right lines _AB and BC touching the one the other, be \|parallells tothe[e'two lines D E and E F touching alfo the one the other, and not being in the felfe fame [uperficies that thelines A B and BC are. The Lfay, that the angle_ ABC is equallto the angle D E F . For let the lines B A,BC, B right lines AD,C F,BE, AC,and DF . And fora{much as the line B A is equall to the line ED, and alfo parallell unto it, therefore (by the 33.0f the firft ) theline AD ise- guall and parallel to theline BE: and 4 the fame reafon 4 alfo the line CF is equall & parallel to the line B E.Wher- fore either of thefe ines AD and C Fis equall cy parallel - totheline EB But right lines which are parallells to one and thefelfe fame right line,and are not.inthe felfe [ame {u- perficies with it, are al(o (by the 9 .of the eleuenth ) parallells the one tothe other Wherefore the line A.D is aparallelt. line to the line C F ..And the lines AC. and D F ioyne them tagether. Wherefore (by the 33.0f the firft) the line AC is.” F equall and parallellto the line DF. And forafmuch as the[etwo right lines AB ey B Care. equall to the(etworight lines D\Eand EF, and the bafe AC alfots equall to the bafe DF: therefore (by,the 8.of the firft) the angle AB C isequall to theangle D E F. If therforetwo right lines touching the one the other be parallells to two other right lines touching theone the other;and not being in one and the felfe {ame {uperficies with the two first : thoferight lines containe equall angles : which was required to be demonftrated. aa This figure here fet more plainly declareth the former Pro- pofition and demonftration, if ye eleuate the fuperficieces DAB E, and FCBE, till they concurre in the line FE, | VY .i: . The Construttion. Dem onfira- £70%., | ‘a , ‘ : , 4 | aH ; Wale ; - | i) 44 : ‘ ' “ A 4 / ' { \ { 4 } er : : wii Tee wae - ' } _-. ia § a 5 { ' q : , i : i , ' ’ uae : { : fi ‘ 1 i] : lt rit F { - t 4 a ‘ ‘ ayy My at } , a i ih Lan P ann - re n ¥ it “Pept 1! 4 Ai Hh 5; abe: 5; Li a : f,4 q rd { } 4 | te 3| iy : ¢ f 7 | { } 4 ' \) . —_—— 88 ————— = —_ eee - = P = £:0%, Lhe eleuenth Booke-- q Ihe 1. Probleme. The 11. Propofition. From a point geuen on high, to drawe Ynto'a ground plaine.fuperficies perpendicular right line. | Confiruttion , : . ee _ a? ppofe that the point geuen on high be A, and [uppofe 4 ground plaine fuperficies, this prepofi- yy vamely, BCG H. It is required from the point Ato draw vnto the ground [uper- tion. i ficies a perpendicular line. Drawe inthe ground fuperficies a right line at aduen- The fir cafe, tures,and let the fame be BC. And (by the 12. of the fir{} ) from the point 4 draw Gnto the line cee” Nobe Dee. ae ee endiedlar line AD. * Now if AD bea perpendicular line to the ops erating psn Sf nel perigee dome mbich ms fnght for Bat fw then pointe A, and thestraight line BC. (Oy the rt oft e fir ft) fr omthe point D raife vp in the ground {uper- And fo helpe your felfe in the lyke ficies vatothe line BCa perpendiculay line DE. And ( by ther 2.0f the = pape F Mathematically imagt- fir) fromthe point A draw vatothe line D Ea perpendicular line AF. chanicalig And by the point F draw(by the 31 .of the firft)unto the pracifing. Hine BCGvsaparalleliline FH Andextend the line. FH. Second caft. from shepoint F ta the point.G.: And forafmuch as the Demonjira- “ling R Gixeretted perpendicularly to either of thefelines cetieyy DiE.dudi DA; therefore (by the 4.of theeleuenth).the Une B Cis erected perpedicularly to the faperfictes wher- ve inthe lines ED and AD are s.andtethéline BCthe. ag. “ine G H is a parallell . But if there be twe parallell right lines of which one ts erected perpendicularly to.acertaine MAGIeAt Soige plaine{uperficies, the other al[o( bythe Sof the elewenth) = Aes Se 2 is erectea perpendicularly to the felfe fame fuperficies. : ee slt Wherefore the line GH 1s eretted perpendicularly tothe plaine {uperficies. wherein the lines E Dand D Aare .Wherfore alfo. (by the 2.definition of the elenenth)\thelineGH is.eretted perpendicularly to all the right lines whith touch it, and arein the plainefaperficies wherein the linesE D and AD are . Buttheline A F touchethit being in the [uperficieswherein the lines E D and A D are (by the 2.of this booke) ‘Wherefore the line G His erethed perpen= dicutarly tothe line F A. Whereforel{o the line.R At is ereched perpendicularly ‘to the line GH: and the line A F is alfo ereéted perpendicularlytothe line DE. Wherefore AF is e- vechéd perpetidicularlyto either of thefe lines H GahdD E. But if aright line beeretted per- pendicalarty from the commen fection of two right lines.cutting the onetheother;it [ball alfo be exectid perpendicularly to. the: plane fuperficies of the [aid two lines (by the 4.0f the eles nenth) Whereforethe line A F is ereéted perpendicularly to that ‘fuperficies wherin the lines E DuanhGH are, Bat the [uperfieces wherein the\ lines E. D.and G H are, isthe ground {- perjicies . Wherefore theline A Eis tretted perpendicularly tothe cround fuperficies “Wher- fore from a point geuen on high namely, fro the point A,ss drawen to the ground [uperficies a perpendicular line : which was required to be done. In this figure fhall ye much more plainely fee both the cafes.of this former demonftrati6. For as touching the firft cafe, ye muft erecte perpendicularly to the ground fuperficies, the fuperficies wherein is drawen the line A D, and compare it with the demonftration, and it willbeclere vnte you. For the fecond cafeye mutt erecte perpendicularly vnto the ground fuperfi- cies the fuperficies wherein'is drawen the line A F,and vito itlet the other fuperficies wherein-is-drawen the line AD, incline,fo that the point A of the one may eoncurre with the point A of theother : and fo with your figure thus ordered,compare it with the demon- ttration, and there will be in'icno hardnesatall. qT he ES 3 ? of Euclides Eilementes. 3290 ath q The 2: Probleme. maner ofie which common fection fhall bearightline,and: ~~ magination hall pafse by the poynt A:let that common fetion™ Raat be thelineD A E.Wherefore(by the 3. of the ele denth) the lines AB,A C,and D AE arein one ..§—-_— | and the [etfe.fame playne fuperficies. And oral. °° <3 much as the line C A is erected perpendicularly othe gro und {uperficies , therfore ( by the | ee ee 3 Sn ale " > a oa8 Se BES a eS a = = 3 -= - SSS 2 tion leading ito ant mpofs bs litiee 4 [ 4 ' | ; ' f tiv 1M j | VBA | aaa a Bh a Demonifra- gt 7 Saaliit } iN} ; ") } ti th! i ‘] a) 44 ; ; : OF | i) Hit ' . i i 4 | : ; 4 { - thefe fuperficieces CD and E F are paral- _.. the 3. of the elenenth ) be a right vine. _ Let that common fection be GH. And inthe line GH takea point at all aduen- T he eleuenth Booke 2 definition of. the eleuenth it ma heth right angles with all thevight lines that touch it, and | are in the ground {uperficies.But the line D A E toucheth it, being in the ground {uperficies. Wherefore the angle C AE is aright angle,and by the fame reafon alfo the angle B A Eis aright angle Wherefore( by the 4 petition the angle CAE is equall ta the angle BAE the le(Se to the more, both angles being in one cy the felfe fame playne [uperficies: which is im- poffitle. Wherefore from one and the felfe fame poynt,and to one and the felfe fame playne fu- perficiescan not be eretted two perpendicular right lines.on one @ the elfe fame fide: which was required to be demonfirated. v > ae f In this figure if ye ere&t perpendicularly the fu- | / perficies wherein are drawne thelines 2 a and.c ato oo | the ground fuperficies wherein is drawn the line p aE, / , MS + | ie and fo compare it with the the demontftratié ofthe for mer propofition it will be cleare ynto you. . 7 ; recy D a Ait “sy “ a r 7 > ? : . Ante bh Sembee |. M. Dee his annotation. — Exclides wordes in this 13,propofition admit two cafes: one, if the poynt affigned be in the playne fuperficies,(as comonly the demonitrations fuppofe)the other, if the poyntafligned be any where with out the fayd playne fuperficies , to which , the perpendiculars fall,is confidered . Contrary to either of which, if the aduerfarie affirme , admitting from one poynt two right lines, perpendiculars to one and ' the felfe fame playne fi pte and on oneand the fame fide thereof, by the 6.of the eleuenth he may ore him to confeffe his two perpendiculars to be alfo parallels. But by {uppofi- | be bridled: which will tion agreed one,they concurre at one and the fame poyne, which(by the definition of parallels ) is im- poffible. Therefore our aduerfary mult recant aud Boe to out propofition. q The 12. I beoreme. Ihe 14. Propofition. Lo whatfoeuer plaine fuperficieces one and the felfe fame right line is ¢e reéted perpendicularly : thofe fuperficieces are parallels the one to the other . ) s RR pp o/c that aright line AB be 845 Mleredted perpedicularly to either [s iN of thefe plaine fuperficieces O50 A CDandEF. Then 1 fay, that lels the one to the other . F or if not, then if they be extended they will at the length meete , Let them mecte, if it be pofible. Now then their common fection {hall (by tures and let the [ame be K . And drawe a right line from the point A to the point K, and an other from the point Bto thepoint- =. K . And fora{much as the line AB is e- | Hb i eee vetted ee ee ae tte Oe me 4%; an of Euchdes Elementes. Fol.330. rected perpendicularly to the plaine fuperficies EF, therefore the line AB is alfo erected perpendicularly to the line B K which is in the extended {uperficies E F Wherfore the angle AB Kis aright angle. And by the fame reafon alfothe angle BAK is avight angle.Wher- fore in the triancle Ab K ,thefe two angles AB K ¢& B AK, areequalltotwo right angles: which (by the 17 .of the fir{t) is impofGble . Wherefore thefe fuperficieces C D and E F being extended meete not together . Wherefore the {uperficieces C D and E F are parallells. Wher- fore to what foeuer plaine fuperficieces one and the felfe {ame right line is erected perpendicu- larly : thofe [uperficies are parallels the one to the other : which was required to be proned. In this figure may ye plainly fee the former demonftra- tion if ye ereéte the three fuperficieces, GD, GE, and | | K L Mperpédiculary to the ground plaine fuperfictes: but yetin {uch fort that the two fuperficieces GD and GE may concurre in the common line G K H, as is required in the demonitration. , A corollary added by Ca | If avight line be eretted perpendicularly to oue of thofe fuperfictes , at {hall alfo be erected perpen- dicularly to the other. For if it fhould not be ereGted perpendicularly to the other , then it falling vpon that other fhall- make with fome one line thereofan angle leffe then aright angle :which line fhould (by the 5 .petition of the firft)concurre with fome one line of that fuperficies whereunto itis perpendicular . So that thofe fuperficieces fhould not be parallels: whichis contrary to the fuppofition . Fer they are fuppfed to be parallels, q The 13. I heoreme. The 15. Propofition. If two right lines touching the one the other be parallels to two other right lines touching alfo the one the other and not being in the felfe fame plaine fuperficies with the two firft: the plaine fuperficieces extended by thofe right lines, are alfo parallells the one to the other. z = 5\ V ppofe that thefe two right lines AB and B Ctouching the one the other be pa- KIC SLS vallells to the{e two right lines D E G E F touching alfo the one the other,and Me, A not being in the felfe fame plaine [uperficies with the right lines AB and BC. S24 SKS Then fay, that the plaine (uperficieces by the lines AB and B C, and the lines DE and EF being extended, fhall not meete together, that is,they areequediflant and parallels . From the point B draw ( by the rr. of the eleuenth ) a perpendicular line to the {u- perficies wherein are the lines D E and E F, and let that per. pendicular line be BG. And by the point G in the plaine fu- perficies palling by D E,and E F draw (by the 31.0f the firit) vntothe line E Da parallellline GH : and likewife by that point G drawe in the fame {uperfictes unto the line E F a pa- 2 : rallell line G K. Aud fara{much as the line B G 1s erected per- Hi 7 IS pendicularly to the {uperficies wherein ave the lines D E ana E F, therefore (by the 2. definition of the eleuenth ) it is alfo erected perpendicularly to all the right lines which touchit, ® and are in the feife [ame [uperficies wherein are the lines D E VV iy. and w Conftructione Demonftra- tion. The eleuenth B ooke and EF .But either of thefe lines G H and G K touch it, and are al{oin the {uperficies wherein are the lines D E and EF, therefore either of thefe angles BG H, and BG K, is a right angle. And forafmuch as the line B Ais a parallell to the line GH (that the lines G H andG K are parallells unto the lines AB and BC it is manifeft by the 9.0f this booke ): there- fore (by the 29. of the fir|t ) the angles G B Aand BG.H are equall to two right angles. But the angle B G H is (by con- firuitio) a right angle,therfore alfo the angle G B Ais aright angle - therefore the line G Bis ereéted perpendicularly to the line B.A. And by the fame reafon alfo may tt be proued, that the line B Gis erected perpendicularly to the line BC . Now forafrauch as the right line BG ts erected perpendicularly to ihefe two right lines BA and BC touchine the one the other, therefore ( by the 4.of the elenenth ) the line BG is erected perpendicularly to the [uperficies wherein are the lines B AandBC.And itis alfo erected : perpendicular) to the {uperficies wherein are thelines GH and GK . But the {uperficies wherein arethe lines G H and G K, is that [uperficies wherein are thelines DE and EF: wherefore the line B Gis erected perpendicularly to the {uperficies wherein are the lines DE and EF. Wherefore the line BG is wetted perpendicularly to the {uperficies wherein are the lines D E and E F, and to the [upes ictes wherein are the lines A Band BC. Butif one andthe {elfe fame right line be erected perpendicularly to plaine [uperficieces, thofe {uperfi- cieces are (by the 14.0f the eleuenth) parallels the one to the other . Wherefore the {uperficies wherin are the lines AB and BC is aparallel tothe [uperfictes wherin are the lines D E and EF. if therefore two right lines touching the one the other ‘be parallels to two other right lines touching alfothe one the othr, and not being tn the felfe fame plaine [uperficies with the two firft, the plaine (uperficteces extended by thofe right lines are alfo parallels the one to the other which was required to be demonflratea. 3 cise By this figure here put,ye may more clerely fee both ' ghe former 15.Propofition and alfo the demonittration therof: if ye ercéte perpendicularly ynto the ground -fuperficies , the three fuperficieces ABC, KHE, and L HB M, and fo compare it with the demonttration. @ A Corollary added by Fig Vntoa plaine {uperficies being geuen,to drawe by « point geuen without Wt, @ parallel plaine fuperfieies. Suppofe asin the former defcription that the fuperficies geué be ABC, 8 let the point eué without itbe G » Now then by the point G. drawe ( by the 31.0f the firft ) vnto the lines AB and B C parallel lines G Hand HK. And the fuperficies extended by the lines G Hand GK fhall be parallel vato the fuperficies AB C, by this t5.Propofition. | a The 14. T heoreme. The 16. Propofition. If two parallel playne fuperficieces be cut by fome one playne fuperficies: their common /eciions are parallel lines. id * ppofe that thefe two plaine (uperficieces A Band D be cut by this plaine fu- NG ZL} perficies E F G H,and let their common fettions be the right lines E F andG Ways | H.Then 1 fay that the line E F is a parallel to the line G H. For if not, then the Sq lines E F and G H being produced, fall at the length meete together either on the of Euclides Elementes. she fide that the pointes F ,H are, or on the fide that the pointes E,G are. F irft let them be produced on that fide that the pointes F,H are,and let them mete in the point K. And forafmuch as the line EF K is in the fuperfucies AB, therfore all the points which are in the line E Fare in the fuperficies A B (by the first of this booke) But one of the sintes which are in the right line E F K is the point K, therfore the point K isin the (uperficies AB. And by the fame reafon alfo the point K is in the fuperfictes C D. Wherfore the two [u- | perficieces AB and C D being produ- ced do mete tagether, but by [upzofitio _ they mete nottogether, for they are [up Fol,331. Demonftra- tion leading to an abfurditte® ofed to be parallels. Wherfore theright lines E F and G H produced [hall not meete together on that fide that the pointes FH are.In like fort alfo may we proue that the right lines EF and G H produced meete not together on that fide that the pointes E,G are. But right lines which being produced on no fide mete together, are rallels (by the laft definition of the firft.) Wherfore the line E F is a parallel to the line G H.If therfore two parallel plaine {u- erficieces be cut by fome one plaine fuperficies their common feciions are parallel lines: “ghich was required to be proued. eter This figure here fet more plainly de \ demonftration,if ye erect perpendicularl}} | fuperficies the three fuperficieces A B,C | }— and fo compare it with the de monitratio E # A Corollary added by Flufvasem If two plaine fuperficieces be parallels to one and the felfe fame playn a ee ~ - a ane < rs e2F Hy > od - at ae a. oe SSS Der PS. EL Py foal alfo be parallels the one to the other,or they {hall nzakg one and the felfe fame plaine faperficses. © rorifthe plaine fuperficieces DG and GH being parallels to one and the felfe fame fuperfi- cies,namely, to AB be not alfo parallels the one to the other,then being produced they fhall con- curre(by the conuerfe of the fixt definition ofthe eleuenth) Let them concurre in the right lineG E. Then I fay that the fuperficieces G D andGH ate in one and the felfe fame playne fuperficies. Draw in the playnefuperficies ABa right line at all aduentures A C.And by that rightlyne & the point Eextende a playne fuperficies, cutting the two fuperficieces D G and G by the right lines > E DandEI. Wherfore the right lines AC and D E,alfo A Cand ELare parallels by this propofiti- on.But the lines D E and El forafmuch as they concurrein the point E are not parallels the one to the other. Wherefore the right lines DF and E I make direétly one right line (by thar which ts added after the 30. propofiton of the firft.) And therfore the plaine fuperficieces D G and G Hare ‘none and the felfe fame playne fuperficies . For \ D { ; eee, G VV. ilil. a ss Wahl) ii \ y . = Q a ° " AY 7 . : . a 7 ~ —- > - - sel = a - —S u es — a ~ ~ _~ »_—- se > —" ee —— = SS ~~ i — Sa RE oy OE = = = amet — — r se = - - es anol 3 e : _ c = We = 6 eo — ss ce z = — — - = = <=: <= = oe = one > —— a —— - = ~ — = = oP ——— = ——— = - ~* = a ae = - -- ——- = - - = Ps a. = ET . = - sr ne > —— = — “= Per - a _- == — = =: alga ane - ee = ~—— SSS er mee = = —_—-— =e Tee — Se — — - — oF = ne SE = —— = : SS SE Se : —— - c oe an = —_—— —— = — —— _ —— - _ a = - ee ES - = os = ame se - - — = —— ee = es —_ - Se aaa Sears sheen = a = 2 em = > me wwe ly. ae —-= ~ _ Pa _ . In this propo- fetson ye muft ynderstand the praportio- nail partes or feGions to be é hofe which are contained betwene the parallel /uper- Jictese Construction. Demon|tra- tion ® The elenenth B ooke if shey benoe,then part of the right line D I,namely,the part D E is in the playne fuperficiesD G, and an other part therof,namely,E I 1s on high in an other fuperficies G H which by the firft of the eleuéth is impoffible. Wherfore the {uperficieces D G and G H are in one and the felfe fame playne fuperficies. But if the fuperficieces D G and G H neuer concurre,then are they parallels by the 6. definition of the eleuenth. In this figure here fet, ye may more plainely fee the former demonftration, if ye eleuate to the ground fuper- ficieces A C DI,the three fuperficieces A B,D G,& GI, and fo compare it with the demonftration. The 15. Theoreme. Ihe 17.Propofition. Tf two right lines be cut by playne fuperficieces being parallels: the partes of the lines deuided {hall be proportionall. a GV ppofe that the[e two right lines A B and C D be deusded by thefe plaine {uper- Ce ficieces being parallels, namely,GH,KL,MN inthe points A,E,B,C,F,D.Theé \Wa5S | 1/2 that as the right line A E is to the right line E B,fois the right line C F to YN the right line F D.Draw thefericht lines AC,B D and AD. And let the line A D and the [uperficies K L concurre in the point X. And draw aright line from the point E to the point X and an o- ther fron the point X tothe point F. And fora{much as thefe two parallel fuperficieces K Land MN ave cut bythe fuperficies E B D X, therforetheir common {ections which are the lines EX and B D, are (by the 16. of the eleuenth) parallels the one to the other. _And by the fame reafon alfo forafmuch as the two parallel (uperficiesG H and K L be cut by the [uperficies AX F C, their common fections AC and X F are(by the 16 .of the eleuenth ) parallels.And fora{much as to one of the fides of the triangle ABD, namely, to the fide B D is drawne a parallel line E X stherfore,( by the 2.0f x% the fixt) proportionally as the line A E is to the line E B,fo is the line AX tothe line X D. Againe foraf{much as to one of the fides of the triangle A D C,namely, to the fide AC is drawen a parallel line X F therfore by the 2.0f the fixt, pro- portionally as the line AX is to the line X D,fois the lineC F to the line F D.And it was proned that as the line AX is to the line X D,fois the line AE to the line E B, therefore alfo (by the 11.0f the fift) as the line A Eis tothe line E B, fois the line C F to the line F D.If therfore two right lines be dewided by plane fuperficieces being parallels the parts of the lines deuided {hal be propor. tionall : which was required to be demonftrated. In ee San eS wt ~ of Euchides Elementes. | In this figure itis more eafy to {ee the former demontftration, ifye e- rect perpendicularly vato the.ground fuperficies AC BD, the thre fu- perficieces,GH,K L,and MN, orif ye fo erect them that they be equedi- ftant one to the other. rs q The 16. Lbeoreme. Ihe 18. Propofition. If a right line be ereéted perpédicularly to a plaine fuperficies: all the fuperficieces extended by that right line, are eretted perpendicularly to the feife fame plaine fuperficies. = | ; Vppofe that a ae line A B be erected perpendicularly toa eround fuperficiesThé ae INES a Ae that all the [uperficieces pafsing by theline AB, are erected perpendicularly MONG to the ground {uperficies . Extend a [uperficies by the line AB, and let the [ame be E D, & let the comon fection of the plaine aS jo Nes Sale | the right line CE. AndtakeinthelineCE VY = | a point at alladuentures, and let the fame ~ a F: and (by the11.0f the first) from the poin F drawe unto-theline CE a perpendicular line in the [uperficies D E, and let the fame be F G.And forafimuch as the line AB ison | — erected perpendicularly to wee Fo See B SORTS | | & He. a ay aa a ficies, therefore (by the 2.definition of thee. lenenth) the line A B is erected perpendicu- 7 r ‘on Te? = - ~ : = ae ” AN: ‘ An Ly Confirnttions | Ha) | Demon ftra- larly to all the right lines that are in the ground plaine fuperficies,and which touch it.Wher- site aes Sore it is erected perpendicularlyto the lineC E. Wherefore the angle AB Eis aright angle. And theangleG ¥ Bis alfoaright angle (by confirutiton) . Wherefore ( by the 28 of the Sift) theline A Bisa parallelto the line FG . But the line A Bis erected perpendrcularly to the ground [uperficies : wherefore (by the 8 of the elenenth) the line F Gis alfo eretted per- pendicularty to the ground [uperficies.And fora{much as(by the 3 definition of: the elenenth) a plaine {uperficies is then erected perpendicularly to a plaine. [uperficies, when all the right lines drawen in one of the plaine {uperficieces unto the common fection of thofe two plaine {ua perficieces making therwith right angles, do alfo make right angles with the other plaine [u- perficies:and it is proued that the line F G drawen in one of the plaine fuperficieces, namely, in D E, perpendicularly to the common {ection of the plaine [uperficieces, namely, to the line C E, is erected perpendicularly.tothe ground fuperficies: wherefore the plaine [uperficies D E is ereched perpendicularly to the ground fuperfictes. In like fort alfo may we proue, that all the plane {uperficieces which paffe by the line A B, are erected perpendicularly to the ground [u- perfictes . If therefore aright line be erected perpendicularly to a plaime (uperficies all the [u- perficieces pafSing by the right line, are erected perpendicularly to the felfe fame plaime {uper- fictes : which was required to be demonstrated. 3 a In T-he-elenenth Book In this figure here fet ye may erect per- | _ -pédicularly at your pleafure the {uperficies wherin are drawen the lines D C,GP,A B, and HE, to the ground fuperficies wherin is drawen the line C FB E, and fo plainly compare ic with the demonftration before put. ae ‘ = ee z = = ail a ~ SSS eS ql he 17. I beoreme. The 19. Propofition. Demon firati- en deading to an impofsrbi-, bite. j fame lines ~ “vight lines befides B D which gs the common fection of the twofuperficieces, AB and BC. If. therefore tmo plaine {uperficieces cuttingthe one the other be erected perpendicktarly to any. plaine [uperficies their common fectiowis alfo crected perpendicularly to the felfe {ame plaine. fiperficies:whith was required to be proucd. =" AA ‘. Herehaue Ifet an other figure which, . . will more plainly fhewe vnto you thefor-: merdemonfttation, ifye erecte perpendi= ~— cularly to the ground fuperficies; A C the»: two fuperficieces A 3. and B C which cut. ; Sheonethe otherirthpline BD. + waa Yan UNS = Y awe = E ass | Reyes ct awa amet || \ Stee ' a : hy : Sok Motte be se. T heoreme, - The 20:Propofition. s if a folide aggle be contayned bnder three playne fuperficiall angles:euery . ) ee two | of Euchides Elementes. Fol.3 33. tivo of thofe three angles, ‘which two fo ener be taken , ave oveater then the third. ) eg GH KR: therfore the angle G H L is Greater then the angle D EF, And forafmuchas thefé twolinesG H and HL areequall to thefe two lines D'E'and EF, and the angle -G:H L is’ greater then the angle D E F therfore (by the 25.0f ial )thebafeG L is greater the the bafe DF But the lines GK and K L are greater then the line GL. Wherforethe lines GK cK L are much greater then the line DF But theline K L 13 equal to the line A C.Wher- fore the lines A Cand GK are greater then the line D Fi like irs alfo may we prone that the lines AC and D F are greater then the lineG K, and that the linesG K and D F are greater then the lyne AC. Wherforeit is poffible tomaken triangle of three lynes equall to the lines AC;D F and G K : whith was required to be demonstrated. V5 44) ‘. An other demonftration. . Suppofe that the three [uperficiall angles be .A BC, D E F,.andG H K, of which angles, tivo how/oeuer they be taken, are greater then the third. And let them be contained under thefe equal right lines AB,BC,D EE F,G HH K,xlich equall right lines let thefe lines AC, DF, andGh | | | : = toyne together. Then I~ * ~ ere Say that it is pofible of | three right lines equal to thelines AC, DF, and G K to make atri- angle,which againe is as much to fay, as that 4 two of thofe lines which two foeuer be ta- ken,are greater then the third. Now againe if the angles B,E,H,be equall,the lines alfo AC | XX. DF, G K D F Two tafesin this propofie iO, 2 firfh cafe, Setind tafe. Conjirndlions Demonfira- tion. . An other den monstration, 14 @ t p) bh a \ +] ' Bb ; ‘i wh , ‘ ; ’ } a i j yy ue % yy i H 7 : "oF i U 5 Hy i) 7 is : a 7,3 a | N te : HM 4 Ate 4 iy a) au | eh, : 7) Yh i , 1) : | oS im) ‘ Mf] The elenenth Booke | D Fyand G Kare equall,and fo two of them hall be greater then the third.But if not,let the angles B,E,H, bevnequall,and let the angle B be greater then either of the angles E and H. Therfore (by the 24.0f the first) the right line A Cis greater theneither of the lines D F cy GK. And it is manifest that the line AC with either of the lines D F orG K is greater . wos then the third. 1 fay alfo that the lines D F and G K are greater then the line AC. Vnto ConSteuttion. the right line A B,and to the point 1 it B smake (by the 22.0f the first) unto the angle GH K an equall angle AB L, and vnto one of the lines AB,BC,DE,£ F,G H or HK, make by the 2.0f the first) an equalllinéBL. And draw aright line from. L the point Atay, point L,and an other frotheA point L to the point C. Demonfita-.,. And foralmuchas we Ci randica thefedne dees A BAK icvehs ees AQ Op. 25 Le Tee BLare equal to theferwolinenG H GH K the one tothe other,and they coutatne equal an- -. gles:therfore( by the 4. of the first) the bale. L ts equal to the bafeG KxAnd forafmuch as the angles E.and.H be greater then the angle A BC,of which the angle GH Kisequal tothe : H z angle A B L,therforethe angle remayning, namely the angle Ets greater then theangle L B C, And forafmuchas thefetwo lines L Band B Care equal tothefe two,lines DE and EF the oneto the other,and the angle DE Fis greaser then the angle L.B.C,therfore(by the 2$. of the first) the bale D.F is greater then the bafe L C : and is proued that the line G K is ¢- guall tothe line ALWherfore the lines D.F & GK are greater then the lines ALG LC. Bus the lines ALand LC are greater then the line A C.Wherfore the lines DF. -G K are much greater the the line ACWherfore two of thefe right lines A C,D F & GK which two focuser betaken aregreater then the third. Wherfarest és poffible of three, right. lines equal to the lines AC,DF and GK to make atriangle, - sohichwas. required to be demonftrated. Mesrsve pL be 3. Probleme: +. Lhe 23.Propofitions sos Of three plaine fuperficiall angles, two of which how foener they betaken, are greater then the third,to make a folide angle: Now st ts neceffary that thofe three fuperficiall angles be leffe then fower right angles. a ppofe that the Superficiall angles genen be ABC, DEF,GHK: of which OG let two how foeuer they,be taken, be greater then the third : and moreouer, let thofe three angles beleffe then fower right angles . It is required of three ep ficiall angles equall to the angles ABC, DEF, and GH K,to makea. lide < * « : SS : o* of Euchides Elementes. 35. or bodily angle. Let the lines 4B,BC, DE, EF,GH, and HK, be'made equall : and ee drawe aright line from the point Ato the point C, & another from the point D tothe point Confirutiione F, and an other from the point G to the point K . Naw (by the'22. of the eleventh) itis pof fible of three right lines equall to the right lines:.AC,D F,and G K,tomakea triangle. Make {uch a triangle, and let the fame be LM N, a fothat let the line AC be equall tothe lineL M,.. and the line D F to the line M Nand the lineG K L to the line LN. And (by the 5.of the fourth about the triangle L M N_defcribea circle L MN, and take (by the 1.0f the third ) the centre of the [ame Three cafes circle, which centre {hall either be within the tri- in this propo- angle L M N, or in one of the fides therof,or with. fition. out it. Firft let it be within the triangle, and let The firft cafe. the fame be the point X, ¢ drawe thefe right lines L X, MUX, and NX. Now 1 fay, that the line AB A uetehaty ss greater then the line LX. For if not, then the line RS A B is either equall to the line L X, or els it is lefe p» ouedd sore thenit . Firft let it be equall. And forafmuch as he proceede the line A Bis equallto the line L X, but the line AB is equall to the line BC, therefore the any farther line LX 1s equall to the line B C : and unto the line L X the line X Mis (by the.15 definition in rhe con of the firft) equall : wherefore thefe two lines A Band B C, are equall to thefetwo lines. LX S rN P : oy of and X M the one to the other : and the bafe AC is {uppofed to be equall to the bafe LM. seebietorss:: Wherefore (by the 8 .of the firft) the angle ABC is equall to the angle LX M .. And by the [ame reafon alfo the angle D E F is equall to theangle (M XN, and moreouer the angle GH K tothe angle N X L .Wherefore thefe three angles ABC, D EF, andG HK, are e- guall to thefe three angles LX M,MXN,&N XL . But thethree angles LX M,MX N, andN X L, are equall to fower right angles ( as it is manifeft to fee by the 13 of the fir [tif a- ny one of thefe lines MX, L X, or N X, be extended on the fide that the point X is.) . Wher- fore the three angles ABC, DEF, andGH K, are alfo equall to fower right angles. But they are fuppofed to be leffe then fower right angles : which is impofable Wherefore the line AB is not equall to the line LX . I fay alfo that the line _AB is not leffe then the line L X: For if it be pofsible, let it be lefse : and (by the 2.0f the firft) unto the line AB put an equall line X O.: and tothe line BC put an equall line X P, and draw a right line from the point O tothe phint P . And forafmuch as the line A B is equallto the line B C,therefore alfa the line X 01s equall to the line X P ‘Wherefore therefidue O L is equall ta the refidue M P.. Wher- fore (by the 2.0f the fixt) the line L M isa parallel to the line 0 P - andthe triangle L MX is equiangleto the triangle O PX . Wherefore asthe line X Lis tothe line LM. [ois the line X Oto the line’ 0 P.. Wherefore alternately (by the 16.of the fift) as the line L Xistothe line XO, foisthe line L M tothe line O P . But the line L X is greater then the lineX 0. Wher- fore alfo the line L Mis greater then the line O P . Butthe line L Mis put to be equall to the line AC : wherefore alfo the line AC is greater then the line O P . Now forafmuch as thefe two right lines A B and BC are equall to thefe two right lines O X and X P,and the bafe AC 4s greater then the bafe O P, therefore (by the 25. us the firft) the angle A B Cis greater then the angleO X P . In like fort alfo may we proue, that the angle D E F is greater then the an- gle MX N, and that the angle G H K is greater then the angle NXL . Wherefore the three angles ABC, DE F,andGH K, are greater then the three angles LX M, MXN, and NX L. Buttheangles ABC, DE F,andGH K, are {uppofed to beleffe then fower right angles : wherefore much more are the angles LX M,MX Nye NX Lleffe then fower right angles. But they are alfo equall to fower right angles:which is impofuble. Wherefore the line A Bis not leffe then the line LX: andit is alo prowed that it is not equall untoit. Sot at XX 4. the * Which how to finde out is taught atthe end of this de monitration, and alfo was taughtin the aflumpt put before the 14. propofition of the céth boke. Demon tra- gran of the firk cafe, Secoud cafe, Lhe elenenth Booke the line AB 4s ¢reater then the line L X. | Now from the point X raife vp unto the plaine fuperficies of the circle L M N a perpendi- cular line X R (by the 12.0f theeleuenth) : And unto that which the {quare of the line CAB excedeth the.fquare of the lineX L* let the {quare of the line X R be equall. And draw aright line from the point R to the. point L, amd an other from the point R to the point M, and an otherfrom the point R to the point N. And forafmuch as the line RX is erected perpendicularly to the plaine [u- perficies of the circle LM N , therefore ( by conuer- fion of the fecond definition of the eleuenth) the line RX iseretied perpendicularly to euery one of thefe lines LX,(M X, and N_X . And foralmuchas the line LX is equall to the line X MU, & the line X Ris common to them both, and is alfo erected per- pendicularly to them both, therefore (by the 4.of the firft) the bafeR Lis equall tothebafe RcM. And by the [ame reafon alfo the line RN is equall to either of thefe lines RL and RM .Where- fore thefethree lines RL, RM, and RN, are equall the one tothe other . And fora{much as unto that which the {quare of the line A B excedeth the [quare of the line L X, the fquare of the line RX. is [uppofed to be equall, therefore the {quare of the line AB is equall to the Squares of the lines LX and RX . But unto the {quares of the lines LX and X R, the [quare of the line L Ris (by the 47.0f the firft) equall, for the angle LX Ris aright angle .Where- fore the {quare of the line A Bis equall tu the [quare of the line R L. Wherefore alfo the line A B is equallto the line R L.But unto the line AB is equall euery one of thefe linesB C,D E, E F,G H,and H K,and unto the line RL is equall either of thefe lines R M and R N.Wher- fore euery one of | thefelines AB,BC,DE,E F,G H,andH K, ts equall to euery one of thefe lines RL,RM,and RN. And forafmuch as thefe two lines. R Land RM areequall to thefetwo lines AB and BC, and thelafe L M is {uppofed to be equall to the bafe AC, ther- fore (by the 8.of the firt) the angle L RM is equall totheangle ABC . And by the fame reafon alfo the M RN is equall to the angle DE F,and the angle LRN to the angle GHK . Wherefore of three [uperficiall angles LR-M,M RN, and LRN, whichare e- quall to three fuperfictall angles geue,namely,to the angles ABC,DEF,¢GHK, is made — a folide angle R, comprehended under the fuperficiall angles LRM, MRNyand LRN: which was required to be done. e But now let the centre of the circle be in one of the fides of the triangle, let it bein the fide MN, and let the centre beX . And draw aright line from the point L tothe point X. I fay againe, that the line A B is greater then the line LX . For if not,then AB is either ¢ guall to . LX, or els it is lefvethen tt. First Tet. it be equall. Now thé thefe two lines AB and of Euclides Elementes:. 336. BC, that is, D E cy E F are equall to thefe two lines , MX andX L., that is, tothe line MN . But the line “ M Nis raphe to be equalltothe line D F.Where- ae fore alfo the lines D E and E F are equall to the line sae Bas . ee re 4 f Va iy . ~ DF :which(by the 20.0f the fir{t)is impofibleWher- re fore the line A B is not equallto the line LX . Ip. like > fort alfo may we prone, that it is not lefe.Wherefore the line AB is greater then the line LX. Andnowif. x as before unto the plaine fuperficies of the circle bee- rected fro the point X a perpendicular line R X whofe {quare let be equall unto that which the (quare.of the line AB excedeth the {quare of the line LX, andif the reft of thé conftruction and demonftration be 0b+ ~ , . feruedin this that was in the former cafe, then hall the Probleme be finz|bed, | But now let the centre of the circle be without the triangle LM N, and let st be in the point X And draw thefe right lines L.X,M X,and NX. I fay that in this cafe alfo the line Thirdcafe. A Bis greater then the line LX «For if not, then isut either equall or leffe. Firft let it bee. Sak | Oe Eaaltd . # quall. Wherefore theferwo lines AB and B ov) equall to.thefetwalines MX and X Ltheone tothe other, and the ca AC isequalliothe bafecM Lh... Wherefore (by the 8.of the fu,ft) theangle ABCis. equalltothe angle M X-L.: and by the [ame reafon .. alfo theangleG H Kis equalltothe angle. LX XX, Wherefore the whole angle MXN. is equallto. thefe two angles AB CandGH.K . But the angles ~*~ AB Cand GH K are greater then the angle DEF. Wherefore the angle M X Nis greater then the an- gle DEF. Andfora{much as thefe two lines D E and E Fare equall to thefe two lines MX and X N, and the bafe D Fisequallto the bafe MN,therc- fore (by the 8.of the firft)theangle MX N isequall totheangle DEF. And itis prowed thatitisalfa. = greater : which is impofsible..Whereforethe line AB isnot equall tothe line LX . Intike fort alfo may we proue, that it is not leffe. Wherfore the line A B is greater then the line L X. And againe if unto the plaine fuperficies of the circle be erected perpendicularly from the poimt Xaline X R, whofe (quare ts equallto that which the {quare- of the line AB exceedeth the (quare of the line LX, and the reft of the confifuction be done in this that was in the for- mer cafes,thenfhall the Probleme be finifbed. | SA ise 1 fay moreauer, that the line AB is not leffethen the line LX . For if it be pofsible, let ™ other demi- it belef[e. And unto the line AB, put (by the 2.0f the firft) the line X O equall: and vnto f ag shag the line B C put the line X P equall : And draw aright line from the point O to thepoimt P.. Bis not leffe thé XX. iy. And the lmeL X. si r) : ‘ : . a = fe ne ee eS er Sn me ae Fe ae eh ——— a SSS — : . ~’_ < . ——we a as ee ai rh: a oo —, = a ——— ——~ = — — : — — = —-—- = - ” c= in _ a e572 a = eee = Ss = - ~~ - - a a: —_— — Sn - = = <= = === ———— re a = = — - — ~ Se SS = — —, = _ - a = => = Ss —— = ao ; eS a ee SS = — , oa ns = =.) oe - - = iy oa es 7 = = = > “ j ss Theeleneath B ooke ind forafmuch asthe line A Bis equall to the line B C,therefore the line XOis equall to the line X P Wherefore the rcfidue O Lis equallto the refidue M P . Wherefore (by the 2.0f the fixt) thelineL M, 1s a parallelto tht line PO . And the triangle L X M is equianeleto the triangle P XO. . Wherefore (by theo .of the fixt) as the line LX isto the line LM, fos the line XO to the line P 0. Wherefore alternately "by the 16.0f the fift) asthe line LX is tothe line 30, fo vsthe line L M'to the line OP... But the line LX is greater then the line X O .Wherefore alfo the lim L Mts gréater then the line OP. But the line L Mis equall to the * line AC :Wherefcre alfo the line AC is greater then the line O PNow forafmench as thfe two lines AB and B C are equall to thefe two lineso X & XP the’ one to the other, and the bale AC isereater then the spp Gate OP; therefore ( by the 25. of thifirft) the angle This was before taught an thetenth booke tn the Affe mp put before thet s. propofttion, ABCs creater then the angle OX ?'. Andin like fort if we put the line X Requall to either ofihefe hnesX 0 oF XP; and draw rrieht line fro the point 0 to the point R, we may prove that the angle G H K ts greater ther the angle OX R. Vatothe right line LX, and vn- tothe point in it X, make ( by the23. of thefirft ) unto the angle ABG an equall angle LXS: and unto the angle GH K make an equall angle LXT . And. by the fecond of the firft ) leteither of thefe line SX yand XT , be equall to the line 0 X. And dvawe thefe lings O 8,0 T ,andST «And fora{much as thefe two lines AB and BC aree- quall to thefe two lines OX and X Sand the angle ABC is equalltotheangleo X'S, there- fore (by the 4.of the firft) the bafeaC,thatis,L oppofite fides are (by.the 16. of the cleuenth) parallels. Wherfore the ‘are parallelo- line A Bis aparallel tothe line c D. ‘Agathe forafrauch \ grammese 4s two parallel plaine [uperficieces ¥ Baka AE wre dénp © ded by the plaine [uperficies AC their common fettions are by the fame propofition,parallels: Wher fore thé lyne >=: AD is4 parallel to the line BC. Anditiealfo proued,* that theline A Bis a parallel to thé ine DC. Wher fore the [uperficies AC isa parallelograrinne: In like fort alfo | | may we proue,that enery one of thefe [wperficices CE;G F} BG; FB, and AE ave parallelos grammes.Draw aright line from the point A,to the pont Hand an other from the point D tothe point F.Aud fora{much as the ling A B és proued 2 parallel tothe lineCD, op debe hne Demonftratio BH 10 the line CE, therfore thefe tworight lites AB and BH touching the one the other, are *at the oppo- parallels tothe/e two right lines DC and CF touching Alfo the ene the other, and not being ~ SH a 1m one ated the felfe [ame plaine upérficies. Wherfore (by thet o. of the elewenth )they compre- — hend equallangles.Wherfore the diigle AB His equall tothe ancle DC F 4nd forafmuch As shel twolines AB and B H are* equal tg the ¢ twolines D C indC F, . : = mee oe | . and Hi ale 428i pred salem D6 Ebbert Ojthe ofthe A Dizewalee DC; beck fitft) the bafe A His equall to the bale D F, and the triangle AB H ise. rallelograme,and by the fame rea- quall to the triangleD CF: And fora{much as (by the yr. of the firft)the fon, isBH equall to C F, becaufe ; ; vee Sa 375 >Jy. the {aperficiesFBwis proved a pa- parallelogramme B G is doubleto the triangle ABH, and-the parallelo- iatindaae Eedipettuareforé the 2¢.0f gramme E is allo double to the tYeancle DG F; therfore the parallelo-the firftis our proofe. oth XX ity . gramme gaia T he eleuenth Booke gramme B G is equall to the parallelogramme C E.In like fort alfo may we prone that the parallelogramme AC is equall to the parallelogramme G F and the parallelograme AE to the parallelogramme F B.Lf therfore a folide or body be contained under fixe parallel plaine [uperficieces,the oppofite plaine fuper ficieces of the fame body are equal c parallelogrammes: whith was required to be demonftrated. Oe I haue for the better helpe of young beginners, defcribed here an other figure whofe forme ifit be defcribed vpon pa- fted paper with theletters placed inthe fame order that it is here,and then if ye cut finely thefe lines AG,D EandC F not through the paper, and folde it accordingly, compare it with the demonftration,and it will fhake of all hardenes from it. T he x2: T heoreme ~The 2s. Propofition.” Ifa Parallelipipeds be cutteof a playne fuperfie cies beyng a parallel to the two oppoftte. playne ‘fuperficieces of the fame body: then, as the bafe is to'the bafe, fo ts the one*folide to the otber foltdesi-is: ee see EAE Vppofe that this folidé'A BCD being contained... ANG 45 y under parallel plaine (uperficteces ( and therfore. ’ BOING called a parallelipipedd) be cut of the plaine fupers. - “\. ficies V E being a parallel to the two oppofite [uperficteces of: . ‘ the fame body,namely,to the fuperficreces ARG DH.The. visvnetnss J fay that as the bafe A E F Wis to the bale. EH CF3fois Confirution. the folide AB FV to the folideE GC.D.Exted the line A . H on either fide,c put vito y lineEH asmany equal lines. as you wil,namely ti M,cy M N:clikewifevntotheline A E,putas many equallines as you will namely, A Kr K Lc make perfect thefe pavallelogrames LO,K WAZ, M S,and likewife make perfect thefe folides or bodies LP, K RD M,andMT..And fora{muchas thefe rightlines,. L.K,K A,aud AE are equall theone.tothe other therfore thefe parallelogrammes LO, K W, and, AF; arealfo (by. » the firft of the fit equall the one to the other : and fo-alfa (by the [ame)are thefe parallelagrammes K X,K B,and A. G equall the oneto the other. And likewife(by the 24.0f the. 1 eleuenth) are the parallelogrammes LY,K P, and A R,e-.. qual,for they are oppofite the one ta the other. And bythe ..“} » fame reafon alfo the parallelogrammes E C,H ZandMS~ are equall the one to the other, and the parallelogrammes — _ HG,H 1,and LN are equal the one tothe ther. And more ver she parallelogrammes D H,U Q and NT are (by the 24.0f the elewenthequall the oneto the other, for they ——- SS - ; : : : ie | \ - ‘ ' ' ti ' ai a] eA | i : Tr ; || ae ; i! i : ine : , ' a :% |e } i q - i? WS a 1) | i} t . nh bp’ i 1 ia be { } i ‘By : : : { + ‘ > ; El ] \ TT its { i i : t ; EH { it = y f 1 N \t {r \ } i : \ ) | Bh ) 4 , nae ; ie i iu i} wile + ‘a t { ' ik ; Teh : a) ie 4 a } | | {8 : iH : ij Lie { t v ' we ‘a wae . q 6 4 1 (3 - a : i at ) vn 1 i! is ir 4 a . Shi.» er i am iG) t } ) } 1 ; f wee \y iy : We, a "4 + ; ae i ial W ) UP iy i ee sae | { ‘2 ne ia) H Ben |e cc | ' i { . (W “d } p f ia 4 WJ ; 1g aq a : wey ¥ \ | ih ‘h ’ 1 } i H ay aay i " t Ps - gy Wu rT ¥ ! 14 oe wih ia \ 7 : it j a iW pel | way ff Pe 7 aang e “ 7 " a f Ban | ye | a) | : 4 ‘ } ih | i ie. : ft ' ji . - ‘ i . ‘ i am im tiie t ( | ’ in oh y : ¥ ‘ , \' | . ; \ iF] T . i 4 uf t IG ; ; ; , i y ; i! ae ati \ { ‘ - if \) 1) i gy b the I a 7 @ Wa! H i 4 it | 4 Hh” Ri | \ lt 1) Ny i ui i | a “a 4 fii uf if , ' : i “ i i! i ‘ am |) } t } i] -j. = ay e a . ie > of Euclides Elementes, Fol.338. are oppofite: wherefore three plaine fuperficies of the folides LP, RK, and AV are equall ta three. plaine [uperficies : but unto eche of thefe three fuperficieces are equall the three oppofite [uperficieces (by the 24.0f the eleutth) Wherfore thefe three folides or bodses L P,K R, and AV, are equal the one to the other, by the 8. definition of the elewéth.And bythe fame reafon alfo the three ‘folides,ED,D Me MT are equal the one to the other Wherfore how multiple x the bafeL F is,to the bafe.A F, fo multiplerxasthefolide LV tothefolide A V. Ana by the fame reafon alfo how multiplex the bafe N F is to the bafe F H,fo multiplex is the folide NV to the folideHV : fothatif the bale L F be equall to the bafe N_F ,the folide alfo LV fhall be equall to the folideV N,and if the bale L F exceede the bale N F the folide alfo L¥ fall exceede the folideV Nand if the bafe L F be leffe then the bale N F the folide alfo LV fhall be leffe then the folideV N_{ by the 1. and 14. of the fift.) Now then there are foure magnitudes, namel 'y ,the two bafes A F and F H,and the two folides or bodies AV and V Hof which are take their equemultiplices,namely,the equemultiplices of the bafe AF, ¢ of the folide AV or the bafe L F,¢y the folide LV ,¢> the equemultiplices of the bale HF ,¢ of the [olide HV arethe bafe N F and the folide N ¥.. And it is proued that if the bale LF excede the bale N F the folide alfo L V-excedeth the [olide N Vc if it be equal,it is equal, and if it be leffe,itis “ot whethea by the 6 definitio of the fift)as the bafe AF is to the bafe F H,fois the folide AV to the folideH V.If therfore a parallelspipedon be cut of a playne {u- perficies being a parallel to the two oppofite playne fuperficieces of the fame body, then,as the bafe isto the bafe,fois the one folide to the other folide: which was required to be proued. RK @) P t haue for the better fight of the coftrudtio 8 = dem Oftration of the for- A | mer 25 .propofitid , here fet another figure, whofe | forme ifye defcribe vp- " | Fl Vv pon patted paper,and finely cut the three lines XI,BS, and T Y, not through the paper but » halfe way, and then fold it accordingly, and compare it with the conftruction and dem6- ftration,you fhall fee that it will geue great light therunto. =~ Here Flufas addeth three Corollaries, Firlt Corollary. If a Prifme be cutte of a playne fuperficies paraltelto the oppofite Superficieces the feftions of the Firft Corolla. ig fhall bethe one tothe other in that proportion, that the fettions of the bafé are the oneto the ry. other. For : i a ; ; : - ’ ‘ eh ' if t f ” ‘ i] ' i} ' i ; : ) \ y | r & y PR: : { ; Kb | : \f 1 oe Ee ' ‘ : ( iP :. ' : : } j ; : & - ’ '@g eae | . ' ‘s i he. 1 iy - i ey : . Tv) (ee a , : i) y : He) at ' > thi i ; L » J ’ it Ho. ; : as | ie 2) wy i , sat Fi A ‘ “ + ‘ aie : ay ae ~ ne a i i t - - ‘ ; Le? : ‘ . b fl Mi . i ; q § 5 4 4 oo : | wi é ; in oa } = te @ me : a ; ‘ 3 ie | ) Pak J ‘ i) : ! t i} i : a c | | * 4 ' : a ; 2 } iv) i th 7 i i ut f ) | t ] i mee an 4 i] : en! on, R - th : : ’ } | : rf ‘ : li : t ‘ | 7 >it) { + ; § : ¥ * iY i if Second Corol- ry. Thefle folides which he Speaketh of tn this Corollary are of foue called fided COLUMN Se Third Corol- barye Const ract#on. T he elenenth Booke For the feétions of the bafes, which bafes(by the r1,definitid of this booke)are parallelogrammes, fhall likewife be parallelogrammes , by the 16. of this booke(whenas the fuperficies which cutteth is parallelel to the oppofite fuperficieces and fhal alfo be equiangle. Wherfore if ynto the bafes(by produ- cingitherightlines)be added like and equall bafes,as was before fhewed in a parallelipipedon,of thofe fettions fhalbe made as many like Prifmes as ye will.And fo by the fame reafon,namely, by the comm6 exceffe equalitie,or want of the multiplices of the bafes & of the fections by the 5. definiti6 of the fifth may bé proued , that every fection of the Prifme multiplyed by any multiplycation whatfoeuer , fhall haue to any other {ection that proportion that the fections of the bafes haue. Second Corollary. Solides whofe.twe oppofitefuperficieces are porsgeveg figures like equall and paraltels the other fa. perficies, which of neceffitie are parallelogrammes being cutte of a playne fuperficies parallel tothe two oppofite fuperficies : the fettions of the bafe are the oue tothe other , as the fettsons of the folide are the one to the other. Which thing is manifeft, for fach folides are deuided into Prifmes ; which hane one c6mon fide, namely,the axe orright line , which is drawne by the centers of the oppofite bafes. Wherefore how many parallelogrames or bafes are fet vpon the oppofite poligonon figures,offo many Prifmes fhal the whole folide be cépofed.For thofe poligonon figures are deuided into fo many like triangles by the 20. of the fixth whichdefcribe Prnfmes. Which Prifmes being cut by a fuperficies parallel to the oppofite fuperficieces,the feGtios of the bafes fhal(by the former Corollary) be proportional with the feds of the Prifmes. Wherefore by the 12.0f the fifth , as the {ections of the one are the one tothe other, fo are the fections of the whole the one to.the other. Of thefe folides there are infinite kindes,according to the varietie of the oppofite and parallel po- ligonon figures, which poligonon figures doo alter the angles of the parallelogrammes {et vpon them according to the diuerfitie of their fituation.But this is no let atall to this corollary , for that which we haue proued willalwayes follow, When as the fuperficieces which are fet vp6 the oppofite like & equal poligonon and parallel fuperficieces are alwayes parallelogrammes, Third Corollary. T be forefayd folides eompofed of Prifmes,being cutte by a playne fuperficies parallelto the oppofite. fuperficieces arvethe one tothe other.as the heades or higher parts cntte are. For itis proued that they are the one to the other as the bafes are. Which bafesforafmuch as they are paraiielogrammes,are the one to the other as the rightlines are vpon which shey.are fet by the frit of che fixth which right lines are the heddes or higher parts of the Prifmes. Bn? The 4.Probleme. Ihe 26.Propofition. V pon aright lynegeuen,and at a point init geuen, to makeafolide angle equall to a folide angle geuen. bers) AV ppofe that the right line geuen be AB, and let the point in it ceuen be A, and let the folide or bodily angle gewen be D being contained under thefe fuperfict- D\S \all angles ED C,E D F and F D C.It is required upon the right line AB, & eet KG) at the point in it geuen Ato make a folide angle equall tothe folide angle D. Take ig the line D F apoint at alladuentures,and let the fame be F. And (by the 11. of the clesenth) fro the point F. Draw unto the fuperficies wherin are the lines ED ¢ D C a per- pendicular line F Gand let it fall upon the plaine fuperficies in the point G, oo draw aright line from the point D to the point G.And (by the 23 .of the firft) unto the line A B,and at the point Amakevnto the angle E DC an equall angle B.A L, and unto the angle E DG put the angle B AK equall : and by the 2:of the firftput the line A K equall to the line D Gand, ( by the 12, of the elenenth ) fromthe point K raife up vnto the plaine [uperficies BALA 2 Set perpendicular line K Hand put the line K H equall to the line G F, and draw aright lyne from er ET of Euclidles Elementes. Fol.339. from the point H to the point A: Now fay that the folide ayele Acontained under the [te perficiall angles B A L,B A H,andH ATL is equall to the (olide angle D contained under the [uperficiall angles ED C,E.D F,and F DC. Let the the lines AB and DE be put e- quall,and draw thefe right lines H B,B K;F E,andE Gi-And foral much as the line F G is erected perpendicularly to the ground [uperficies,ther fore by the 2.d efinition of the elenenth, the line F Gis alfo erected perpendicularly to all theright limes that arein the ground [uper- ficies aud touche it. Wherfore either of thefe angles F'G D and F G Eta right anele, and by the fame reafon alfo either of the anglesH K A and HK Bisa right anele. And fora{- much as thefe two lines K AG AB are equal to the{e two lines G D & D E, the ene tothe other, ana they containe equall RS : angles( by conftruction): Wher ~~~ fore (by the.4.:of the firft) the. .. .. bafe mf 1s wets bye y. ee *. G, and the line K His equall to | the lineG F, and they coprehtd right angles. Wherfore the line» ~* B His equall to the line FE. °° Agayne,foralmuch as thefe two lines A K and K H are equal to thefetwo lines DG and GF, and they containe right angles. / Dd Wherfore y bafe.A H is (by the 4.of the firft) equall to the bafe D F.And the line AB is equall tothe line D E.Wherfore thefe | two lines A B and A H are equallto the[e two lines F D and D E,and the bafe B His equall to the bafe F E.Wherfore (by the 8.of the first) the angle B.A H is equall to'the angle EDF. And by the fame reafon alfothé angle H K Lis equall tothe angle F G C.Wherfore if we put thefe lines A L and DC equall,and draw thefe right lines K L, H L,G C,and F C: for. afmuchas the whole angle B-A-Lis-equall to the whole angle E D C,of which the angle B A K is fuppofed to be equall tothe angle E D G, therfore theangleremayning,namily, K Ak 3s equall to. the angle remayning G.DC.And foralmuch as thefe two lines K A and Ab are equal to.the[e two lines GD and D C,and they containe equall angles, therefore by the 4. of the firsts the bale K Lis equalltothe bafeG C, and theline-K.H. isequalltotheline GF, wherforethele two lines. LK andK H are equall tothefetwolines CG and G F,and they ¢0- taineright angles Wherfore the bale H Lis (by the 4.0f. oe ) equal to the baleF C.And forafmuch as thefe two lines H A and A Lare equalltothe{etwe F D and D Canad the bafew H L isequall tothe bafe F C,therfore( bythe 8 -of the firft) the angle H A.L is equallto the angle F D Cand by conftruction,the angle BAL is equall tothe angle E D.C. Wherefore vuto the right line geuen,and at the point in it geuen,namely Ais madeafolideangle equal tothe folide angle geuen D.:.which was required to be done, . ‘ ____In thefe two figures here put,you may more clearely conceiue the for- mer conftmudtion andde- . monftratio, ifye erect per- pendicularly vnto the ground fuperficies the tri- anglesAL BandDC Ey eleuate the trianglesA LB and DC F that. the lym | | —s LA Cemonslra~ 120% Confirnctions \ The eleuenth Booke LL AandC Dofthem may exaélly agree with thelines LA and CD of the triangles ‘erected: For foet- dering them,if ye compare the former conftruGtion and demonitration with them, they will be playne ynto you, Althonegh Euclides demoftration-be touching folide angles Which are contained only vnder three {uperficiall angles, that is, whofe bafes are triangles syet by it may ye performe the Probleme touching folide angles contained ynder fuperficiall angles how many foeuer, that is, hauing to their bafes any kinde of Poligonon figures.For etiery Poligonon figure may by the 20. of the fixt,be refolued into like tringles. And fo alfodhall the folide angle be deuided into fo many folide angles as there be triangles in the bale. Vnto euery oneof which folideangles you may by this sicdipoditian defcribe an equal] folide angle vpona line geuen, and ata point in it geuen. And fo atthe length the whole folide angle after this marierdefcribed fhall be equall to the folide angle geuen. The s.Theoreme, The 27. Propofitio. Vpon a right line geuen to defcribe a parallelipipedon like and mn like fort fre tuate to a parallelipipedon geuen. V ppofe that the right line cenen be AB, and let the parallelipipedon geuen be CD. Itis required upon the right line genuen AB to defcribes parallelipipe- don like and in like fort fituateto the parallelipipedon geuen , namely, toc D. Vso the right line AB aud at the poynt in it A defiribe ( by the 26 of the ele. L ~ 7 Y . acnth( a folide angle equall to the folide angle C , and let it be contayned umpder thefe {uperfi- cial anglesBAH;H AK, andK AB , [0 that let the angle BAH be equall to the angle E © P and the angleB A K to the angle EC Gand moreouer the angle K AH tothe an- gleGC FE. And as the line EC is to thelineC G,fo let the'A B beto the line A K (by the 52.0f the fixth ) and asthe line GC isto the line CF , fo let the K A be tothe line AH: Wherefore of equalitie ( by thé 22. of theft ) as the line E C is tothe line CE [0 7s the ine - . - BA to the line AH Make pin fe the parallelogramme BH ,and alfo the folide AL. Now for thatas the lincE.C is tothe ineC G, fois the lineB A tothe line AK , therefore the fides which rt a the equallangles ; namely, the an elesE CG andB AK are proportio- nall:wherefore(by the firft definition o the fixth)the parallelogramme G Eis like to the pa- rallelogramme K B.And by the famereafon the parallelogramme K H is liketothe paralle- logramme G F and moreouer the parallelogramme F E to the parallelocramme HB-Wher- fore there are three parallelogrammes of the folideC D like to the'three parallelocrammes of the folide AV. . But the three other fides in eche of thefe folides are equall and like to their oppofitelides. Wherefore the whole [olide C D is like tothe whole folide A L: Wherfore vpon the right line genen A Bis de[cribed the folide A 1. contayned under parallel playne [uperfia cieces like and in like fort fitwate to the [olide gewe CD contayned alfo vader parallel playne Superficieces: which was required to be doone.: A esas This demonftration is not hard to conceaue by the former figure as it is defcribedin a playne, if ye | auc > 's. > _ of Euchides Elementes. Fol.34.6 that imagination of parallelipipedons defcribed ina layne which we befote taught in the difinition ofacube. Howbeit I hauehere for the more eafe of {uch as are not yet acquainted with folides , def | z NA A ee \ 3 : cribed two figures , whofe formes firft defcribe vpon pafted paper with the like letters noted in thent P and then finely cutte the three midle lines of eche figure,and fo fold them accordingly , and thar doone compare them with the conftructionand demonttration of this 37. propofition , and they will be very caly to conceaue. The 23.T heoreme: T he 28.Propofition. Ifa parallelipipeds be cutte by a plaine fuperficies drawne by the diagonal lines of thofe playne fuperficieces which are oppofite : that folide is by this playne fuperficies cutte into two equall partes. | V ppofe that the parallelipipedon % A B be cutte by the playne [uper- (| ficies C D EF drawne by the dia- LaNEZOK | gonal lines of y plaine [uperficieces which are oppoftte,namely, of the [uperficieces C Fand DE. Then 1 fay that the parallelipi- | pedon A Bis cutte into two equall partes by the i {uperfictes C DE F.For forafmuch as ( by the 34.0f the firft ) the triangle C G F is equall to _ the triangle C BF ,and the triancle A DE to a the triangle DEH, and the parallelograme . C Azsequall to the parallelogramme BE, for they are oppofite,and the parallelogramme G E is alf{o equall to the parallelogramme C Hand the parallelogramme C E,is the common fection by [uppofition: Wherfore the prime contai- ned under the two triangles C GF ,andD AE, and under the three parallelocrammes GE,A Cyand C Eus ( by the 8.definition of the eleuenth ) equall tothe pri{me contayned under the two triangles C B and D EH and under the three parallelogrammes CH, BE,and C E.For they are cotayned under playne [uperficieces equall both in multitude and in magnitude .Wherefore the whole parallelipipedon A Bis cutte into two equall partes by she playne [uperficies CD EF which was required to be demonftrated. | Demonfiras On, yd. A dia- The eleuenth Booke. A diagonall line ts aright line which in angular figures is dravene from: one angle and extended to his contrary angle as you'fée in the figure AB. Defcribe for the better fight of this demonftration a figure ypon pafted paper like vnto that which you defcribed for the 24. propofition onely altering the letters: namely,in fteade of the letter A put the letter F,and in fteade.of the letter H the letter C : moreouer in fteade of theletterC puttheletrerH , and the letter E for the letter D,and the letter A for the letter E,and finally put the letter D forthe letter F. And your figure thus ordered compare it with the de- monttratio, only imagining a fuperficies to paffe through the body by the dia- gonall lines C FandDE, q Ihe 24. I’ beoreme. L he 29. Propofition. « Parallelipipedons confifting ‘ypon one and the felfe fame bafe , and vn- 5 Tooke ceils der one and the felfe fame altitude swhofe* Jtanding lines are in the felfe end of the de- Jame right lines are equall the one to the other. won {tration whatisn= = (FAV ppofe that that thefe parallelipipedens CM and CN doo comfist upon one derStanded by \(\C and the felfe fame bafe, namely AB , and let them be under one and the felfe fading lines. | an DS" fame altitude, whofe flanding lines, that is,the fower fides of eche folide which ihe ehi7a fallupon the bafe,asthe lines AF,C D,BH,LM of the folideC M, and the lines CE,BK,A Gyand LN os | of the folide © N, let bein the D at felfefame right lines or paral- Inhn Dee his lel linesE NJ DK. Then I fay aii, that the folide C M is equall to ae fr By this figure the folide CN. For forafinuch st appeareth as either of thefe fuperficieces why fucb Prif’ CBDH,C BEKisa paralle- septs 2 a- logramme,therefore( by the 34. fed wedgessof of the firft ) the line C Bis e- the very fhape % of a wedoe,as quallto either of thefe lines D : : | | ssthefolide Land Kk. Wherefore alfothe : 7 / DEFGA- line DH 4s equall to the line | CGC EK .Takeaway EH whichis - A common tothem both , where- | ; | Sore the refidue namely D E is equall to the refidue HK. Wherforealfo the triangle D CB is equall to the triangle AK B. And the parallelogramme D Gis equall to the parallelo- gramme V1 N.And by the fame reafon the triangle A G F is equall to the triangle MLN, and the parallelogramme C F is equall to the parallelogramme B M.But the parallelograme C Gis equall to the parallelogrammeBN , by the 24.0f the tenth for they are oppofite the one to the other. Wherefore the prifme contayned under the two triangles¥ AGand DCE and under the three parrallelogrames AD,D G, andC Gis equall to the pri{me cotayned under the te triangles M LN and HBK , and under the three arallelogrames, that is, BM,N Hand BN.Put that folide common tothem both, whofe bef isthe parallelograme t A.B, and the oppofite fide unto the bafe is the fuperficies GEH M. Wherefore the whole pa- rallelipipedon CM is equall to the whole parallelipipedon C N:Wherfore parallelipipedons confifting upon one and the felfe fame bafe,and under one and the felfe fame altit ude,whofe panding lines arein the felfe fame right lines, are equall the one to the other: which mas re- quired to be demonstrated. Although A ee i ae = a or T= ie | ae . 2" i a a? a SS sisal of Euclides Elementes. 341, Although this demonftration be not hard to a good imaginati- On to conceaue by the former fi- gure (which yet by af, Dees refor- ming is much better then the figure ofthis propofiion commonly def- cribed in othercopyes both greake aad lattin) :yetfor the eafe of thofe which are young beginners in thys matter of folides , I haue here fetan other figure whofe forme if it be defcribed ypor patted paper, with the like letters to euery line as they be here put,and then if ye finely cut not thorough but as it were halfe Way the three lines t a,w m G F,and KHED, & fo foldeit accordingly, 8& compare it with the demontitratid, it will geue great light thereunto. Stading lines are called thofe fower right lines of euery parallelipipedon which 1oyne together the angles of the vpper and nether bafes of the fame body. Which according to the diuerfitie of the angles of the (olides , may either be perpendicular vpon the bafe,or fall obliquely . And forafmuch as in thys propofition andin the next propofition following,the folides compared together are fuppofed to hae one and the felft fame bafe , itis manifeft that the ftanding lines are in refpeét of the lower bafe in the felfe fame pirallel lines, namely,in the two fides ofthe lower bafe . But becanfe there are put two folides vpon one and the felfe fame bafe,and ynder one and the felfe fame altitude,the two vpper bafes of the folides may be diuerfly placed. For forafmuch as they are equall and like(by the 24.0f this booke) either they maybe placed betwene the felfe fame parallel lines: and thé the ftanding lines are in either folide fayd to bein the felfe fame parallel lines,or right lines: namely,when the two fides of eche of the vpper bafes are contayned in the felfe fame parallel lines : but contrariwife if thofe two fides of the Vp- per bafes be notcontayned in the felfe fame parallel orright lines,neither fhal the {tanding lines which are 1oyned to thofe fides be fayd to be in the felfe fame parallel or right lines. And therefore the ftan- ding lines are fayd to be in the felfe fame right lines,when the fides of the vpper bafes , at the leait two of the fides are contayned in the felfefamericht lines: which thing is required in the {uppofition of this 29,propofition.Butthe ftanding lines are fayd not to be in the felfe fame right lines , when none of the two fides of the rpper bafes are contayned in the felfe fame right lines, which thing the next propofiti- on following fuppofeth. gq The 25. T heoreme. The 30. Propofition. | Parallelipipedons confifting bpon one and the felfe fame bafe, and Ynder the felfe fame altitude, whofe Standing lines are not in the felfe Jame right lines, are equall the one to the other. Gi V- po that thefe l arallelipipedons om and cw , 4o confi/t upon one and the felfe Rs | [ame bafe, namely, » », and under oneand the felfe fame altitude, whofe fanding “ec2 i lines namely, the lines F.C D,B Hand i M, of the Parallelipipedon cm, and the ftanding lines a G,c 8,3 x and 1 N, Of the Parallelipipedon c x, let not bein the felfe fame right lines .T hen I fay, that the Parallelipipedon c mis equall to the Parallelipipedon cx. Forafrauch as the upper fuperficieces ¢ u fs G x ,0f the former Parallelipipedons, are in one ana the felfe fame [uperficies ( by rea{on they are {uppofed to be of one and the felfe fame alti. tude): Extend the lines + p andw H, till they concurre with the lines x c and k + ([uffi- ciently both waies extended : Sor in diners cafes their comcurfeis diuers). Let ¥ » extended, mecte with » G, and cut it in the point x: and with x x inthe point p . Let lkewife m x extended, meetewith » ( {uffrctertly produced) in the point 0, andwith x & in the point R « And drawe theferight lines x x,1 05 2, ands x. Now (by the 29.0f the elewenth the folide c ws, whoje bafeis the parallelogramme » c 2% Land the parallelogramme oppofite un- Ty to Stading lines, Confirutlione T he eleuenth Booke _ tosis FD Hu, sequal to the folide c 0, whofe bafe is acBL and the oppofite fide the ae pe parallelogranoe xP Ro, for they conjifte ‘upon one and the felfe fame bafe, namely, vpor the parallelogvime 4 c » 1, whofe flanding lines s ¥,A X,LM, LO, CD ,CP,B HANd8 R, K BR L.Dees figure. are in the fele {ame right lines = p and wx. But the folide c 0, whofe bafe is the paralle- logramme Ac 8 1.,and the oppofite fuperficies voto it is x PR 0, is equall to the folide cn, whofe bafe is'he parallelogramme a c 2 1, and the oppofite [uperficies unto tt is the {uperfi- cies GB K N.for they are vpon one and the felfe fame bafe, namely, ac BL, and their flan- ding lines 4.344%, C By CP, LN, L 0,8 K, and B R, are in the felfe fame right lines \ X, and v x . Woerefore alfo the folide c ,isequall to the folide c x . Wherefore Parallelipt- pedons confifing upon one and the felfe fame bafe,and under the felfe fame altitude, whofe Standing line are not in the felfe {ame right lines, are equall the one tothe other : which was requiredto be proned. @ This demonftra- tion is fomwihnt har- der then the former to conceaue b' the fi- gure defcribec in the plaine(and yeibefore M. Dee inuened that figure which i:placed for it, it wa: much harder)by reaon one folide is contaned in an other . Anc there- fore for the clerer light therof, D 4 cone “S oy thefe rosa a 20) AEand mie! C F, being under one & the fel fame altitude, The 1 fay, that the folide ‘AB. és equall to the folide C F . Firit let thé ftan- ding lines, namely,H K,B E,A G,LM,0 P, D F,CX, andRS, be erected perpendicular- ly tothe bafes AB and C D, and let the an- gle ALB not be equall tothe angleC R D. Extend the lineC R tothe point T . And ( by the 23.0f the firft ) upon the right line RIT , and at the point init R, deferibe vn- tA * y al 4\.iag * | e . * . > ~< 4 : — —% raye ak : : . a ~ ‘ \ 4 “ » ABS VGQ% 9 " .% * ,.itn =e RA ‘ ok he > i, : : ~ “ts . 3 > PSA ~ 5 : . : ‘ee we, ~CeR | = 5 4362? : * : an | tO ger cars ar. ° 2 ww Sth etd :s ard €, Fol. 342. a Two tafes in this prepofi~ £1097, The fit cafe, Cc onfirntiion. We ure bebol- ding to M.Dee {or inuenting thas figure, with other, which sid bis reforming were as much mifbappen as this was,and fe both inthe Greeke and Lae time copses re~ MIdIRE , j v i? ar Ay “A y a . 4 4 a " J ' , ee | ie % A A & am. f " } ' é » i aA { . a q Fy rh q is od d ’ 4h , i be, ; a if oi ed a ap) tales ak! *)) ‘he F > 5 ot , ty a) - " : vw ¥IN % ae By Ln ony, . ) dd | ks } ae \ Bi ‘i Pe i Y hy 1 ‘ >: ?'Re aA , \ e }) ef ma he my. 1y Me aE a nae Demon itra- C20M- Sadad sxe 3 Fi $3 + M92 Qe grit [PARA Fo . duvet {Pere ae , Fiat dvdatpmdte geaneatartsk Agerrtt ta VUISe ta ROCA AT As e\ aan eT | to the angle AL B an equallangleT RV. And ( by the third of the firs ) pusge line RT equail to the line LB, ana’ line RV equall to the ine AL. nf (by the 3 1,0f the firft) by the point uy 1aw unto the line RT a parallel tine V, W :and make perfecte the bafe RW, and! the folide EY Now foralmuch asthefe tia lines TR dnd RV are equal to thefeiwo lines B Lund L A, and they containe equa anglessther- forethe parallelogramme RW, is equall and like'to the patallelogramme AB . Againe, forafmuch asthe line L B is equall to\the line, RT, andthe line L DM tothe lineR ( for the lines Lana RS ave the altiqnde of the Paraflelipipedons A E apd C F fohich altitudes Are {uppofed to be equall ) dnd they ri 2 right angles by [uppofition , there- Yoré the parallelogramme RY is equall and Cid cle conlldogcamns 398, tnd by the fame reafon alfa the ee ee LG is equall and like to the parallelograme SV. Wherefore three parallelogrammes of D4 . w ont oa ee ae » oe Wor IOV SISGOO Oo o : : hae > ety eee 7 : et #& pee eo ee few OF AMOGOV NTE A TKO , 3 et (perk aie Oke Ge mai 91 a\s\s102 EMO IAIG| Shin’ y rth +4 * ‘ % . . > : ty ‘S at neer arhdt LEDS BEE shi 5 % > SHY ac\4 eos ek + 2 +2) a - a~ Ne . es q i) hy Z : \ > ee : : % * : f BW oa aed , % Vo eee y “4 7 ern : A echelabban he \ eked i £77 SG WSEAS 94 ne } : : "¢ Ss te mn BMIBA 4 ies oy ob ' + Pata’ SAAAgk < 4 . prec es us ¥ 7 ‘ “ee \ Ges ae - : . . | a a ws 2 s . & fe i ie JIANG) IWS TAWA, {% : SA kh BPM -* "Wa 5 *4 om : aMslieat Set ‘ En ws sHAIU i 3 v4 7 ~ vin .% «4 ~ 7 wes VG v4 ‘ x % os ‘ik ak j : . Y bam a | r a »¥ ’ ne ma tgp , A OU. OA EA pisses % +” PAIL MOAB AS , Py ae oS BAG S34 RAV —o eet ake «teed rs a a a ee eo of Euchites Elementes.. \ Fol.343. thefolide AE are equall and like to the three parallelocrammes of the folide ¥ P - Bue thefe three parallelogrammes are equall and like to the three i bpofite’ fides . Wherefore the whole Parallelipipedon CLE 1 equall and like to the who | Extend ( by the fecond petition’. thé lines DR and W VP, Ontill they concurre ; and les them concurrejn the point O . And ( by the 31. of the firft) by the point T drawe'vnte the line RQ a parallel line T a i And extend duely t é lines T a and DO wntill they RI . Now thefolide 2 Y, whofe * bafe isthe parallelogramtme RT and the oppofite fide voto the bafe the parallelogramme Db, is equall to the folide ¥ V, whofe balers the paral- concurre, and let them concurre ip the point % And mate pegtete Wee folides QY and . "Note now, how le Parallelipipedon TV. . & : & the bafe re/pecs tinely as taken > lelogramme RY, and the oppofite fide unto the bafe theparallelogramine ¥ Z:For they con- for fe may alte Jifte vpon one and the felfe [ame bafe;namely, RY; and areonder one andthe felfe [ame al- titude, and theirStanding lines, tamely, RQ ,RV,T a,TWiSN SSA: 1b, and TZ ; are in the felfe [ame right lines, namely, OW, and NZ? Rut the folie TV is proned e- quall to the folide AE . Wherefore alfo the Solide T 2 weaquatl to the folide AE . Now forafmuch as the parallelogramme RV WT is equall to the paralleloeramme “OT (by the 35.0f the firft) and the parallelogramme AB is equall to the parallelogramme RW: there- fore rhe parallelogramme QT 3s qual tothe parallelocramme CAB, and the parallelo- grarbene CD: is eqauall vo she payallelogramme AB | by fuppofition) . Wherefore the'paralle- _logramme CD is eywall to the parallelogramme OT . And there is a certdine other [uper- ficies, namely, DT . Wherefore (by the 7.0f the fi ft) as the bafe CD is tothe bafe DT, fo “asthe bale’ ‘OT tothe bife DT . And forafmich as the whole Parallelipipedon C1 is cut by the plauefuper ficies RE ,which is a parallel to either of the oppofite plaine {uperficieces ther- oforeiasthe baft CDistothe bafeD T, fois the folide CF’ to the folide RI (by thé 25.0f the ‘elehenth )*: tnt by the fame redfon' allo, forafmnch as the whole Parallelipipedon QI is cut by the plaine [uperficies RY whichis a parallel to either of the oppofite plaine [uperficieces, therefore as the bafe QT istothebale DT, foisthe folide QTY tothe folide RI . But as the bale C D is tothe bafe D r »foisthe bale QT tothebafe T D. Wherefore ( by the 11 .0f the fift) as the folide C F isto the folide R1, fe is the folide QTY to the folide R1 .Wherefore either of thefe folides C F and 9 Y ,haue tothe folide R I one and the on proportion.Wher- fore the folide C F is equallto the folide 2. But itis prowed that the folide RI is equall tothe folide AE. Wherefore alfo the folide C F is equal to the folide AE. « ; But now {uppofe that the flading lines, namely, AG, HK, BE,LM,CX,0P, DF, and RS, be not erected perpendicu- larly tothe bafes AB and CD. Then alfo Ifay, that the folide A E is equall tothe folide C F. Draw (by thett.of theeleuéth) ~ unto the cround plaine [uperfi-. .. cieces A 2 and a ane . pointes Ky E,G,M,P, F,X,S, | thefe perpendicular lines K N, L ; / ET,GV,MZ,PW,FT,X Q 3 - IN / : and SI. And draw thefe right : if af \/ _ linesNT,NV,ZV,Z7,WI,. + shat mbschrbath before bere pro one re wed on this: 32: Propofition:} the’. BSS eter , Sie BIN (2sbil cLosi2 33h f DUVETS) aOSIIAs 230! YY tty. [olde S OnE PS | “Ss ratio o if refpects alter the name of the Lowndes eyther of folides or playnes, Setoud ¢4fes | } i il . , ip » J h ~~ ‘4 h . i eit j J tend wa; “A : a.) “a ; mt re | Ny id ] » yi ie H Lee et Se & There you erceaue haw the bafe ts d= uerfly conf- dered & tht- en: a8 before (reubaerfel Jee “ “them in maner following. T heeleuenthBooke folide K Z is equall to the folide Pir s P I, for they confift upon equail | * bafes; namely, K M,andP S, and are under one and the felfe fame altitude , wh sf fanding ofe ines alfo are erected perpendicn-_ larly tothe bafes . But the folide KZ is equall to thefolide AE (by the 29.0f the eleuenth):ana the folide P I.is (by the fame) €- O quall to the folide C F, for they confift vppon one and. the felfe . eit fame bale, and are coke hen, ry gata cy the felfe [ame altitude,whofe aT Mc tee phate ts are uponthefelfe C | QO ate fame right lines .Wi atl f | Gee es the folide A E is equallto the folideC F. Wherefore Parallelipipedons confifting upon equal bales and being under one and the felfe fame altitude,are equall the one to the other : which sas requireato be densonftrated. eat | : The demonftration of the firft cafe of this propofition is very hard to conceaue by the figure def- eribed forit.in a playne.And yet before M. Dee inuented that figure which we haue there placed forit, -it was much harder.For both in the Greke and Lattin Ewe/sde, itis very ill made,and itis in.a maner im- sages to conceaue by it the conftrudtion. and. demonitration thereto appertayning. Wherefore I haue gre defcribed other figures, which firft defcribe vpon paited paper , or {uch like matter and then order As touching the folide A E in the firft cafe , I neede not comake any new defcription . For itis playne inough to conceaue as it is there drawne. Although you may for your more eafe of imagination defcribe of pafted paper a parallelipipedon hauing his fides equall with the fides of the parallelipipedom AE before defcribed,and hauing alfo the fixe parallelogrammes thereof (contayned ynder thofe fides) equiangie ih pe of Euclides Elementes. 34.4.0 equiangle with the fixe parallelogrammes of that fioure,ech fide and eche angle equall to his correfpon dent fide,and to his correfpondent angle. But concerning the other folide. When ye haue defcribed thefe three figures vpon pafted pa- per: Where note for the proportion of eche line, to make your figure of palted paper eqnall with the fi- gure before defcribed vpon the playne,let your lines 0 p,c x, x s,p F, &c. namely,the reft of the ftan- ding lines, of thefe figures, be equall to the ftanding lines 0 p,c x, s,p F, &c. of that figure . Likewife let the lines o c,c RR D,D 0, &c.namely,the fides which cétayne the bafes of thefe figures be equal to the lines o c,c R,R DD 0,&c, namely,to the fids which cétayne the bafes of that figure. Moreouer let f the lines p x,x s,s F,F p,8c.namely,the reft of the lines which cétaine the ypper fuperficieces of thefe figures,be equal to the lines p x,x s,s F,F P,8¢c.namely,to the reft of the lines which cétaine the vpper fuperficieces of that figure(to haue defcribed all thofe forefaid lines of thefe figures equal to all the lines of that figure,would haue required much more {pace then here can be — I haue made them equall —- ; onely to the halues of thofe lines, but by the example of thefe ye may, if ye will defcribe the like figures | hauing their lines equall to the whole lines of the figure in the playne,eche line to his cortefpondent line). When I fay ye haueas before is taught defcribed thefe three figures, cut finely the lines x'c, s R; F p of the firft hgure,and the lines s x,y r,and 1» of the fecond figure: likewife thelines s x, Q,Z VY, and y r,of the third figure , and fold thefe figures accordingly , which ye can notchufe but dooif ye marke well the letters of euery line. i , The three former figures being after this fort defcribed,fet them ypon this figure here defcribed ypon a playne,as vpé their bafes,namely,the lines O.C,C R,RD,D O:R T,T h, «D;, DR: VR,RT, T W,W a, V Q, and Q R of thefe three figures vpon the lines correfpondent vnto them in this figure. And they fo ftanding compare them with the conftruction and demon ftration of the firft cafe, 0 and they will geue ee vito it. Thisalfo . ‘h ye muft note,that if ye make the lines of the fore | fayd three figures equall to the lines of the figure Conftrutlions of the plaine defcribed before in the deméftrati- ¢ on of the firft cafe: then muft ye make a new bafe for them like vnto this, which is eafy to doo , if DemuiSira- ye draw a pallelogramme equall and like to the tion. parallelogramme O C T +h, and thé cut of from zg. Vv Ww the fame a parallelogramme DRT equall . and in like fort fituate to the parallelogramme DRT of of that figure: & vpon the line RT defcribe two parallelogrammes, the one equall like , and in like fort firuate to the parallelogramme R T Q a of thatfigure, and the other equall, like and in like fort fituate to the parallelogramme R T.V W ofthe fame figure . The lines of this bafe which I haue here put are equall onely to the halues of the lines of that figure in the demonftration. : As touching the fecond cafe ye neede no new figures , for itis playne to fee by the figures ( now reformed by 4.Dee) defcribed for itimthe playne , efpecidlly if ye remember the forme ofthe figure of the 29.propofition of this booke, Only that which there ye coficeaue to be the bafe,imagine here in both the figures of this fecond cafe to be the vpper fuperficies oppofite to the bafe, and that which was there fuppofed to be the vpper fuperficies conceaue here to be the bafe. Ye may defcribe them vpon pafted paper for your better fight, taking hede ye note the letters rightly according as the conftruétion reguireth. ) | ; - TERS ; i, ~ _ Fluffasdemonttrateth this propofition an other way taking onely the bafes of the folides,and that after this maner. } Take equall bafes(which , yetfor the furer vnderitan- ding let be vtterly vnlike) namely, AEB FandADC Hand let one of the fides of eche concurre in one & the fame right line AE D,& the bafes being vpon one and the felfe fame playne let ee there be fuppofed to be fet ) vpon thé parallelipipedons , yonder one & the {elfe fame | altitude. Then I fay that the folide fet vp6 the bafe AB is equal to the folide fer ypon the bafe AH. By the poyne E-draw ynto the liane ACa SESE A A parallel T he elenenth Booke parallel line EG, which if ie fall without the bafe AB, producethe right line HC to the poynt 1. Now foraf- muchas AB and AH are pa- rallelogrmaes , therefore by the 24. of this booke , the triangles ACT and EGL 4, fhall be equaliter the one to the other: and by the 4.0f the firtt, they thal be equian- gle and equall: and by the firit definition of the fixth, and fourth Propofition of the fame, they fhall be like, : Wherfore Prifmes ereted 2 £ A vppon thofe triangles and vnder the fame altitude that the folides AB and A Have, fhall be equal] and like, by the 8. definition of this booke. For they are contayned vnder like playne fuperficieces equall both in multitude and ma nitude.Adde the folide fer vpon the bafe A.C LEcommon to them both. Wherefore the folide fet vp- pon the bafe AE G C;is equall to the folide fer vpon the bafe A EL I.And forafmuch as the {uperficie~ ces AEB F,and AD H C are equall (by {uppofition): and the part taken away A Gis equall to the part taken away AL : therefore the refidue B I fhall be equall to the refidue G D . Wherefore as A Gis to G DasAListoBI (namely, equalls to equalls), But as A G is to G D, fo is the folide fet vponA Gto the folide fet vpon GD by the 25, of this booke, for itis cut by aplayne fuperficies fet ypon the line G E,which fuperficies is parallel to the oppofite fuperficieces. Wherefore as ALis to BI » fois the folide fet vpon A L to the folide fet vpon B L. Wherefore ( by the r1.of the fifth) as the folide fet vpon AG ( orvpon AL whichis equall vnto it jis to the folide fer vpon G D,fo is the fame folide fet vpon A GorAL to the folide fer vpon B I. Wherefore(by the z -part of the 9.of the fifth) the folides fet vpon G D and BI fhall be equall . Vnto which folides if ye adde equall folides, namely , the folide fer vpon A G to the folide fet vpon G D,and the folide fet vpon A Lto the folide fer vpon B I: the whole folides fet vpon the bafe A 'H and vpon the bafe.A B thall be equall. Wherefore Parallelipedons confifting vpon equall bafes and being vnder one and the felfe fame altitude are equall the one to the other: which was required to be proued. q Uhe 27. T heoreme. Ihe 32. Propofition. Parallelipipedons being nder one and the felfe fame altitude, are in that proportion the one to the other that their bafes are. = iV ppofe that thefe parallelipipedons A B and C D be under one cy the felfe fame jy | altitude.T hen I fay that thofe a kc (AB andC D are in that pros | portion the one to the other that their afes are,that is,that as the bale A Eisto _ aed the bafeC F fois the parallelipipedin A B to the parallelipipedon C D.V pon the Confiructions Line FG defcribe (by the 4s. 3 of the first) the parallelo- gramme F H equallto the 2 parallelogramme A E and e- x : quiangle with the parallelo- grammeC F, Andvponthe | © ‘ a 7 | ; 4 D | , : t " i is a} : \ } 3 : | - ig : 4 y : : $ ce : i | i] | et : é : Wea t } : , a 7 i } by r ) Dy ’ ig ll ye f i! s ; iy $ = , | j i Ze ij U ne) | oe ; . ‘ie 4 ' 1} ‘ \ | at { (i 4 | a ! | D K B => = = = z = rier = =~: =: niu — “ wae: i es race: “4 . > -" 4 ‘y oe a OA A OR I A aa ee ee Su ——— So eo bafe F H defcribe a parallels pipedo of the felfe [ame alti- F tude that the parallelipipedo C Disc let y fame be G K. C = Ze, Demonftre. Now (by the zr. of the ele- | fel tion. wenth) the parallelipipedon A B is equall to the parallelipipedon G K.,for they confist upon e- gnall of Euclides Elementes. Foliyas. gutall bafes,namely,A E and F H,and are under one and the felfe fame altitude. And ‘foraf- much as the parallelipipedon G-K is cut by a plaine {uperficies D G; being parallel to either of the oppofite plaine [uperficieces, therfore (by the 25. of the eleuenth) as thebafe il F is tothe bafe F C,fois the parallelipipedon G K to parallelipipedon C.D: but the bafe H Bis equal to the bafe-A E, and the parallelipipedon G K 1s proued equall to the parallelipipedon A B. Wherfore as the bale AxE is to the bale C F,fo is the parallelipedon A B, to the parallelipipe.. don CD .Wherfore parallelipipedons being under one and the felfe (ammgaltitudeyare in that proportion the one to the other that their bafes are: which was required to be demonftrated. Ineede not to put any other figure for the declaration of this demonftration, forit is eafie to fee by the figure there defcribed. Howbeit ye may for the more full fight therof,defcribe folides of pafted paper according to the conitruction there fet forth, which will not be hard for you to do, if ye remem-~ ber the defcriptions of fuch bodies before taught, A Corollary added by Flufias. Equall parallelipipedons cotained under one and the felfe fame altitude ,haue alfo their bafes equal, For if the bafes fhould b e vnequall,the parallelipipedons alfo fhould be vnequal by this 32 propofitié. end equall parallelipipedons hauing equall bafes, bane alfo one and the felfe fame altitude. For if. they fhould hane a greater altitude, they fhould exceede the equall parallelipipedons which haue the felfe fame altitude: But if they fhould haue a leffe they fhould want fo much of thofe felfe fame equal parallelipipedons. T he 28, I heoreme. Ihe 33.Propofition. Like parallelipipedons are in treble proportion the one to the other of that in which their fides of like proportion are. R= V ppofe that thefe parallelipipedons AB and C D be like, & let the fides A E and S( EX a be fides of leke proportion. Then I fay, the parallelipipedon A Bis unto the NS ‘A parallelipipedon C D in treble proportion of that in which the fide-A E is to the fide C F. Extend the right lines AE,G E and H E to the pointes K,L,M. And (by the 2. of the firft ) unto the es line C F put the line E K equal, SESS Ne and untotheline F N put the line EL equall, and moreouer H vuto the line F R put the line E c M equall, and make perfect the parallelogramme K L, and the parallelipipedon K O. Now for- SEES PS x : : ‘= é EN. K afmuch as thefe two lines EK and EL are equall to thefe two linesC F and F N, but the angle KEL is equall to the angle CF N ( for the angle AEG is equall to the angle CF M by reafon that the folides A Band CD are like) .Wher- fore the parallelogramme K L is equall and ltke to the pa- rallelogramme CN, and by the fame reafon alfo the parallelogramme K M is equall and like oO Confiruction, Demonfirae ton. : en —— - — a Tbe elenenth Booke like to the parallelogramme C- R, and moreower the parallelo- gramme O E to the parallelo- gramme F D. Wherefore three parallelogrammes of the paralle- Be gies B ee a ee Ls! ee , lipipedon KO are like and equall to three parallelogrammes of the parallelipipedon CD: but thofe , bh three fides are equalland like to the three oppofite fides: wherfore A E ; ‘ uy x the whole parallelipipedon K O is equal and like to the whole pa- | rallelipipedon C D.Make perfect L the parallelogramme GK. And vpon the bales GR and KL make perfect tothe altitude of the parallelipipedon A B,the pa- vallelipipedons EX & L P.And foraf{much as by reafon that the ea parallelipipedons A Bc C D are like, as the line A E 1s to the line C F, fo is the line E G, te the line F N, and the line E H to the line F R. But the line C F is equall to the line E K and the line F N to the tine E L, and the line F R tothe line E M, therefore asthe line A Eis ta the line E K fois the line G E tothe line E L, and the line H E tothe line E M (by construc- tion). But.as the line A E isto the line E K, [ois the parallelogramme AG to the * mea gramme G K (by the firft of the fixt). And as the line G E is to the line E L, fo is the paralle- logramme G K to the parallelograme K L. And moreouer as the line H Eis tothelineE M, fois the parallelogramme P E to the parallelogramme K M. Wherefore (by the 11. of the fift) as the parallelogramme AG is to the parallelogramme G K, fo is the parallelogramme G K tothe parallelogramme K L, and the parallelograme P E to the parallelogramme K M. But as the parallelogramme AG is tothe parallelogramme GK, fois the parallelipipedon A B to the parallellpipedon EX, by the former propofition, and as the parallelogramme GK is to the parallelogramme K L, fo by the fame is the parallelipipedon X E to the parallelipipe- don P L:and agayne by the fame, as the parallelogramme P E is to the parallelogramme K- M, fois the parallelipipedon P L to the parallelipipedon K O.Wherfore as the parallelipipedo A Bis to the parallelipipedon E X, (0 is the parallelipipedon E X tothe paralleipipedon P L, and the parallelipipedon P L to the parallelipipedon K O. But if there be fower magnitudes in*continuall proportion, the firft fhalbe unto the fourth in treble proportion that it is tothe fecond (by the 10. definition of the fift). Wherefore the parallelipipedon A B is unto thepa- rallelipedon K O intreble proportion that the parallelipipedon A B is to the parallelipipedon E X.But as the parallelipipedon A B is to the parallelipipedo E X, fo is the parallelogramme AG tothe parallelogramme G K, and the richt line AE to the right line EK. Wherefore alfo the parallelipipedon A B is tothe parallelipipedon K O in treble proportio of that which the line A E hath to the line EK. But the parallelipipedon K 0 is equall to the parallelipi- pedon C D, and the right line E K to the right line C F. Wherefore the paratlelipipedon A B is to the parallelipipedon C D in treble proportion that the fide of like proportion,namely, A- E isto the fide of like proportion,namely, to C F. Wherefore like parallelipipedons are in tre- ble proportion the one to the other of that in which their fides of like proportion are: which was requirca to be demonftrated. - Beca - ee —— ae ——-— rer eer ) of Euclides Elementes.. Fol.346. q Corellary. en hz ofome Hereby it is manifeft that if there be fower right linesin *continuall proa portion, as the firft is to the fourth, fo fhall the Parallelpipedon defcribed of the firft line be to the Parallelipipedon defcribed of the fecond, both the Parallelipipedons being like and in like fort defcribed, ... Fot the firkt line is to the fourth in treble proportion thatit is to the fecond : and it hath before béne!: ai, bacied: proued thatthe Parallelipipedon defcribed of the firft, isto the Parallelipipedon + [Zil to neal: defcribed of the fecond, in the fame proportionthat the firft line is to the fourth, “nr Becaufe the one of the figures before, eta defcribed in a plaine, pertayning to the Gao * | demonftration of this 33. Propofition,is | not altogether fo eafie to a younge begin-~ ner to conceaue, I haue here for the fame defcribed an other figure, which if ye firft drawe. vpon pafted paper , and. afterward cut the lines & folde the fides accordingly, will agree with the conftruction & demon- {tration of the fayd Propofition. Howbeit this ye muit note that ye muit cut the lines OQ & MR onthe contrary fide to that = which ye cut the other lines. For the fo. | = lides which haue to their bafe the paralle- : logramme-L K are fet on vpwatd-and the other downward : Ye may if ye thinke good defcribe after the fame manet of patted pa- per a body equall to the folide'C D:though- that be eafie inough:to. conceaue by the fi- gure thereof defcribed in the plaine. ae oe ee ee : q Certaine moft profitable Corollaries, Annotations , T heos remes, and Problemes, with other practifes, Logisticall, and «Mechanical, partly upon this 33.and partly vpon the 34. 36. and other following, added by easter Iohn Dee, €| 4 Corollary. 1. 1. Hereby itis manifest, that tivo right lines may be found, which Shall bane that proportion,the one to the other, that any two like Parallelipi- pedons, and in like fort defcribed, ceuen,haue the one to the others Suppofe Q and Xto betwolike Parallelipipedons, and in like fort defcribed . OF Q take any of the three lines, of which itis produced : as nam ely,R G. Of X, take that rightline of his production, waich line is aunfwerable to R G in proportion (which moft aptly, after the Greke name, may be AAa.]. called ae ietengnhetian amine es iearas aS x — « t oe Se 2 é . ¥ > —_ -—-* — ee = ; - " : al ae -— ———— ——— = > ; ny = At — —— Fe ee — Sarees or == . = —— ss - = > - - ; r . 2 7 —— : = - s-ves t — Shire etinhaninmaian: trrapetiiliain tna ama enttcnte oo - as om - = eae nm — — A , me ——- 7 te a ne SO — atime 8 : —- - nas - Ee - = . . 7 eee -~ eee : if P| eS AE _ i ee, mb | * Note this famous Lem- wide The doubling af the Cube, * Note. * All their BLES 0 fexeth= zine the Lem- wea are i Euto- tius comment a- rses O06 Archie medes booke de Sphera 5 Cy- lindro: where Archimedes G- feth the fame Lemma. called Omologall to R G)&& let that be T V. By the 11.0f che figechj to-RG and TV; let the third line n propor tion with them be founde, and let tha: be Y.. By, the fame 11. ofthe fixth,to TV. and Y, let che thirde right line be foiid,in the fayd pro portion of TV to ¥: & let that be Z.I fay now thatRG hath that proportion to Z, which Q hathto X. For by ) conftruction, we haue fower right lines in continual! pro- R. _ piotion,namely,R G,T VY, , and Z. Wherfore by Euchdes Corollary, here before, RG is to Zjas Q tsto X.Where- V fore we hauc foiid two right lines hauing that proportion the one to the other,which any two like Parallelipipedons of like defcrip- tion,geuen, haue the one to the other : which was requiredto be done. qu Corollary. 23 es a Conuer{e, of my former Corollary, doth it followe : To finde two like Parallelipipedons of like defcription, Which fhail haue that proportion the one tothe ether ,that any two right lnes.genen, bate the one ta the other. | Suppofe the two right lines geen to be A and B : Imagine of feure right lines in continuall pro- portion,A,to be the firit, and Bto be the fourth : (or contrariwife, B to be firft, & Ato be the fourth). The fecond and third are to be found, which may, betwene A & B, be two meanes in continuall pro- portion: asnow,* fuppofe fuch two lines,found : andJet them be Cand D. Wherefore by Ewelsdes Corollary, as Ais to B (if A were taken as firft ) fo fhall the Parallelipipedon defcribed of A, be to the like Parallelipipedonand in like fort defcribed of C : being the fecond of the fower fines in continual proportion : it is to wete,A,C,D,and B . Or, if B fhall be taken as firft, (and that thus they are orderly in continuall proportion, B,D,C,A, ) then, by the fayd Corollary, as B is to A, fo fhalt the Parallelip1- pedon defcribed of B, be vnto the like Parallelipipedon and in like fort defcritbed of D . And ynto a Pa- rallelipipedon of A orB, at pleafure defcribed, may an other of C or D be made like, and in like fort fi- raced or defcribed, by the z7.0f this elenenth booke. Wherefore any tworight lines being geut,&ez which was required to be done. : Thus haue Tmoft briefly brought to your vnderftanding, if (firft ) B were double to A, then what Parallelipipedon foeuer, were defcribed of A,the like Parallelipipedon and in like fort defcribed of C, fhall be double to the Parallelipipedon defcribed of A. And folikewife (fecondly) ifA were double to B,the Parallelipipedon of D, fhoulde be double to the like,of B defcribed, both being like fituated. Wherefore if of A or B,were Cubes made, the Cubes‘of C and D ateproucd double to them + as that of C, to the Cube of A: and the Cube of D to the Cube of B : inthe fecond cafe. * And fo of any proportion els betwene A and B. Now alfo do you moft clerely perceaue the Mathematical occafion, whereby (firft of all men ) Hippocrates, to double any Cube geuen was led to the former Lemma: Betwene any two right lines genen,to finde two other right tines which fhall be with thetwo firft lines, in conts- nuall proportion: After whofe time (many yeares) diuine P lato,Heron,P hilo, A ppoltenius, Diocles,Pappus,Sporus, AL enechmus,eArchytas T arentinus (who made the wodden doueto flye ) Eratofthencs,N icomedes,with many other (to their immortall fame and renowme} publifhed, * divers their witty deuifes, methods, and-engines ( which yet are extant) whereby toexecute thys Probiematicall Lemma.But notwithftanding all the trauailes of the forefayd Philofophers and Mathematiciens, yea and all others doinges and con- triuinges(vnto this day) about the fayd Lemma,yet there remaineth fufficient matter, Mathematically foro demonftrate the Janse that moft exattly & readily ,it may alfo be Mechanically practifed: that whofoewerthallachiene that feate,{hall not be counted. fecond oo = y medes, ee atte’ F : — of Euchides Elementes. 34.9, medes, but rather 2 pereles Mathematicien,and eMathematicorum Princeps.I will fundry wayes (in my briefeadditions and annotations vpon Euchde) excite you thereto, yea and bring before your eyes fundry new wayes,by mcinuented : and in this booke fo placed,as matter thereof,to my inuentions appertayning,may geue occafion:Leauing the farther, full, & abfolute my concluding of the Lemma, to an other place and time: which will,now, more cépendioufly be done : fo great a part therof, being before hand in thys booke publifhed. ¢ A Corollary added by Flufas. Parallelipipedows confifting vpon equall bafes,are in proportion the one to the other as their alti- tudesare. For ifthofe altitudes be cut by a plaine fuperficies parallel to the bafes : the fections fhall be in proportion the one to the other as the fections of the bafes cut, by the 25.0f this booke . Which feétions of the bafes are the one to the other in that proportion that their fides or the altitudes of the folides are ,by the firft of the fixt . Wherefore the folides are the one to the other as their altitudes are. But if the bales be vnlike, the felfe fame thing may be proued by the Corollary of the as.of this booke, which by the 25.Propofition was proued in like bafes. q The 29. T heoreme. The 34. Propofition. In equall Parallelipipedons the bafes are reciprokall to their altitudes. And Parallelipipedons whofe bales are reciprokall to their altitudes, are equall the one to the other. ye es KV ppofe that thefe Parallelipipedons ABE CD be equall the one to the other Then “ a5 7 ay; that the bafes of the Parallelipipedons AB and CD are reciprokall to their : & altitudes, that is, asthe bafe E His tothe bafe N P, fois the altitude of the folide xe C D to the altitude of the folide AB. Firft let the flanding lines AG, E F,L B,H K, of the folide A Bc the flading lines CM; NX, 0 D;and P R, of the folide C D be wea 0 dicularly to the bafes E H c> NP. The I fay,that as the bafe E H is to the bafe N P,fois the line C M tothe line A G.Now if ne - the bafe E HH be equal to the bafe — N P and the folide A Bis equall to the folide C D, wherefore the line CM is equall to the line AG* . Forif the bafes E H and N P being equall, the altitudes AG andC M be not equall, net- ther alfo fhall the folide AB be eguall to the folide CD, but they are [uppofed to be equall. Wherefore the altitude C M is not unequall to the altitude AG .Wherefore itis equall. And therefore as the bafe E His tothe bafe P N, [ois the altitude C M to the altitude A G.Wher- fore it is manife/t, that the bafes of the Parallelipipedons AB and C D are reciprokallto their altitudes . But now fuppofe that the bafe E H be not equall to the bafe N P But let the bafe E H be the greater. Now the folide A B is equall to the folideC D . Wherefore alfothe altitudeC M is greater then the altitude AG * . For if not, then againe are not the ites ABandCD equal : but they are by [uppofition equall . Wherefore (by the 2.0f the Sirf ) put unto the line AG an equallline CT . And vpon the bafe N P and the altitude being CT, make perfecte a folide cotained under parallél plaine {uperficieces and let the fame beC Z. And _— AAY. asthe Note whet is jet lacking ree guifite to the doubling of the b Cube, Twotafes in the firft part ofthis prope- tz0n. First cafe, which alfa may betwo Wayes. Firft ways * This follow eth alfo of the Corollary ad- ded of Finfvas after the 32. propofition of this booke. Second way. * This alfa followeth of | the former Corollary. Thecounsrfe of bath the partes of the ferft cafee J. The elenenth Booke asthe folide ABis equallto the ‘folide € D, and there isa certaine other folide,nameh,C Z, but unto one and the felfe fame magnitude equal magnitudes haue one and the felfe fame proportion (by the 7 .of the fift ) - Wherefore as the folide A Bis to the folide C Z, fois the fo- lide CD tothe folide CZ. But. | : asthe folide AB is tothe folide CZ, fo is the bale EH to the bale N P ¢ by the 32. of the ele- uenth ) - for the folides A Band CZ arevnder equall altitudes. And as the folide C Dis to the folide CZ, fois the bafe M P to the bafe P T and the line M GC to the line CT. Wherefore (bythe r1.of the fift ) asthe bafe E H is to the bafe N P, fois the line C Mtothe line CT . But the line CT is equall to the line AG. Wherefore (by the 7 .of the fift)as the bafe E H is to the bafe N P, fois the altitude CM tothe . altitude AG.Wherfore in thefe Parallelipipedons AB and C D'the bafes are reciprokall to their altitudes. Es But now againe {uppofethat the bafes of the Parallelipipedons AB andC D be recipro- ~ ww Av kall to their altitudes, that is, as the bafe E His tothebafe N P, fo let the altitude of the fo- lide C D be'to the altitude of the folide AB. Then I fay, that the folide AB is equall to the . folide CD . For againe let the ftanding lines be erected perpendicularly to their bafes. And now if the bafe E H be equall to the bafe N P : but as the bafe EH is to the bafe N P, fo isthe altitude of the folide. C.D tothe altitude of thefolide AB. Wherefore the al- titude of the folideC D is equall to the altitude of the folide AB. But Parallelipipedons con- filing vpopequall bales aud under one and the felfe [ame altitude , are ( by the 31.0f the eleuenth ).equallthe one tothe other . Wherefore the folide A B is equall to the folide CD. But now fuppofe that the bale E H be not equall to the bafe N P : but let thebafeEH be the greater . Wherefore alfothe altitude of the folide C D, that is, the lineC M is greater then the altitude of the [olide A B, that is, then the line AG. Put acaine (by the 3. of the frft ) theline CT equallto the line AG, and make perfecte the folide CZ. Now for that As the bafe E Histothe bafe N P, fois the line MC tothe line AG. But theline AG ts e- quall tothe line CT .Wherefore asthe bale E His tothe bale NP,foisthelineCMtothe | lineCT . But as the bafe E H isto the bale N P, fo (by the 32. of the elewenth )-is the [olide A B to the folideC Z,for the folides A B and C Z are under equall altitudes. And as the line CM istothe line CT, fo ( by the x. of the fixt) isthe bafe.M P tothe bafe PT, and ( by the 32.0f the elenenth ) the folide C D to thefolide CZ .Whereforeal[o ( by the 11.and 9. of the fift ) as the folide A Bis tothe folide C Z, [vis the folideC D to the folide C Z .Wher- _ fore cither of thefe folides ABand CD hane to the folide C Zone and the (ame proportion. Setoud cafe,e Con Erultiote Wherefore (by the 7 .of the fift ) the folide A B is equall to thefolideC D : which wasrequi- ved to he acmonftrated.. uaa But now fuppofe that the Standing lines namely, F E, BL,G A, KH:XN,DO,MC, and RP, be not erected perpendicularly to their bafes . And (by the 11.0f the elenenth ) from the pointes FG, B, K:X, At, D, R; draw unto the plaine [uperficies. of: the bafes E H and NP perpendicular lines , and let thofe perpendicular lines light vpo the pointes S,T PF» . £2: W,%,d, and Q , and make perfette the ParallelipipedonsF Z; and X Q., 1 fay that eutn ia thiscafealfo, if the folides AB and CD be equall, their bafes are reciprokall to: their altitudes, thatis; as the bafe EH is.to the bafe NP, [ois the altitude of the folide CD to the altitude of the folideA Bs. For forafmuch as the folide AB is equall to. the. folide CD TR TS & Emer ae ae of Euclides Elementes.\ Fol.342. CD, but thefolide AB isequallto the. folide BT ¢ by the 20.0f thees leuenth ) for they arevpon oneand the felfe fame bafe, mamely, the pas rallelogramme K F , and under one and the felfe {ame. altitude, whofe standing lines are in the felfe fame right lines; wamely; HZ AT, and | LV ES: «and the folide CD is by the fame reafon equall to the folide DT, for they both confist upon one and thefelfe [ame bafe, namely, the parallelogramme X R, and avevn- der one and the felfe fame altitude, whofe ftanding lines are in the felfe fame right lines, namely, P OEY, andOhNW. Wherefore the folide BT is equall to the folide DY . But inequall Parallelipipedons ; whofe altitudes are -ereited D perpendicularly to their bafes, their bafes are re- ciprokall! to their altitudes (by the firft part of this Propofition) . Wherefore as the bat F K is to the bafe X R, fo is the altitude of the Hig DY tothe altitude of the folide BT. But the bafe F K is equall.to the bafe E H, andthe bafe X R tothe bal NP. Wherefore asthe bafe E H is to the bafe NP fe is the altitude of the folide DY to the altitude of thefolide BT . But the altitudes of the folides DY & BT , and of the folides D C GB Aare one and the felfe fame. Wherefore as the bafe E His tothe bafe NP; {ots the altitude |. of the [olideC D tothe altitude of the folide.AB.| — Wherfore the bafes of the Parallelipipedons AB and C D are reciprokall to their altitudes. . Againe {uppofe that the bafes of thé Paralleli- r 2 SS - pipedons AB and C D bereciprokall to their al- i 29 Demonfiyta StOfig , The tonnerfe titudes,that is,as the bafe EH is to the bafe NP fo let the altitude of the folide CD be to the + ee altitude of the folide A B . Then I fay, that the folide A B is equall to the folide C.D. For the fame order of construction remayning, for that as the bafe E His to the bafe NP, fo és the altitude of the folide C D to the altitude of the folide AB : but the bafe E H is equall to the parallelogramme F K, and the bafe NP to the parallelogramme X R : wherefore as the bafe F Kis to the bafe X R, fois the altitude of the folide CD to the altitude of the folide AB But the altitudes of lf A Band BT are equall, and {0 alfo arethe altitudes of the folides DC and DY .Wherefore as the bafe F K is tothe-bafe-X-R, fois the altitude of the folide DY tothe altitude of the folide BT . Wherefore the bafes of the Parallelipipedons BT and DY are reciprokall to their altitudes. But Parallelipipedons whofe altitudes are erected pers pendtcularl to their bafes, and whofe bafes are reciprokall to their altitudes, are eguall the one to the other (by this Propofition ) . Wherefore the folide BT is equall to the folide DT. But the [olide BT ts equall to the folsde B A (by the 29.0f the elenenth) for they confit upon one and the felfe {ame bafe,namely, F K, and are under one and the felf {ame altitude,whofe A Aa.ty. landing > T&e generall couclufione Confiructione T heeleuenth Booke’ | __ftanding lines are in the felf Jame right lines. And the folide D ¥ is eqnall to the folide D C, for they confiste upon one ard the felfe [ame bafe, namely, &Ryand are under one and the felfe fame altitude, whofe finding lines are in the felfe fameright lines. Wherefore alfo the folide AB 1s equall to the {ilide C D_.Wherefore in equalParallelipipedons. the bafes are reciprokallto their altitudes. And Parallelipipedons whofebafes are reciprokall with their altitudes,are equall the one‘a the other : which was required to be proued.. \ | BY ah The demoriftration of the fift cafe of this Propofition is eafie'to conceaue by the figure esicsalehri- bed in theplaine. Butye may or your more full fight defcribe Parallelipipedons of pafted paper, ac- cotding as the-conftruction techeth you. oe Us ses Ve And for the fecond cafe, if ye reméber well the forme of the figures which you'made for the fecond cafe of the 31-Propofition: anl defcribe the like for this,taking heede-to the letters tharye place them like as the coftruGtion in this cfe requireth,ye thall moft eafily bythem come te the full vnderltanding ofthe conttruétion and demorftration of the faid cafe. Pri ry See | SS LR | M.Iobn Dee his findry I nuentions and Annotacions very neceffa ry . hereto be added and confidered. | ATheweme. — If fower right lines be in iontinuall proportion ,and,vpon the jquare of the firft,as a bafe , be erec- ted arectangle parallelipipedes , whofe heith is the fourth line: that rectangle parallelipipedon ts eguat sothe Cube madeof the feconl line, And i upon the {quare of the fourth line, as a bafe,beecretted a retlangle parallelspt pedon , woe besth isthe firft line, that paralleltpipedon is eqnall to the Cube made of the third line, Suppofe aBy CD,EF,andGH to be fower right linesin cétinuall proportion : and vpo the fguare of a 8£( which let be a t) asa bafe,let be ereéted a rec- tagie parallelipt- ed6, hauing his heith 1K, equall to cu,the fourth line. And lerchat allelipipedon aa x .OF the fe- cond line c n,let ACube bemack ks Whofe fis ‘ *" hbfestet be noted withic @andlec. his heith be no- tedby qi? & let the whole Cubes: , bhefignifiedby.c= ) \s 5 xi lfay thata.« is... equalltocr- Let the like coftpudti= ‘lon be fer the: > cube of the third line: thatis,vpon the fquareéfc x ( which fappofe «| co be cm) Jeta- reCtangle paralle- - lipipedon be e- ‘ai eed er fan . oe of Euclides Elementes.. Fol.34.9. rected,hauyng his heith n o,equall to a s,the firftline : which parallelipip:don let be noted with @6. And fuppofethe cube of the thirdiine (z F )to bes mswhofe-{quarebafe,bt be noted by # & 7 and‘hys heith by rm. I faynow (fecondly) that c o isequaltox m. For the fir part confider, that a'r (the {quare bafe of A K ) isto c q , the fquare bafe of c 1,as a 8 isto the third lire z F by the A.Cerallaty of the 20; of the fixth, Butas as, isto F, fo(by alternate proportion) is cp to oH, toe D:Thé ctibes roote, is Q L,the fame cubes heithequall:andto c nis 1K ( by conftrnétin)equalt: wherefore,as.a 1 isto € eyfoisettol K. The bafés therefore and heithes of A Kand ¢ 1, ze reciprocallyimproporti- on: wherefore by the fecond part of this 34. propofition, AK and c 1 are quall. For proofe of the fe- cond part of my theoreme, | fay, thatasa 8, c D,£ F, and un, are in contnuall proportié#ferward, fo are they backward in continuall propottion, as by the fourth of the fift nay be proned’s' Wherefore now confidering c u to beas firft,and fo 4 8-to.be the fourth: the {quare bife c x, is to the Ryaare bafe ER, aS GH isto c D, by thez. corollary of the20. of thefixth: Butas Gu & to ¢ D, fo isé P't'A By by alternate proportion: to the Cubik roote £ F,isR m (the heith of the fam: Cube £ m} equall, And to 48,18 the heith w o equall, by conftruction: wherfore as G n iSto ER, foisimtown o. THerfore by the fecond part of this 3 4. propofition, c ois equall to z m. Iffowretight line: (therefore) be in continu- all proportion &c. as in the propofition: which was required to. be demorttrated. | A. Gorollary logifticall. | Of my former T heoreme it followeth: eAny twonumbers being geust, betwene Which two Wwe would haue two otber numbers middle, ix comtinuall proportion : T hat i} Wwe multiply the fquare of the firft number genen, by the other geuen number (as if it were the fourtl):the roote Cubik of that ofcome or produtt, {hall be the fecond number fought: e4nd farther, of we multiply the /quare of the other number geuen, by the first geuen number , the roote Cubike of thai ofcome {halt be the thirde number fought. - _ aes a4 | . For¢ by my T heoreme) thoferetangle parallelipipedos made of the fuares of the firft & fourth, multiplied by the fourth & thefirst accordingly, are equall tothe Cubes made of the fecond c& third numbers: Which Wwe make our. tWo middle proportionals. Wherefore of ibofe parallelipipedons (as Cubes ) the Cubskrootes, by good and vufuall arte fought and found, gene ihe very two middle num- bers defired. And where thofe numbers, are wot-by logisticall confideratin accounted Cubik, num- bers,ye may ufe the logiftical fecret of approching nere to the precife veryte: fo that therof most eafily you (hall perceaue, that your fayle.ss.of the fence neuer to be perceaned: wt ito Wete,asina lyne of an inch long; not to Want or exceede the thoufandthoufand part: or farther yn may ( infinitely approche at pleafure, O Mechauicallfrend, ve of gaodcomfort, put to thy hand: Labor improbus, om- d disi0 A Probleme. .1. | ‘Popon F right lined playne fuperficies genen,to apply a rettanigle paralelipipedon gener, Orwemay thus expreffe the fame thing. 0 2 Hi V ppon avight lined playnefuperficies geuen, to erett ayettangle paralelipipedon, equall toarec- tangle parallelipipedon geuen. | | _ .») :Sauppofe the right lined playne fuperficies geuen to be.z.: and the reCangle parallelipipedon. ge- a to-be.4.m. Vppon s, asa bale mult 4m be applyed; that is, are@angle jarellipipedon.muft be ere- | . Yppon.By 1... | : | asabaleawhi ey ee 7 SR che fhall bee-= ©” — Z quall ‘toa Mm. | ‘By the lafte of 1S | | | thefecond,to. NA 7 the right ly- O | >< ned figure , it madate: | , Jet an equall {quare be | made : which 5 gi fuppofe to be | Bes FRX."Pproduce * a a | Fmy i | . yet 4 ny \ xf i) ' ‘A 7, 7? i ah eal ye 4 4 | & Pe. | et ‘ a . 1 f ‘ - a z } } ae Tat: ( ? Att , , f ( Via Ah eas ity > NATE ie a eu wo ‘| La 1) Mt ‘ : | I : ‘ * hein) ae ith fy , | 7 "i wh V 4 mee Beh Tl) , { iF hi ie " ty ! Leet Oy ae im tHE a ty ' I ¥ Le mee eh! ‘. . yt" | f < ee eee Se ~~ Tf there be two fuperficiall angles equall,and [from the pointes of thofe ane Sa een gles nl a ER SS | Pa ae of Euchides Elementes. 350. gles be eleuated onhigh right lines, comprehending together with thofe right lines which containe the Juperficiall angles equall angles ,eche to his coreSpodent angle ,and if m eche of thé elenated lines be také a point at all anentures , and from thofe pointes be drawen perpendicular lines to the round playne fuperficieces in-which are the angles geuen at the begin- ning, and from the pointes which are by thofe perpendicular lines made in the two playne fuperficieces be ioyned to thofe angles which were put at the beginning right lines: thofe right lines together with the lines eleude ted on bigh {hall contayne equall angles, ASI V ppofe that thefe two rectiline fuperficiall anglesB AC, and ED F be equall KVMS the one to the other-and frome the pointes A and D let there be eleuated upwara AIX thefe right lines AG and D M, comprehendinge together with the lines put at POSK MY the beginninge equall angles, ech to his corre/pondent angle,that is,the angle M D E to the angle G A B, and the angle M D F to theaneleG AC, and take in the lines AG and D M pointes at all auttures and let the {ame be G and M.And (by the rr.of the elenéth) from the pointes G and Madras vito the ground playne fuperficieces wherein are the an- gles BAC and E | D F perpendicular linesGL andm N and lerthem fall inthe {ayd playne fiperficieces 17 ‘the pointes N and L, and drawe a righ Line from the point L to the point.A and an other from the pointe N ta the 3 pointe D.T hen I fay that the angleG A Lis equall tothe ancleM DN. Fro the ereater of the twa lines AG and D M, ( which let be AG ) cut of by the 3. of the first the line AH equall vato the line D M.And (by the 31. of the firit)by the point H,drawe unto the lineG L aparallel line,and let the fame be H K..Now the line G L is ereéted perpendicularly to the grounde playue fuperficies BAL : Wherfore al{o( by the 8. of the eleuenth ) the line H Kis ercited perpedicularly to the [ame grounae plaine uperficies BAC. Drawe ( by the r2. of the jirft) frothe pointes K and N unto the right lines AB,AC,D F,¢y DE perpedicular right lines and let the fame be K C, NF, KB, N E. And drawe thefe right lines HC, CB, MF, F E.Now forafmuch as (by the.47. of the firft ) the [quare of the line H Ais equall to the [quares of the linesH K and K A,but vnto the {quare of the line K A are equall the [quares ™> Conftru€ion. Demonffra- of the lines K Cand C A:Wherefore the fquare of the line H A is equall to the {quares of *#°%- the lines HK,KC and CA. But by the [ame unto the {quares of thelinesH K and KC 1 equall the (quare of the line HC : Wherefore the {quare of the line H Ais equall tothe [quares uf the linesH CandC A ‘wherfore the angle HC Ais( by the 48. of the firft)a right angle. And by the fame reafon allo the angle MF Dis aright angele. Wherefore the angle HC Ais equall to theangle UL F D.But the angle H AC is ( by [uppofitio equal to the angle Mt D F Wherfore there are two triangles 4D F andH AC hauing two an- gles of the one equall to two angles of the other , eche to his corre/pondent angle,and one fide of the one equall to one fide of the other namely, that fide which fubtendeth one of the = angles, T he eleuenth Booke angles, that is,the fideH Aisequall tothe fide DM by conftruction. Wherefore the fide remayning are (by the 26 .of the first equall tothe fides remayning . Wherefore the fide AC és equall to the fide D F In like fort may we proue that the fide A Bis equall to the fide D E, if ye drawe aright line from the point H tothe point B,and an other from the point M to th point E.F or forafmuch as ‘ait quare of the line AH is ( by the 47. of the firfte equall to tht [quares ofthe lines AK and K H, and (by the fame ) unto the {quare of the line A K ar: equall the {quares of the lines AB and BK.Wherefore the fquares of the lines AB,B K, and K H are equall to the (quare of the line A H . But unto the (quares of the lines B K ant K Hisequall the {quare of the line BH (by the 47. of the firft) for the angle HK Bs a right angle,for that the line H K is erected perpedicularly to the ground playne [uperficies: Wherefore the’ fquare of the line AH is equall to the {quares of the lines A Band Bh. Wherefore (by the 48.of the firft) the angle AB His aright angle. And by the fame rea fonthe angle D EM aright angle. Now the angle B AH is equall to the angle E DM, for it isfofuppofed, and the line A H is equallto the line D M . Wherefore (by the 26. the firfte) the line A Bis equall tothe line D E. Now fora{much as the line AC 4s equal to the line DF ,.and the line AB to the line DE, therefore thefe two lines AC and A Bare equalltothefetwo lines F Dand DE. But the angle alfo C AB is by {uppofit- onequall to the angle F D E.Wherefore(by the 4.of the firfle ) the bafe BC is equall to the bafe E F and the triangle to the triangle,and the rest of the anGles to the reste of the angle., Wherefore the angle AC B is.equall to the angle D F E. And the right angle AC K 1s equel to the right angle D F N.Wherfore the angle remayning, namely,B C K,ts equall to the ar. gle remayning, namely, to E F N.And by the fame reafo alfa the angle C B K is equal to tle angle F E N.Wher fore there are two triangles BC K <7 E F N,hauing two angles of the one equalto two angles of the i Be to H G SSL et Se his correfpondent angle, and one fide of the one equallto one fide of the o- ther, namely , that | $ fide that lieth betwene the equall angles , that is the fide BC 1s equall to the fide E F :Wher'- fore (by the 26. of the first )the fides remaininge are equall to the fides remayning.Wherf ore the fide C K ts equall to the fide F N :but the fide AC # equall to the fide D F . Wherefore thele two fides AC and C K areequallto thefe two fides D F and FN , and they contaynt equal angles: Wherefore (by the 4.of the first ) the bafe A K wequallto the bafe D N, And forafwuch as the line AH ss equallto the lineD M, therefore the [quare of the line A Hs equallto the [quare of the ine D M. But unto the {qnare of the line AH are equall the {quares of the lines A K and K H(by the 47.0f the firft for the angle AK Hus aright angl. And to the {quare of the line D M are equall the {quares of the lines D Nand N M, for tle angle D N M is aright angle. Wherefore the {quares of the lines A K ana K H are equal io the fauares of the lines D N and NM:of which two,the [quare of sheline AK 1 equal io the {quare of the line D N(for the line A K ts proued equall tothe line A N ) Wherefore the refidue, namely the {quare of the line K Hw equal to the vefidue, namely ,to the [quare of the. line NM .Wherefore the line H K is equall to the lineMN . And foral{much as thefe tno lines H Aand AK are equall to the[e two lines M D and D N,the one tothe other,and the | | | bale i ‘ i I - j H , "} Aa : ‘ White f ah i ' ia | : : - : wE a we | : i | 1 i) ey / I bh) ae ; : ] } 4 i}! } ; ’ | 2 | f | ; a ,' | y | } 8 At ; oe F | i Fi : ay f : { " } A | } } yi ; | tig Ff ity ” o i! 2 ae wi) 3 4 i : i Mh, ty ei 4 Pa) t ‘ ) Pf Say ' ; MH a | ’ Veith t ' : ’ nyt i : Neh \ ; ya : iy ' Sols 7' : h i ; | ait 4 | 1 ‘ : : q tha ; : ) ) 4 - { i Seu, | { *t 4 ; ; ’ 1 = Ne ai } i » | - ‘ ; ' i | | oa nee f iy ; : ; : { ih | ; 5 W i: | ; | 1 : walt a iy : : ' 4 san | 4 nm | 4 ’ ; 4 i i | mm} ’ ’ ait H I | r| j | 4 | : | i» | i inn | t) 4 tee = - 7 ' 4 { , , ae ol : AL EE ; a ete Maan BET |: : ‘Thy im | ’ . 4 : i TS PR BR : : if » | : 4 iL) : f : } i | ' An OB 1 ig wi! if : : ait ’ t ith W3 & | hae , \ | if pb 4 i fy WY { h ea ! ‘a ) M a Re: : ’ ¢ | 7 - Many |e ; a / : : } 4 ; f | - " OF | DER N é) M a i : i if; } - bey ; ’ / oa) | aa aL ae > i at mis) oe : .) im | : i?) Tea 7 } i , ( f ; at hy H Al) i an be ; Thal Se . q it 4 : ) a ; ; y ’ } 1 Mea { fel: “pa at Ww | > ; AY) ae i| eine | | i i ait j | , te : { ; fee |e ' § c) ag i YS a ie ' =) wea : ’ 4 y : ‘Vn BLS Ge - 5 ia ae ; a4 We 7 | ; , ' rp ith) 4 a a Po Ee wt) mit + } / aye : ee Rae 3 i hh if f ' i F ; 1) Ye q ’ pe : 3 4 OF : Di hy i I ‘ as ar of Euchides Elementes. \ — Foljsi. bde HK is equall ta the bale. MN :,therfore (by the 8.0f thé first) the Angle H A K is equal tothe angle M.D N . If therefore there be two fuperficiall angles equal, and fro the pointes of thofeangles beclenated uxhigh right lines comprehending together with thofe right lines which were put at the beginning, equallangles,ech to his core/pondent angle, and if in ech of thi erected lines be taken a point at all aduentures,and from thofe pointes be drawen perpen- diular lines to the plaine {uperficieces in whith are the angles geuen at the beginning, and frm the pointes which are by the perpendicular lines made in the two plaine fii perficieces be toned right lines to thofe angles which were put at the beginning, thofe right lines {ball to- geher with the lines elewated on high make equal angles: which was required to be proued. Becaufe the figutes of the former demonftration are fomiewhat hard to ¢onceaue as they are there draven in 2 plaine,by reafon of the lines that are imagined to be eleuated on high, E haue here fet o- the figures; wherein you mutt e- reée perpendicularly to the ground fuperficieces ‘the two triangles BEK,and EMN, and then ele + uat the triangles DFM,& ACH, in dich fore that the angles M and H ¢ thefe triangles,may concurre Wit the angles Mand H of theo. the erected triangles . And then im-gining only a line to be drawen fron the pointG oftheline A Gto thepoint L in the ground fuperfi--~. 7 cie!, compare it with the former contruction demonftration, and it will make it very eafye to con €Ccale. q Corollary. By this it is manifeft, that if there be two rettiline Juperficiall angles es quall, and vpon thofe angles be eleuated on high equall right lines contaye ning together with the right lines put at the b: ginning.equall angles: pers pendicular lines drawen from thofe elenated lines to the ground plaine fue » perficieces wherein are the angles put at the beginning, are equall the one to the other. Foritis manifett, chat the perpendicular lines HK, & MN, which are dra nee the endes of the equall eleuated lines A H, and D M;to the ground fuperficieces, are equall, | 7 The 31. T heoreme. The 36. Propofition. If there be three right lines proportional: a Parallelipipedon defcribed of thofe three right ines, is equall to the Parallelipipedon defcribed of the middle line, fo that it confifte of equal fides, and alfo be equiangle to the forefayd Parallelipipedon. ——- be V ppofe that thefe three lines A,B,C, be proportionall, as Ais to B, fo let B be to C. GS T hen I fay, that the Parallelipipedon made of the li | a, Tfay, ‘lipspedon made of the lines A,B,C, is equall to the Pa- rallelipspedon made of the line B, {0 that the folide made of she line B confist of ¢- gual SE CSS a a =o > = ~ ar . a a a ————— ie * . coms —————— = ee eee I me — -—s —_ om - —— sr —_ we. — > 2 ie . | : : ) , t ‘2 7 } | ORF ; ; ; } ; \ } " Phy : } 4 ‘ : “ ; ; } be | fi : - gy ; ie { A ; : t : 4 i Li . ' J LT i q ” =. ' + : : k } > : H : if i : i) i a) } : 4 : , : : | q ES. @ f : , hy \ | eee ’ : t - i M4 . =) i : # i : ae ry ‘ ; : , By F » ty ‘ ‘ = y ig im | ii he ¢ ; i tr i 4 7 7 y 7 : ' ‘ ae |} (a Al evn weotan sary" " _ es — —— — ——w_ ee oe Se oo -_- ~ — = ae ~ nae —— re ; SS - = = a : Sa ~ = 5S > 7" , * ee io ——— —— ~ Mita te. 2. me. - a ¥ betow #8 Ses ; - ees or E ; Fe te ames Sit 4. = ees oe Conftrnttion. Demon ftra- 680M. X It is ensdet that thofe per- pendiculars are ailone with the fta't- ding lines of the solzdes,ef their folide an oles be made | of fuperft- ciall right ane© gles onely T he-elenenth Booke qual fides, and be alfo equiangle to the folide made of the lines A,B,C ° Defcribe (by the 23. of the eleuenth) a folide angle E contained under three (uperficiall angles, thatis, D EG, GEF, and F E D : and (by the 3 of the firft) putvnto the line B eneryone of thefe lines DE,G Evcy E F; equalt:s YUN ® VOT UNS ahh as > ae seine and make perfecle the fos | lide EK . And vata thes. 9% + So C line A let the line L Mibes alate tle 50 iiadalabeetitcartas equal . And (by the:20.0f xo 0 iy Ny VOT Sy Ge inieal shane ts the eleueth)vnte thereght. ia fC. ®o ung odawed line L M, and at the point age init L, de{cribe vnto the d folide angle Ean equall fo- 0%: forvorno} a 10 lide angle, cotained under’) °° ©) (Ns! thefe plaine {uperficiall an- gles NLX, XLM, and N LM, and vnto the line B put the line LX equall, ey the line LN_to the line C.Now for that as the line A isto the line B, fois the line B tothe line C:but the line A is equall to the line LM, and the line B to ene- ry one of thefelines LX, EF,EG, and ED,and the line Ctotheline LN. Wherefore as LM is to EF, fois DEWLN: So then the fides about the e- gnal angles ML.N,cy D- E Farereciprokall: Wher- fore(by the 14. of the fixt) the parallelograme MN. ee oe , 7 is eyuall to the parallelogramme D F And for afmuch as two plaine Juperficiallangles,name- ly, DEF and N L M are equall the one tothe other, and upon them are erected upward e- . quall right lines, L X and E G, comprehending with the right lines put at the beginning e- guallaneles the one tothe other .Wherefore™ perpendicular lines drawen from the pointes X and G tothe plaine {uperficieces wherin are the angles N L M, and D E F, are(by the Co- rollary of the former Propofition) equall the one to the other : and thofe perpendiculars are the altitudes of the Parallelipipedons L H and E K, by the 4.definition of the fixt. Wherfore the [olides LH and E K, are under one and the felfe fame altitude. But Parallelipipedons confifting vponequall bafes,and being under one and the [elfe fame altitude,are (by the 31. of the elewenth) equall the one to the other. Wherefore the folide LH is equall to the folide EK . But the folideL H is deferibed of thelines A,B,C, and thefolide EK 1s defcribed of i the line B.. Wherefore the Parallelipipedon defcribed of the lines A,B,C, 18 equall to the Pa- _ rallelipipedon made of the line B, which confisteth of equall fides,and 15 alfo equiangle to the fore{aid Parallelipipedon. If therfore there be three right lines proportionall,a FP arallelipipe- do defcribed of thofe three right lines is equall to the Parallelipipedo defcribed of the middle lize, {0 that it compst of equall fides, and alfo be equiangle to the fore{aid Parallelipipedon: which wasrequived tobe proud... : | The nee F.. of Euchdes Elementes..\ Fol.352%- The conftru@ion and dethonttration of this Propofition; and of che next a following; may eafily be conceaued and vnderftanded: by the figures deferibed’ in the plaine Onging tothem, But ye may forthe more full fight of them ,defcribe fuch bodies of patted paper,hauing their fides pro- portionall, as is required in the Propofitions. neh Mahe | «ff New inuentions ( coincident ) added by Master Iohn Dee. $M Corollary. ns Flereby it is enident, that if three right lines be proportionall:the Cube produced of the widdle line, is equall to therettangle Parallelipipedon made of thofé three lines. ‘For a'Cubeis a Parallelipipedon of equall fides: and alfo reftan ed : as We fuppofe the Parallelipi- pedon; made of the three lines to-be likewife rectangled. 2 = er Up a} 4 Probleme. b "eA Cube being geuen, to finde three right lines proportionall, in any proportion genen betwene ye right lines : of which three lines, the retkangle Paralelipipedon produced, {hall be equall to the ube geuen. MoS aL | — Suppofe AC to be the Cube. geuen::whoferoote, fuppofe tobe AB. Lerthe ._ proportion getien, be that’ - which is, betwene the two... tight lines D and £., I fay now, three right lines are to be found, proportionall, in the proportion of Dto E, of which, thereCargle Pas. {x . rallelipipedon “produced, ) \ fo Ahall be equalltosA C.. By the r2vof the fixtdeta line Be 7" | found, Which 46 AB. haue Pe that proportion that D hath : to E. Let that line be F: and by the fame. 12. of the fixth, Jet an other line be found, _ yh to which, AB, hath thae #H proportion that D hath to E : andlet that line found be H . Leta rectangle Paral- lelipipedon mathematically be produced of the three right lines F, AB, and H, which fuppofe to be K:1 fay now, that F,A B, and H,are three right lines found pro- portionall in the proporti- onofD to E, of which, the rectangle Parallelipipedon K , produced , is equall to AC the Cube geuen . Firft it is euident that F,A B,and | a Revie H, are proportionall in the proportion of Dto E.. For, by conftruction; as Dis to E, fois to.A B: and by conitruction likewife, as D is to E, fo is AB to H. Wherefore Fisto AB, and AB isto H, as Dis to E . So then itis manifeft, F, AB, and H,to be proportionall in the proportion of D to E,and A B to be the middle line . By my former Corollary therefore, the reCtangle parallelipipedon made-of F,A B, and H, is equall to the Cnbe made of AB . But AC, is (by fuppofition)the Cube made of AB : and of the three lines F, AB, and H, the reCtangle parallelipipedon produced, is K, by conftruétion : Wher- fore, K, is equallto AC : ACube being geuen, therefore, three right lines are eases in, at f BB je any Cc . ” » ‘ hn, ‘1% nea % - > any propdintiang produced; ts: O19 2986 Mote : oA reibaihgle Playallelipipeddon b bing venient te finde three ig i lines propertiohall -oftthe which, the rettangle Parallelspipedon produced, 1s equall to therettangle Parallelpipedon genen. Doubling of the ~ Liften to this new deuife, you couragious Mathematiciens : confider, how nere this crepeth to Cube €$e. the famous,Probleme of doubling the Cube . What hope may (in maner)any young beginner céceiue, ~ Demonfrati- by one meanes or other,at one ume or other, to execute thisProbleme? * Se:ng toa Cube may in- ox of pofsrbelstic finitely infinite Parallelipipedons be fourid'equall : all which Parallelipipedons Fatt be produced of inthe Problem. three rigntlines proportionall, by the former Probleme : butto any rectangle Parallelipipedon geuen, formic ons Cabe isequallasis cafe to demonttrare::: We Gan not doubr; but vnto our rectangle Paral- leliptpedon-geuen, many other rectangle Parallelipipedons are alfo equall; hauing their three lines of produgtion,proportionall. In the former Probleme, infinitely infinite Parallelipipedons may be found of three proportional lines produced, equall to the Cube geuen : it is to wete, the three lines to be of all proportions, that aman can deuife Betwene two rightlifies : andhere any one will ferue : where alfo is infinite varietie : though all of one quantiti¢ : as before in the Cube . Ileaueas now, with thys praketicte feo. vptothoore at.vHitsrwhoan oo) si grs os dishua 62 3 Ks Sin 4 \ t ¢ ch Vas i} \ 28) OS RRADD BY AS Wh mdIeERs &* q Ihe 32.Theoreme. . The 37.Propofition. = es 5 » | : An other arpu- gent to comfort the feedious ; . - e-pro as ABistoC D, foleeE F betoGH, and upon the lines ABj CD, EF, and te H, defcrabe thefe Parallelspipedons K A LC, ME,and N G, being luke and Weg in like fort defcribed . Then I fay, that asthe folide K Aistéthe folide L Cfo is the folde M E to the folide PiAE ess ROMOqON 1809 DemonSiratie NG. Farforafimuch as the Pa- : mgt | on of the firft part. oc 2S 7 as the Parallelip ‘tithe Paral vas of Euclides Elementes. Fol.353. the Parallelipipedon L C, fais the Parallelipipedon M E to the Parallelipipedon N.G.Then s ; ; me OA ght ae cae ore see emontraq I fay, that asthe right line AB is to theright tine:ODy {ois the vightline@ EF to theright fthe line GH . For againe fora{much as thefolide K ‘Ais tothe [olide LCinireble proportion of fecond part, that which the fide A b isto the fide CD, andthe folide, ME alfe,as:tethefolide NG in whichis the treble proportion 6f that which the line E F is tothe lineG H, and.as the folide K Hows tothe conuerfe of folide LC, fois the folide M E to the folide NG . Wherefore al{a.as the line ABistothe line *h« fit part CD, fot the line EF tothe line G H . If therefore there be fower right lines proportionall: the Parallelipipedons defcribed of thofe lines, being like & in lke fort defcribed,fhall be pro- portionall. And if the Parallelipipedons de{cribed of them,and being like.and in like fort de- Scribed, be proportional: thoferightlines alfo [hall beproportionall. which was required to be proued. q The 33. Eheoreme.\\v'T beige: Propofition: Ifa plaine fuperficies be erected perpendicularly to a plaine fuperficies, and from a point taken in. oneof the plaine fuperficieces be: drawento the other plaine fuperficies a perpendicular line : that perpendicular line fhall ow i fall ypon the common feétion of thofe plaine fuperficieces. Rie AV ppofe that the plaine feperfisies C:D-be eretted perpedicularly to the plaine fuperfi- wife : ees) take a point at.all adwenturesyand let the fame be E Then I ay, that a perpendicu= lar line draiven from the point E tothe sal oe Sale | plaine [uperficies A B, fhall fall Upon the | AR TS right line DiA. For if not,thén let it fall : WI Va tthout the line D A, as thé live EF doth, and let it fall upon the plaine {it~ perficies AB in the point F . And (by the 12. of the-firft) fromthe-point:F draw unto the line D A, being inthe fuperficies A B a perpendicular line F- G, which line alfois erected perpendicu- larly to the plaine tperae CD: bythe third diffinitio: by reafon we pre[uppofe C D and AB to be perpendicularly erec- ted ech to other. Draw.arightline from the point E tothe point G . And foraf- ! much as theline F G is erected perpendi- BS UINGATR cularly tothe plaine fuperficies€ D, and FSR Pen the line E G toucheth it being in the [uperficies C D. Wherefore the angle F G Eis (by the 2. definition of the eleuenth) a right angle. But the line E F ts alfo erected perpedicularly to the Superficies AB : wherefore the angleE F G is aright angle. Now therefore twa angles of the triangle E F G, are equall to two right angles : which ( by the 17. of the firft) is tmpofable.. Wher fore a perpendicular line drawen fro the point E to the fuperficies AB,falleth not witha out the ine D A.Wherefore it falleth vpon the line D A : which was required to be proutd. DemonFra- bon leading te ‘ @n tmpofsibi« dsttee eq Notes,» Campane maketh this as a Corollary,following vpon the t3: and very well, with fmall ayde of other Propofitions he proiteth it:whofedemonftratia there,F/uff~ hath in this place, and.none.other:thongh he fayth that Campane of {uch.a Propofitid,as of Exelides, maketh no mention, | \ BBb.ij. Ta T he elenenth Booke -In this figure ye may more fully fee the former Propofi- 4i0n and demonftration if ye erecte petpendicularly vnto. the ground plaine fuperficies AB the fuperficies CD, and __ imagine 2 line to be extended from the point E to the point £,inttede’ whereof ye may-extend if ye walla thred. | q The 34. T heoveme. | d he 30. Proposition. If the oppofite fides ofa Parallelipipedon: be. deuided into two equall partes, and by their common feétions be extended plaine fuperficieces: the 2s cond feétion of thofe plane fijperficieces, and the diameter of the Parale oct .dedepipedon./hall denide the one the other into two equall partes. SON V ppofe that AF bea Parallelipipedon; and let the oppofite fides thereof C F and WSS oS AH be deuidedintotwo equall partes in the pointes K,L,M,N, and likewife let Ko 0) theoppofite fides.AD apd GF be denided into two equall partes\in the pointes IE SCLY PO; Rand by thofe fethions extend thefe two plaine [uperficieces K N GX R, and letthe commen fection of thafe-plaine [uperficieces bethe line V S, and let the diagonall line of the folide AB be the line D K Roti . «:. DG.ThenI fay,that thelines V § ) Contrainte 444 DG dodenide the onethe b- ther into two equall partes,that és, « that the line VT is equalt tothe a dine T 8,andtheline DT to the » Line T Gs, Drawe thefe right lines DVWVE, BS, ana 8 G. Now for- afmuch as theline D X is a parallel- to the line O E,therfore(by the 29. of the first) the angles D XV and V OE being alternate angles, are equal} the one tothe other . And forafmouch as the line D X is equall tothe line OE, andthe lineXV to the line V 0, and they comprehend equall angles : Wherefore the bafe DF iseqaall tothe bale VE (by. the g.of the fir) and the triangle DXV is equall.to the triangle V OE, andthereft of the anglesto thercfi.of theangles Wherefore the angle X V D is equall to the angle OF £ . Wherefore DEE is onevight-hue, and by the [ame reafon BSG ts alfo one right line; and the line BS as equall to the line SG. And fora{much as the line C A is equallto the line D Band is un- to it a parallel, but theline C Ais equall to the lineG E; and is unto it al{o a parallel: wher- fore ( by the fir(t common fentence) theline D Bisequall tothe line GE, cts alfo aparallel vata it > but the right lines DE and BG doioyne thefe parallel lines togetber : Wherefore by the 33-0f the firft the line D Eis a parallel unto the line B G.And in either of thefe tines Demon ftra- t i0 We eon ns o SS RR RE 8 See me —- ~ - Aw ee = = = = ee ‘ ae aie = e ave of Euclides Elemente. 354, are taken pointes at all aduentutes, namely, D3V,G-8, and a right line is drawen from the point D tothe point G, and an other from the point V to the point S. Wherefore (by the 7.0f the cleuenth) the lines DG and ¥'Sarein oneand the felfe Td plaine fuperficies . And for- afmuch as the line D.E is aparallel to.the line BG, therefare (by the 24.0f the. firft) the an: gleE IT is equall to the angle BGT; for they ave alternate angles, and hikewife' the angle DTV is equalltothe angle GTS. Now then there avétm triangles, that is, DT V and GT S, hauing two angles of the. oneequall to two Angles of the other, and one fide.of the one equall to one fide of the other, namely, the fide which Jubtendeth the equall angles,that isthe fide DV tothe fide G S, for they are the halfes of the lines D E and BG: Wherefore the fides remayning are equall to the fides remayning.Wherfore the line DT is equall to the lineT G, andthelineVT tothelineT S. if therefore the oppofite fides of a Parallelipipedon be de- uided into two equall partes, ana by their ‘[ections be extended plaine [uperficieces, the com- mon fection of thofe plaine {uperficiecesand the diameter of the Parallelipipedon, do deuide the one the other intotwo equall partes : which was required to be demonftrated. A Corollary added by Flufas. Euery playne fuperficies extended by the center of parallelipipedon, dinideth that folide into two equall partes:and {0 doth not any other playne Juperficces not extended by the center. For euery playne extended by the center, cutteth the diametet of the sei Aran in the cen ter into two equall partes, For iris :proued; that playne'fuperficieces which cutte the folide into two equall partes,do cut the dimetient into two equal! partes in the center, Wherefore all the lines drawen by the center in that playne fuperficiés fhall make angles.with the dimetient. And forafmuch as the di- ameter falleth vpon the paralict right lines of the folide, which deferibe the oppofite fides of the fayde folide, or vpon the parallel playne fuperficieces of the folide, which make angels at the endes of the diameter: the triangles contayned vnder the diameter, and the right line extended in that playne by the cegter, and the right line,which being drawen inthe oppofite fuperficieces of the folide,ioyneth together the endés of the forefayde right lines, namely, the ende of the diameter, and theendé of the line drawen by the center in the fuperficies extended by the center, fhallalwayes be equal; and equi- angle, by the 26. of the firft. For the oppofite right lines drawen by:the oppofite playne fuperficieces of the folide do make equall angles with the diameter, forafmuch as they. are parallel lines, by the..2s.. of this booke. But the angles at the céter are equall, by the 15. of the firft, for they are head angles: & one fide is equall to one fide, namely, halfethedimeticnt. Wherefore the triangles contayned: ynder e- uery right line drawen by the center of the parallelipipedon in the {uperficies, which is extended alfo by the fayd center, and the diameter thereof, whofe endes are the angles of the folide,are equall,equi- Jater, & equiangle(by the 26. of the firit) -Wherfore it followeth that the playne fuperficies which cut= teth the parallelipipedon, doth make the partes of the bafes on the Oppofite fide,equal!,and equiangle, and therefore like, andequall both in tultitudeand iin or ews whereforethe two folide fections of that folide,fhalbe equall and likesby the 8. diffinition of this booke.. And now that no other playne fuperficies ,befides thar which is extended by the center ,deuideth the parallelipipedon into two equall partes, itis manifeft: if vnto the playrte fuperficies which is not extended by the center, we'extend by the centera parallel. playne fuperficies. by the Corollary ‘ofthe 15\:of this booke ). For forafmuch as that fuperficies which is extended by the center,doth denide the parallelipiped6 into two equall parts itis manifeit, that the other playne fuperficies (which isa parallel to the fuperficies which deuideth the folide into two eqitall partes) is in one of the equalt partes of the folide : wherefore feing' that the whole is euer greater then his partes, it muft nedes be that oneof thefe f{eétions is leffe then thie halfe of the folide, and therefore the other is greater. For the better vnderftanding of this former propofition, & alfo of this Corollary added by Flufeas, it fhalbe very nedefull for you to defcribe of pafted paper or {uch like matter a parallelipiped6 or a Cube, and to deuide all the parallelogrames therofinto two equall parts, by drawing by the céters of the fayd parallelogrammes (which centers are thé poynts made by the cutting of diagonall lines drawen fré the oppofite angles of the fayd parallelogrames) lines parallels to the fides of the parallelogrames:as in the former figure defcribed in a plaine ye may {ce,are the fixe parallelogrames D E,E H,H A,A D, DH sand C G, whom thefe parallel lines drawen by the céters of the {ayd parallelogrames, namely, XO, O R, PR, and P X;dodeuide into two equall parts: by which fower lines ye muft imagine a playne {uperfi- cies to be extended, alfo thefe parallel lynes KL, LN, NM, andMK, by which fower lines likewife ye muft imagine a playne fuperficies to be extended ye: may ifye will put within your body madethus of pafted paper,two fuperficieces made alfo of the fayd paper,hauing to their limites lines equallco the borefayde parallel lines! which fuperficiéces muft alfo be deuided into two equall partes by ss BDD.iij. nes A . ip J - tia Vy He JE a Ps i 1 i | \ Pt | } } eens Pode se x ome > . — > een ee COMRIEMEN So TL. | ie ES A se —— =I oe Smit Tae : ee oe —_ = ee a Se SS ES — ne : a eae Ses same = — wes ee — ~w 7 ‘ if Ay S if " ‘ ¢ : ‘ { ma i a! y | , } a i 7 q 4 | i } q : » | | \ ' } f t / f 4 a | #¢ | ; ' ") a i} a 2 i | . 0 ; mn a\) ' y q | i} ' i} | Tei i Ht { it 7 ny ft h { heey! } Wa | ! o| : } VW / | : ' i} j " \ it} : i } i ’ | P| ‘ mn | 1 i, oF i i : : | i] i a iif . ; hl Wel ti e : ~t nh r : f ’ \ ell} 4 " } i h \ ij iy al 1 . ot ; } . { 3. i it : - a 4 ‘ y +a 4 \ t ; : iit : nih ’ ] H HY | 7 : a i fl i | » ’ | ai 4 ‘ a ' ' }) } : > | ; 4] | ' . a | ‘ Fi - hk ; | } : 7 ’ : ; q 7 HM ’ i oi ;' : tT . aa) ar BE. q ih | A ' \ ; ‘5 : { ft : a} é i TF mY oF - 1 ' £ q Hide : Hr t , w 5 | , t lyf! : : ’ faa a bl pee j , iT 1 4 : Fi 4 | : : pew - 7? 0) Fa ; ; y 4 4 a0)! : : ’ 4 rt M 7, Ab ua : Vp } : “ § " : 4 . ty | ¥ | \ i 1a ee bf ! y K i . rn : Mh a } he | ; i ie ; 7 : ' / \ q : aH ' fe § = 16 ; t 7 it. . } ; ! y ; . rh ee : f ; bi - + | t N 5 f . F Bg | : é } ‘ uly H “} @ 4 ’ 7 ee . re ! © iT . By! r te | | as : it t it I , rik ; Pat val | ie ) } ‘ > } 7 ad } i . Lh ee ae A i ‘ Bat tt 4 : itt [ey ‘ ve J H ; si a , ir" Ma q 1 \ 19) Rae i? im) t i i= : — ae > ow - = ~ —* _ —. Sea se Ae ee Confiruction. Demonftra- tron. T he elenenth Booke Ynes drawem by their centers, and muftcut the one the other by thefe parallel lines. And for the dia- metet of this body,.extéd a thred from one angle in the bafe of the folide to his oppofite anglej which $hall pafle by the tenter of the parallelfpipedon, as doth the line DG ip the figure before defcribed in thé playne: And draw in the bafe and the oppolite fiperficies vita it, Diagonall lines,from the angles from which is extended the diameter of the folide: as in the former defeription are the lines BG and DE. And when you haue thus déferibed this body, compareit with the former demontftration, and ic will make it very playne vnto you, fo your letters agree with theletters of the figure defcribed in the booke. And this defcription will playnely fet forth vnto you the corollary following that propofition. For where as to the vnderftanding of the démonttration of the*propofitiou the fuperficieces put within the body wete extended by.parallel lynes drawen bythe scéters,ofthe bafes of the parallelipipe- don: to the ynderftanding of thefayd Corollary, ye may extende a {uperficies by any othér lines dra- wen in the fayde bafes,fo that yet it paflethrough the middeft of the thred, which is {uppofed to be the center Of the parallelipipedon. , | | « ™~ q Lhe 33... heoreme...... The. 40, Propofition. If there be two Primes onder equall altitudes, ¢7 the one haue to his bafe a parallelogramme and the other'a triangle, andif the parallelogramme be double to the triangle : thofe Pyifmes are equall the one to the other. | ‘AY profe that thefe two Prifmes ABCD EF, GH KMON,be under equall UF fall studes, and let the one haueto his bale the parallelogrammé AC, and the o- : S*l:ther the triangle Gt K:and let the paralieloeramme AC be double to the tri- Merle angle G HK. Then 1 fay, that the Prifme ABCD E F is equalltathe Prifme GHKMON... Make perfecte. pesncapuemeeio the Parallelipipedons AX GG 0. And foralmuch as the paraltelo-° gramme AC is double to the tri-.— angle GEH-K ,-but the parallelos). granime GH is alfo (by. the Zr. | of the frit) double to the tridngle GHEE 5: wherefore the parallelo- gramme AC isequall tothe pa’ ralielicramme GH . But Parallelipipedans confifting vpon equal bafes andunder one and the felfe fame altatude, areequall the oneto the other (by the'31.0f the elenenth) :Wherefore the folidesA 8 is equall to the folideG O° But the balfe of the folide_A X is the Prifme A B- CDE F, and the halfe of the folide G 0 is the PrifmeG HK M ON .Wherforethe Pri{me “This Propofition and the demonftration thereof are not hard to conceaue by the former figures: but ye may for your fuller vnderftanding of thé take two equall Parallelipipedons equilater and equi- angle the oneto the other deftribed of patted paper or fuch like matter, and in the bafe of the one Pa- rallclipipedon draw a diagonall line, and dtaw an other diagonal! line in the vpper fuperficies oppofite vnto the faid diagonal line drawen in the bafe ,. And in one of the parallelogrammes which are fet vp- on the bafe ofthe other Parallelipipedon draw a diagonall liné, and drawe an other diagonall line in the parallelogramme oppofite to the fame . For fo ifye extend plaine fuperficieces by thole. diagonal lines there will be made two Prifmes in ech body . Ye tuft take heede that ye put for the bafes of eche of thefe Parallelipipedons equall parallelogrames . Awd then note thé with letters according to the let- ters of the figures before defcribed in the plaine. And copare the with the demonftration, and they will make both it and the Propofition yery clerc vato you.. They willalfo geue great light to the Corollary following added by Flafas?* © Sever o8 ee By a Ae of Euchides Elementes. 352, A Corollary added:by-Féufiass 0 20% 910) s..s10 paidowes | Srl 2 bs il ore cia aeqiguousie sic 2s dads one . By thss.and the former propohtions st ts manifeft thar Prifnes and jfalides® ses vnider ty poltgonon figures equal ,like,wird parattels, and the reft paralleligrammes:may be Compared the ane to the oth er after the félfe fame.maner that patallelpipedonsare,. > i Oi . 0" , , Five tie ‘ For forafmuch as (by this propofition and by the fecond, Corollary of the 442 0fthisbooke) itis manifeft, that euery. parallelspipedon, mayberefolued into two like, atid equal Prifmesjof one:and the fame altitude, whofe! bafe thalbéione aind-the felfe fame with the bafe-of the paratlelipipedomsor:the halfe thereof, which Prifines alfo thalbe-contayned vnderthe feifeame fides withthe patallelipipeds, the fayde fides beyng alfo fides of like proportion? I fay thar Prilmies tay be compared together after the like maner that their ParallelipipedonsareiForifwewottld-denide Prifmelike vnto his foli. e by the a5.0f this hooke,ye hall findeinthe Corollaryesofthe zs, propofirid; that that whiclvis fet forth touching a parallelipipedon,, followeth-not.onel yina Prifme}but allo in any fided columne whofe Op- pofite bafes are equall, and like, and bis fides parallelogrammies, $)s inc | If it be required by the27.propofition vpona mgheline gewen todefcribea Prifine like and in like forte fituate toa Prifme geuens defcribe frit che whole paralleliprpedén whereof the prifme geuen is the halfe (which thing ye fee by this 40. propofition may be done ). And ynto that parallelipipedé de- {cribe vpon the right line geuen by the fayd 27. propofition an other parallelipipedon like ‘ and the halfe thereof fhalbe the prifme which ye feeke for, namely, halbe a prifme defcribed vpon the right line geuen,and like vnto the prifme- geuen,: 4 nS Soe | In deede Prifmes can not be cut according to the 23.propofition. For that in their oppofite fides can be drawen no diagonall lines: hawbéit by that 28: prépofition thofe Prifmes are manifeitly con- firmed to be equall and like, which are the halues of one and the {elfe fame parallelipipedon. And as touching the 29. propofition, and the three following it,which proueth that parallelipi- pedons vnder one and the felfe fame altitude, and vpon equall bates, or the felfe fame bafes,are equal: or if they be vnder one and the felfe fame altitude,they arein proportion the one to the other,as their bafes are: to apply thefe comparifons vnte'Prifm és, it is to*be required,that the bafes of the Prifmes compared bopetier be either all parallelogrammes, of all triangles . For fo one and the felfe altitude remayning, the comparifon of thinges equall is ener oneand the felfe fame, and the halfes of the bafes are euer the one to the other inthe fame proportion, that theit.wholes are. Wherfore Prifmes which are the halues of the parallelipipedons, and which haue the fameproportion the oneto the other that the whole parallelipipedons hawe, whichate vnder one and the {elf fame altitude: mutt needes caufe that their bafes being the halues of the bafes of the parallelipiped@s.are in the fame proportio the one to the other,that their whole parallelipipedons are. If therefore the Whole parallelipipedons bein the proportion of the whole bafes, their haluesalfo (which are Priffnés) fhalbe in the ‘proportion either of the wholes if their-bafes be parallelogram messorof the haluesifthey be triangles, which,is euer all one bythers.ofthefiueth. — # eee SSE ci ‘ And forafmuch as by the 33. ptopofition, like parallelipipedons which are the doubles of their Prifmes are in treble proportion the oneto the other that their fides of like proportion are, it is mani- feft, chat Prifmes being their halues (which haue the one to the other the fame proportion that their Wholes haue,by the 15 of the fiaeth) andh suing the felfe fame fides that their parallelipipedons haue, are the one to the other in treble proportion of that which the fides of like proportion are. And for that Prifmes are the one to the other in the fame proportion that their parallelipipedons are, and the bafes of the Prifmes(being all either triangles or parallelogrames)are the one to the other in the fame proportion that the bafes of the parallelipipedons are, whofe altitudes alfo are alwayes e- quall, we may by the 34. propofition conclude,that the bafes of the prifmes and the bafes of the paral- lelipipedons their doubles (being ech thé one to the other in one and the felfe fame proportion ) are to the altitudes,in the fame proportion that the bafes of the double folides, namely, of the parallelipi- pedons are. For ifthe bales of the equall parallelipipedos be reciprokall with their altitudes sthen their halues which are Prifmes fhall haue their bafes reciprokall with their altitudes. By the 36. propofition we may conclude, that if there be three right lines proportional], the an- gle of a Prifme made of thefe three lines( being common with the angle of his parallclipipedon which is double)doth makea prifme,whichis equall to the Prifme defcribed of the middle line, and contay- ning the like angle, confifting alfo of equall fides: For as in the parallelipipedon, fo alfo in the Prifme, this one thing is required, namely,that the three dimenfions ofthe proportionall lines do make an an- gle like vnto the angle contayned of the middle line taken three tymes. Now then ifthe folide angle of the Prifme be made of thofe three right lines, there thall ofthem be madean angle like to the angle of the parallelipipedon which is double vnto it. Whereforeit followeth of neceffitie,that the Prifmes which are alwayes the halues of the Parallelipipedons, are equiangle the one to the other, asalfoare their doubles; although they be not equilater : and therefore thofe halues of equall folides are equall the oue to the other: namely, that which is defcribed of the middle proportionall line is equall to that which ts defcribed of the threc proportionall lines. By the 37. propofition alfo we may conclude thefame touching Prifmes which was concluded BBb.iiij. touching * Which of fome.are cal- led hided Cq-~ lumnes, Sj T he eleuenth Booke eouching Parallelipipedons, For forafmuch as Prifmes,defcribed like 8 in like fort of the lines geuen, are the halues of the Parallelipipedons which are like and in like fort defcribed,it followeth that thefe Prifmes haue the one to the other the fame proportion that the folides which are their doubles haue. And therfore if the lines which defcribe them be porportionall,they fhalbe proportionall, and fo con- uerfedly accord ing to the rule of thefayd 37. propofition, | ’ But forafmuch as the 39. propofition fuppofeth the oppofite fuperficiall fides of the folide tobe parallelogram mes, and the fame folide to haue one diameter, which thinges a Prifmecan not haue, therefore this propofition can by no meanes be applyed'to Prifmes, Butas touching folides whofe bafesare two like, equall,and parallel poligonon figures ,and their ; fides are parallelogrammes, forafmuch as bythe fecond Corollary of the 25. of this booke it hath bene Sided Co- declared;that {ach folides are compofedjof Prifmes, it may-eafely be proued that their nature buttehete is fuchasis the nature of the Prifmes, whereof they are compofed . Wherefore a paralle- lipipedon being by the 27. propofition of this booke defcribed, there may alfo be de- bribed the halfe thereofswhich isa Prifme : and by the defcription of Prifmes, there may be compofed a folide like vnto a folide geuen compofed of Prifmes. So thatit is manifeft, that that which the29. 30. 31. 32.33. 34. and 37. propofitions fet forth touching peri Be may well be applyed alfo to thefe kyndes of folides. ~~ “Ses ee SS Se SE ee et — > Nee BE Se - ~ Pras ee St ee aes Theend of the eleuenth booke of Euclides Elementes. : | ; ' } F t s | ‘ \ : ; ; ; i ’ ‘ J | : : : - : ; : > | | mb " | ee | : - HE : ’ MN) . fi } : § - . 4 i 4 } as t 7 7 ’ } 4 : Ee - ; 4 i i | 1 i t } ty ’ { } i | ap ; ¥ a } - ; . f 7 \t ; : - \ ; T - | ; a! q ae ui 4 Fa i] - Lie : Shy 1 4 : ‘ \ aay tl: *) i; f # i ) | : oi a hi : a " a al Vy 1 mu Y f | " F ie, | a b4e > hy ua HW “a ie ie Har? ‘ HAP Rae v a) : | oa | Be h i] j ; q { } i] oo a Fol.353. @ | hetwelueth booke of Fu- clides Elementés: * A N rHis rvververu BOOKE,EVCLIDE fetteth forth the paffions and proprieties of Pyramids, Prifmes,Cones,Cylinders,and Spheres. And com pareth Pyramids, firft to Pyramids,then toPrifmes: fo likewife doth he Cones, and Cylinders. And laftly he com pareth Spheres the one to the other.But before he goeth to the treatie of thofe bodies, he proueth that, like Poligonon figures in{cribed in circles,and alfo the circles thé felues are in proportion the one to the other, as the fquares of the diameters of thofe circles are . Becanfe that was ne- ceflary to be proned, for the confirmation of certayne paffions and proprieties of thofe bodies. = s§ a. 4 2 q Ihe 1. TU heoreme. Ihe 1.Propofition. Like Poligonon figures defcribed in circles : are in that proportion the one to the other, that the | quares of their diameters are. | Vppofe that there be two circles CABCDE;, and FGHRTI, and in them let there be deferibed like Poligonon ficures,name- ly, ABCD E,and FGHKL, and let the diameters of the circles be BM, andG N. Then 1 fay, that as the {quare of the , Aico), line BM is to the [quare of the line GN, fois the Poligonon fi- / PT AWN! gure ABCD E to-the Poligonon figure FGHKL . Drawe Sa J thee righi lines BE, AM,C L,and F.N.. And forafiuch as the Poligonon figure ABC DE is Like tothe Poligonon figure FGHKL, therefore the angle BAE is equall to the angle H —e C ee D BK — M G ‘ N \ Poll Sate E Le aA, F GF L, and asthe line B Ais to the line AE, foisthe line GF tothe line F L ¢ by the defi- nition of like Poligonon figures ) .. Now therefore there are two triangles B AE and 4 FL, auing Confiruction, Demonfira- ti 073, The twelueth Booke bauing oncangle of the one equallto one angle of the other, namely, the angle B AE equall tothe angleG F L, and the fides about the equall angles are proportional. Wherefore ( by the jirft definition of the fixt) the triangle A B E is equiangle to the triangleF GL. Wherefore the angle A E B is equall to the angle, F. LG . But (by the 21.0f the third) the angle AE B as equall to the angle A M B, for they confifte upon one and the felfe fame circumference:and =, Solo ae Seats See ee Sar z _ — => = > : : — a EE TS 5 aE SE 2 a SS ee es SAGE AAO EE SOE SI ——E—EE——— Se ee a by the fame reafon the angle F L G is equallto the angle F NG Wherfore the angle AMB is equalltothe angle F NG. And the rightangle B AM, 15 { by the 4g. petition ) equall 40 the right angle G F N_. Wherefore the angle remayning, 23 equallto the angle remayning. Wherefore thetriangle.A M B47 equiangle to the triangle FNG. Wherefore proportionally as the line B-Mis to theliae GN, fois the line B A tothe lineG F . But the{quare of the line B nh istothefquare of théline G N in double proportion of that which the line BM isto the line G'N_(by the Corollary of the 20.0f the fixt) , And the Poligonon figure ABC D Eis to the Poligonon figure F GH-K Lin double proportion of that which the line B A is to the line : * 'G Eby the 20.of the fixt) . Wherefore (by the 11.0f the fift)-as the {quare of the line BM is to vbe [quare of the line GN, [ois the Poligonon figure ABCD E, to the Poligonon fi- gure FGCH KL. Wherefore like Poligonon figures defcribed in circles, are in that pro- portion the one to the other, that the {quares of the diameters are: which was required to be demonfirated. ~~ — ow qlobn Dee his fruitfull nfiruétions, with certaine Corollaries, and their greatvfe. ~ V Ho.can not eafily perceaue,what occafion and ayde, ¢rchimedes had, by thefe firft 8 fecond Pro- pofitions, to finde the nere Area,or Content ofa circle: betwene a Poligonon figure within the circle, and the like about the fame circle,defcribed ? Whofe precife quantities are moft eafily knowen : being comprehended of right lines . Where alfo (to auoyde all occafions of errour) it 1s good in num- bers, not hauing precife {quare rootes,to vfe che Logifticall proceffe, according to the rules, with /F712, 4/3719, and fo, of fach like. Who can not readily fall into Archimedes reckoning and ac- count, by hismethod? To finde the proportion of the circumference of any circle to his diameter, to be almoft triple, and one feuenth of the diameter : but to be more then triple and ten one & feuen- tithes : that is to be leffe chen 3—and more then 3. And where Archimedes y{ed a Poligonon ee al ’ ' ’ ; #43 L Te 4 € ai ; Fie | to Sal 3 i) } tf ‘ fs , : b 4 : : vy be 4 ’ y . & Ite: } fe |. } yt r tt ’ ? sy } , : ; ir * f ’ i ey ar - ® re ¥ roe hy i bon ' i ' A i é q i Sf H 1 ‘ re Ab th 4 { : : Vie} t Pai ir : * 1] i : é ia | { } : et; : y ' ! f : U Pal y i - ¥ \e yy ie: hi ? ut ; —— —~o Ss + i : 7 ° figure'of 96 ides: he thar; for.exercife fake,or for earneft defire of a more nerenes, will vfe Polygonon fissures of 384,fides (or more) may Well trauaile therein, till either wearines caufe him ftay, orels he finde his labour fruitles . In deede' Archunedes concluded proportion, of the circumference tothe dia- meter, Fol.3$7. meter, hath hitherto ferued the vulgare and.Mechdnicall'workém en ; wherewith, who fo isnotcon« tented, let his oun en Sa trauaile fatiffie his defire : or let him procure other therto . For,nar- rower termes (of greaterand leffe) found, and appointed to the aaUTRae ilalfe winne.to the Area of the'cirelé a neater quantitie: féing, it ven dem oftiaci eras eae te a triangle reétan- gle,of whofe two fides (contayning the right angle ) one is equalf tothe Yerhidiameter oF the circle, and the other to the circumference of the fame, is equall to the Area of thatcircle. Vpon which two Theoremes ;it followeth;:thatthe {quare made‘of the diameter is iv that proportion tor theicirclé (ve- ty neate) itt Which 74 fis: to "ir Wherefote' etiery circlé'ts"29 Clete! fowyerten thes’ well néare ) of the fquare about him defcribed . The one fide,then, of that fquare, denide into 14.€equall parte? : and from that point which endeth the eleuenth part, drawe to the oppofite fide, a line, parallel to the other fid-s, and fo make perfecte the parallelesramitie) Phen, by the lat Propofition ofthe fecond booke, vnto that parallelopramme(whofe one fide hath thofe 11 -equall partes), make a fquare equall . Then is it euidgas, th: feauats.to becquall cathe circle, about which she fit (quarcis delenbed.. Ag ye may heré beholdé in thefe figures . sil ati ee Se, STE I . srt sf #12 &S \Gce-s.- ~ + * o* + . +4 _ . % vasewe -_. bs ~ es i? _ . +2 ae Pee the great defire. which I hane,that both with plea(are and al(o profite, thou mayeft fend thy-time in thefe excellsot ftudies, dork canfe me here to furnithe thee -fomewhat Cextraordina- Ty Asbo unt cichs » not onely by pointing vnto thee,the wellpring of archrmedes his fo, much won- dred at, and iuftly commended trauaile (in the forin er 3 Theoret here repeated), but alfo to make thee more apt, to vnderftand and praétife thisandotherbookes following , where, vfe of the circle may be had in any confideration : as in Cones, Cylinders,and Spheres, &c. + °“~< : ry , : >. op SA | e oe 4 > tern te the) } . . . \ « ‘“ it . : \ ‘ 3 4 7 so g tA Corollary: 44 oT. 2 \ i - .* > . ° J Xa . 7 a vy. ¥ + A \ . ‘A - > . 4 ° a ww G4 Corollary, . : 2. _ " ' \ 5 : % : ' > ' : . oe ots La Likewi €it4s entdent, that the parallelogramnecontayned under the femidiameter,and halfe of any portion oft ecucumference of acwcle geuen,iseqnal tothat fector of the fame circle, ta Which “the whéle portion “4 he circumference genen, doth peteoe » Or you may ufethe halfe femsdiawseter, and th e whole port mi Of the circumference as fides of the faid parallelogramme, ~~ } The farther withing, and'inferri ne, Tcommittéto your fkill,care,and ftudye. But ian other fore will getleyouneweayde, andinfttuétion here. +: to tf sheen aaah " ih. eh | «| A Theoreme. 26 3° The [quaring of the circle. * : ' ¥ i q : : ; Y - J y 2 9 ’ ° oie - % - ; | y r 7 tit ake ; : ban 5 5 ’ - : : } ef ie : i | " if a] | i} if yt : " it 7 i ; i ; : ! »} C8 ; } wi ‘in : : : ; irae f ’ 7 an teil - : be ye 4 . t ; b eS ; f | } eer { - bh ; ’ la J | Y b : ~ : : b 4 4 a tt hie iW ’ } : { . ‘ : i eu a 7 Bit ‘ i y of 1 Vy , eu : ’ : 2 ip , te Ve ; fai ty a i. , : ' A “J é ke » ri 7-H t 8 : ' ; : ia Vif ‘ * \ 4 oa) i bi} wie | ih 0 : ' 4 : : - y 7 : ti i! yt ‘ ; ty ‘ i , } ‘ei og i , | ii fet } bt» & ti 4 : : i a et i ’ 4h : te ee. : +: ae he i 1 , | { : | a y | tea . > ; ite ei en iI { iE ie = ‘faa : : \ is, f a i "i | } . ea eae : ‘ . int) i ee | ; ; i Lin ' aS) { { tf ;. EW ——— . = eat en Demon ftrati- on feading to an impofstbi- litte. Twocafes in this propofi~ LIG ile The firf cafe, T hetweluethBooke ial ory ming: | MG AT heoreme.s oxo: un Of all circles, the dercunsferences to thei owne diameters, hawe one andthe Some proportion in that one circle foener, they ATE Afigneds 5 6 hones ne ee Bi / , ati Hilo aptA ens OF Liswp? z1, : ito odds « | i \Thatis (as Aedbimedes hath demonftrated) almoft, as 22. to 7 : or nearer, ifnearer be found: vn- till the very precife proportion be demonttrated, .. Which, what foeuer. it be, in all circumferences to « their proper diameters, will be demonftrated oneand the fame... bg , sig %.4 Corollary. de ot Wherefore if twe circles be © gamete tobe AandB asthe circumference of A #5 to the circumference of ‘B, fots git 91989 AL Fblo, | diameter of Ato the diameter of B. For by the former Theoreme, as thecircumference of A,is to hisown diameter, fois the circumference of B, to his own diameter : Wherfore, alternately, as the circumference of A, is to the circumference of B, fo is the diameter of A,to the diameter of B : Which was required to be de- monftraced. % A Corollary. 2. Ye ts now then enident, that We can gene two circles Whofe circumferences ene to the other , [oak haue any proportion geuen iv two right lines. | iyi ~~ > # a ~ witht 4 wet ¢ i; of bis il 2, q The 2. T heogemer ss... >t he 2. Propofition. ___ Circles arein that proporgion the one tothe other. that the {quares. of their ~ a ediameter sare. | niiededrehus tokhs bs siniaee saree = al V ppofe that there be two circles ABC D,and E F GH,and let their. diame- oq) ters be B Dand FH.T hen 1 fay, that as the {quare of the line DB is to the ORK {quare of the line F H; fo isthe cirele-A BCD tothe circle EF GH. For if LK MAY the circle A BC D be not unto the circle E F G H,as thef{quare of the line B D Ys 10 tbe iar of the line FH: then thé fquare of thé line B.D hall beto the {quare of the line E Haas ibe cle A BC Dio fips eer eeabin te circle FG Ta gree ‘ter . Firff letthequare of the linesB D be to the [quare of the line F H.,as the circle ABC D is to.a fuperficies le(fe then the circle E F GH, namely, tothe fuperficies S. Defcribe (by the bof the fourth 9 in the circle EF GHa [quare E FGH, Nowstss Jqware thus deferibed figren- serrhenshe halfe of the circle EFGH. Forif by the pointes E, F,G, —— oo lw touching : y ~~ =“ sweet = > of Euchides Elementes, Fol.3s8, touching the circle, the [quare EF GH, is. the halfe of the fquave defcribed about the circle, but the {quare defcribed about the circle, is greater then thé circle. Wherefore the Jquare E F GH, which is infcribedin the circle, is greater then the halfe of the circle EFGH. Deuide the circumferences E F, F G,G H, and H E, into two equall partes in the pointes K,L, M,N. And drawe thefe right lines EK, K F,FL,LG, GM, H, H N, and NE . Wherefore enery one of thefe triangles EK F,F LG,G MH,andH N E, is greater then the halfe of the fegmet of the cir- cle which is de{cribed a- bout it. For if by the pointes K,L,U, N,be drawen lines touching the circle, and then be made perfecte the paral- lelogrames made of the right lines E F ,F G,G- H,¢y H E, enery one of the triangles E K F,F - L6,G MH, HNE, is the halje of the paral- lelograme which is de- Seribed about it ( by the 41.of the firit): but the fegmet defcribed about it is leffe then the paral- lelogramme . Wherefore enery one of thefe triangles EK F,F LG,GMH,and H NE, is greater then the halfe of the efi: of the circle which is defcribed about it. Now then dewiding the circumferences remarning into two equall partes and drawing right lines from the pointes where thofe diuifions are made, Cy {o continually doing this,we hall at the length (by the 1.of the tenth ) leaue certaine fegmentes of the circle,which fall be lefce then the ex- cefse, wherby the circle E F G H excedeth the fuperficies S. For it hath bene proned in thé Jirit Propofition of the tenth booke, that two vnequall magnitudes being geuen, if from the greater be taken away more then the halfe, and likewife againe from the refidue more then the halfe, and fo continually, there fhall at the length be left a certaine magnitude which fhall be lefethen the lefve magnitude geuen . Let therebe fuch feementes left, ¢ let the fegmentes of the circle E F GH, namely, which are made by the lines. E K,K F,F L,LG,GM,MH, H N,and NE,be leffe then the excelfe, whereby the tircleE F GH excedeth the fuperficies S. Wherefore the refidue, namely, the Poligonon figure EK F.LGMH Nyis greater then the Juperficies 8. Infiribe inthe circle ABC D a Poligonon figure like ta the Poligonon igure EKFLGMHN, and let the famebe AX BOC PD R. Wherefore ( by the Propofition next going before ) asthe fquare of the line B D is tothe [quare of the line F H, fois the Po- ligonon figure AX BOC P DR to the PoligononfigureEKF LG MHN. But as the Square of the line B D is to the {quare of the line F G, fois the circle ABC D fuppofed to be to the {uperficies S . Wherefore (by the 11.0f the fift) asthe circle AB C D is to the fuperfi- cies S, foisthe Poligonon figure AX BOC PD R tothe Poligonox figureE K F LG M- HN . Wherefore alternately (by the 16.of the fift) as the circle ABC D is tothe Poligonon figure defcribed in it, (0 is the fuperficies S tu the Poligonon figure EK FLGMHN. But the circle ABC D is greater then the Poligonon figure defcribed in it. Wherefore alfo the Superpicies 8 ts greater then the Poligonon figure EK F LGH MN: butit is alfolefe : which is impofible .Wherefore as the {quare of the line B.D is tothe [quare of the line ; H, CCC.t. 0% Pry That 4 {ouave ‘Wethin any circle defer} bed is bigger than balfe the circle, That the lfof- celes trian~ gles,without the /quare,aré greater then balfe the feg- ments where they are. Second cafes "This Afsapt és afterwara atthe end of the demiitra- tion proned= T he twelueth Booke 'y is not thecircle ABCD to any [uperficies leffe then the circles EF G H. In li ke fort alfo may mproue,that as the {quare of the line F His to the fquare of the line BD, fois not the circle EF G H to any {uperficies leffethen the circle ABC D.1 {ay,namel), that as the {quare of the | line BD isto the {quare of the line F H, fo not the circle ABC D toa- ny fuperficies greater the then the circleEF GH. For if it be pofsible, let st be to a greater, namely ,to the fuperficies 8. W. her- fore by conuerfion, as the {quare of the line F H isto the {quare of the line BD, fo is the fu- perfities 8 to the circle ABCD .* But as the fuperfictes 8. is to the ctr- cle ABC D, (ois the cir- cle EF GH to fome [u- perficies leffe the the cir- | . cle ABCD Wherefore (by the 11. of the fift) as the {quare of the line F H is ta the [quare of the line BD, forts the circle E F G H, tofome fuperficies lefe then the circle ABCD: which is in the fixft cafe proued to be impofgble. Wherefore as the (quare of the line BD 1 to the [Guareof theline F H, [ois not the circle ABCD to any [uperficies greater then the circle EF GH. Andit is alfo proued that itis not, te any lee. Wherefore as the [quare of theline B Dis to the {quare of the line F i,fo ws the circle A BC D to the circleE F G H. Wherefore circles.are in that proportion the one to the other, that the (quares of their dtame- ters are. which was required to be proued. ¢, An Affumpt. Ffay now, that the fuperficies S be- ane ereater then the circle E F G Has the fuperficies S is to the circle ABC» D, foisthe circle EF GH to {ome {u- perficies leffethen the circle ABCD: For, as the fuperficies S is ta the circle \ ABCD, folet the cirtle EF GH be to the (uperficies T . Now 1 fay, that the fuperficies T is lefethen the circle ABC D.For for that as the uperficies S isto the circle ABC D, fo w thectr- cle EF GH tothe [uperficiesT ,there- fore alternately (by the 16. of the fift ) as the [uperficies S 1s to the circle E F - GH, fois the circle ABC D tothe {u- perficies T. But the fuperfictes S 1s grea Sak SOS | ter then the circle E F G H (by {uppofition) . Wherefore alfo the circle ABC D is greater then the fuperficies T (by the 4h #8 fift) . Wherefore as the {uperficies Sas to the circle sie’ ABCD, AA of Eviclides Elementes.. " Folas9. ABCD; fois the circle E F G Ht0 [ome {uperficies leffe then the circle ABC D : which was required to be demonstrated. | Gg A Corollary added by Fliufas, Circles hane the one to the other, that proportion,that like Poligonon figures and in like fort de- {cribed in them haue, For, it was by the firft Propofition proued,that the Poligonon figures haue that proportié the one to the other,that the {quares of the diameters haue,which proportion likewife , by this Propofition the.circles haue. | — | gen needefull Problemes and Corollaryes by Master Ihon Dee | inuented ; whofe wonderfull vfealfo, he partely declareth. 4A Probleme. tr. T wo circles being geue: to finde two right lines, which haue the fame proportion,one to the other, that the geuen circles bane, one tothe other, . sal Suppofe A and B, to be the diameters of two circles geuen: I fay that two right lines are to be foiide hauing that proportié,that the circle of A hath to the circle of B,Let to A & B(by the 11 of the fixth) a third proportionall line be found,which fuppofe to. be C. 1 fay now that A hath to C,that pro- portion which the circle of A hath to the cir- cle of B. For forafmuch as A,B, and C, are (by 4 te conftruction) three Rear oo lines, the {quare of A isto the fquare of B,as Anis to C, Bons (by the Corollary of the 20. of the fixth)*but 7 as the {quare of the line Ais to the fquare of © —enrenens the line B,fois the circle whofe diameter is the line A,to the circle whofe diameter is the line . B, by this fecond of the eleuéth. Wherfore the circles of the lines A andB, are in the proportion ofthe right lines AandC. Therefore two circles being geuen,, we haue found two right lines hauing the fame proportion betwene thé, that the circles geuen, haue one to the other: which ought to be done. A Problemes: 2. T wo circles being genen , and a right line:to findean other right line, to which the line geué {hall haue that proportion, Which the one circle bath to the other, Suppofe two circles geué: which let be A & B, &aright line geué, which let be C: Lfay-thatan, other tightline is to be founde,to which the line C fhall haue that proportion that the circle A, hathto the circleB. As the diameter of the circle A, is to the dia-~ . meter of thecircle B, fo lettheline ~ C be to afourthline, (bythe 12-0f . .. . | the fixth’) !er that fourth line beD. Anid,’by the 11, of the fixth,let a thirde line proportionall be found, to the lines C & D, which let be E: I fay now, that the line C hath to. the line E, that proportion which the circle A, hath to the... circle B. For (by conitruétion) the lines C,D, and E, are. | q proportional] : therefore the fquare of C, is to the {quare : | of D, asC is to E,by the Corollary of the 20, of the heh, But by conftruction , as the diameter of the circle A, is to the diaméter of the circle B, fo is C, to D: wherefore as the fquare of the diameter ofthe circle A, is to the. {quare of thediameter of the circle B, fois the {quare of the line C to the fquare of the line D, by the 22. of the z fixth. But as the {quare of the diameter of A, the circle, is to the fquare of the diameter of thé circle B, fois the cir- : sack 34 cle A, to the circle B, by the fecond of the twelfth: wherefore by rr. of the fiueth, as the circle - is to Pe st CCc.ij. the D Couftruttion, Demonfira- tion, ConsiruGlion Demonfira- t30M%e ' ay % oa GaP) { ota obey Nhele i iit ala cia Pat t ] ith | $8) Pek ie) ya EG Bis hide " ' $e) Bai ul vane ’ rare tit ! ’ . : t el a | 2 En \ - = r ae | a | > : | Ag 0 ’ t, ; : oe % oi f a Diffreence betwene the fir probleme and the fe- cond, Conffruciion. Deuten ftra- $ 26 Ne Confiruttion. Demonftra- $400. "Lhe rwelueth Booke the circ fore, anda right line, we haue foun which the one circle hath to theo Note. The difference betwene this Probleme, and that next before, is this: there,although we had two circles geuen, and two lines were found in that proportion the one to the other, in which the geuen circles weres: and here likewife are two circles geué,. and two.lines alfo are had inthe fame Proportié, that the geué circles are:yet there we tooke at pleature the firft of the twolines, wherunto we framed the fecond proportionally, to the circles geuen. But here the firlt ofthe two lines, is afligned poynted, and determined to ys: and not our choyfe to be had therein, as wasin the former Probleme. A Probleme. 3. ef circle being geuen, to finde an other circle, to which the &euen circle ss in any proportion geué ba tworighe lines, : Suppolethe circle « 8 cgenen, and therefore his femidiameter isgeuen: whereby his diameter alfo is geuen: which diameter let be a c. Let the proportion geuen, be that which is betwene z »F two right lines«I fay, acircle is to be found, Vito which az c hath that proportion thatz hath to r. As £ isto F, fo let ac the diameter, be to an o- ther rignt fline, by the 12. of the fixth . Which line lipeteec be x. Betwene ac and H, finde a middle . proportionall line, by the r3. of the fixth: which let be LN. By the ro. of the firft, deuide x N, Into two equall [4 partes: and let that be done in the point o, Now vpon |—~~ o L, (o being made the center) defcribe a citcle:which \" ler. be tan, lfaythatarnc,istorm N,asE isto F.For feing thata cjx~,and Hare three right linestin'con- tinuall proportion (by conttruction) therefore (by the Corollary of the 20. of the fixth) as a c isto H, fo is the {quare of a c tothe fquare oft x. But a cis to H, as B is to F, by conftruétion. Wherefore the {quare of a c is to the {quareofin,assise: butasthe {quare of the nf diameter 4 c, is to the {quare of the diameter 1 N, fois ponenenincmeoose the circle as c to the citcle rm x » by this 2. of the twelueth, wherefore by the 11. of the fiueth, the circle 48 Cistothecircle tM n,as& istor. A circle bein geuen (therefore) an other circle is founde, to whic the geen circle is in any proportion geuen betwene two right liness which ought to be done. 4 Probleme. | be T wo circles being genen, to finde one circle equall to them both. = Suppofe the two circles gené,haue their diameters as. 81 ¢ p. I fay thata circle muft be found equall to the two circles whofe diameters are‘a & and c p: ynto the | | line 4B, atthe point a, ereét a perpendicular line a #+ from which (fufficieutly produced) cut a line equall to c 5, which Jet be a F. B the firft peticion draw from F to ra right line: fois F a 8 made a tri. angle rectangle. I fay now thata circle whofe diameter is F x, is equal to the two circles whofe diameters are a p and Fol.360, % A Gorollaryen st. 0s (SHO A Ages ales ed; Hereby it is taade euidént,thar in all triangles rettangle, the circles , Jemitircles ; ious 3 OF any other portions of circles deferibed Upon the fubtendent line. ,isequall tothe two cire Cs femicircles,. » quadrdhts or any two other lke portions of circles, defcribed on the two lines comprehendingshe right angle,like to like beizg compared. \.. dw sw paid daid Ww. Lega) 3 For likepartes haue that proportion betwenethemfeluessthat their whole‘magnitudes haue , of which they arelike partes;by the r5-of the fifth + Burofthe wholecireles; in the former problenie itis , “ euident:and therefore in the fornamed like portions ofcircl:s;itisacrid confequent.: | |) tails: % A Corellary. 2, . By the former probleme, itis ale manifefr ,untdicircles three one Will geue,one citcle may be genen equal, | SHIBIEQ Sih: Sower, fine sor to bow many foener:: For iffirft, toany two’, by th e former probleme ; youfirde one equall;and then‘vrito your found” circle and the third of the geen circles,as two geuen circles; finde one other circle‘équall, and then to that fecond fountdicircle, and eoxrhe fourth of the fit geuen circles 5 as two circles, one new-circle be? found equalljawdfo proceéde tilfyou haue oncecuppled Giderly, eucty One of your propotided’circles’ (except the ifirft and fécond already doone) with the new €ircle thus found*for fo the laitfound circle is equall to all the firitgeuen circlés:Ifye doubtsorfificiently Viiderttand me riot: helve your felfe’by the . difcoutrfe anddemonftration of the lat prapofidon in the fecond booké;and alfo ofthe 31. inthe fixth- bookew suse Q 99 2lo1n ow! moG OT ILI ep 4 Probleme? 3." . : ote bine, paequall circles being gchenstofinde a circleequallto the exceffe of the greater tothe leffe. fea cae vnequal circles geué, to bé AB C&DEF,& let ABC bethe greater: whofe dia- meter fuppefeto be A.C: & the diameter of DE F fuppofeto beD F.1 fay acircle.muft be found equal to that excelfein magnitude,by which ABC is greater thé D E F.By the firftof the fourth,in the circle ABC, Apply a right line equall to D F: rritesaB by. 8 : whofe one end let be at C,and the other | . let be atB. Frd BtovA draw ‘a rightline;*” By the 30.of che third! it may appeares\s.s. sus “A that A B.C a eM thereby ss or fof er ABC, us triangle isredtangled:. wher. >. isis fore Ede aoe two corollaries, A Ve here before,the circle A B Cisequall to TD F the circle DE F, (For BC by con{trudti-._ on isequall to DF ) and more ouer to. the circle whofe diameter is AB “That. circle therefore whofe diameter is A B, ett ian is the circle conteyning the magnitude,by which ABC isgreater then’D EF. Wherefore two ynequal circles being geuen, we haue founda circle equall to the excefle of the greater to the lelle: which ought to be doone. . A Probleme. 6. : | dA Circle being geuen to finde He Circkes equal 3 tothe fame : which found Circles shall haue the. one to the othtr,any proportion geuen in twaright lines, Suppofe.A B C ,acircle geuentand the proportion — geuen,let it be that,which is betwene the two right lines, Diand E.I fay , that two circles are to be found’equall to™ AB C:and with al,one to the other, in thé proportié of A Dito E. Let the diameter of ABC be AC. As DastoE, fo let A C be deuided, by the 10.0f the fixth,in the poyne F.AtF,totheline AC let a perpédicular be drawne FB, and let it mete the circiiferéce at the poynt B. From the poynt B tothe points A and C,let rightlines be drawne: BA and BC .1 fay that the circles whofe diametes are ro Hines A act Care equall to the citcle A B C:and phat thole circles hauing to their diameters thelinesB A _. “Penary: and BC are oneto ihe other in the proportion of the Pye line D to the lineE . For, firft that they are equal, it is e- i Conflruction, ~~ Confirntlions Demonflraa i eee * \» Sa ee T hetwelueth Booke ib uident:by reafon that AB C isa triangle rectangle: wher) \-.. fore by the 47?of the firlt the fquares of B A, and BC are B equail to the fquare of AC x And fo by this fecond itis...» | manifeft, the two circles to be equall to the circle ABC. . Secondly'as Dis to E; fois AF to FC > by conftruction, > Andasthplne AF is 00 the line F Cfo isthe [aware of the ine 2» BAtothe {quare of the line BC. [. Which thing, we will - biiefely -proue thussicThe parallelogramme: contayned 4 mie er: vader AC and: AsByis equall to the fyuare of BiArby the 4* well: for s Lemma after the 32,0fthe tenthibooke:and by thefame«, #8 of great Lemma or Afflumpt,the parallelogramme contayned vn- ve. der AC and FC, is equall to the fquare_of the line B Ca, 22 Wherfore as the firft parallelogramme hath it felfe'to the fecond:fo hath the fquare of B A { equall to the firit pa- _ ralielogramme )it felfe,to the fquare, of B. C,equalktothe » \\\ TOT fecond parallelogramme. But both theparallelogrames ‘ioppy ses" Fr aso, vary § haye-one heigth, namely,the line A.C 3),and bales »the ytrrts yh POT 201 lines AF and FC :.whereforeas A.RistoyE C,,fodsthe o,, parallelogramme contayned vnder AC, AF » tothe parallelogramme contayned ynderA C; oF. C,by the frit of the fixth. And therefore.as.A.F. isto F C,fo.isthefquare of BA to the {quatre of BC.) And asthe {quare of B Aas to the {guiare of B.C: fo is the circle whofe diameter is B A,tothe circle whofe diameter is.B,C,by.this fecond of the, rwelfthy.. Wherefore the circle whofe diameter is B A,is\to the circle whofe diameter isB C.,as Dus to E .. And before we. proved shem equallitethe circle) AB C. Wherefore a circle being geuen, we haue found two circles equal! to the fame : which haue the one to the other any proportion geuen in two rightlines.W hichought to bedone. N ote. | An other way Here may you perceiue an other way how to execute my firft probleme, for if you make a right an- of démonftretia glecontcyned of the diameters geue, asin this figure fuppofe them B A and B C:and then fubtend the ofehefietires, rightaigle with the line A C:and'from the righrangle, let falla line perpendicular to the bafe AC: bleme of this that perpendicular dt the point of his fall ;deuideth’A C into A F and FC, of the proportion required. | addition. : er ie aE eT nee ih oe cdacit «ol ee 29 ) oA Corollary, 7°’ ijt ES : « It followeth of thinges manifeftly proued in the demonftration of this probleme, that in atriana Neardhivprost gle rettangletf fromthe right angle tothe bafe , a perpendicular be-det fall :' tbe fame perpendicular pertie of a tries cutteththe bafe into two partes, in that proportion ,one to the.other; that the fquares of the right anglerectangle, linestontcyning the right angle sare in ,one to the other: thofe on the one’ fide the perpendicular , being compared tothofe omshe other bor [quar and fegment. (: Eeanaeerede ; se A Probleme 7. - Betwene two circles geuen,to finde a circle middell proportional. . Let the two eircles geuen,be Ar AC DandBEF:I fay,thatacir-__ Pr Lae cle is to be fottd which berwene A CD and BE Fis middell pro- Conftrudson, — portionall. Let the diameter of ACD, be AD, and of BEF, let BE bethe diameter: betwene A+ D andBF, finde a line. middell proportionall , by the 13.0f the ~~. fixth: which le¢be H K:I fay that’. ~ acircle,whole diametéris HKis — middell proportienall betwene ACDandBEF.ToAD,HK, and B F, (three right lines in con . tinual{propertion:; by conftruc- tion) let'a fourth line be found: to whichiB F fhal haue that pro- portion , that A D.hath to H K: by the 12.0f.the fikth, && let that . Demonftra- » linebeL . Jt ssmanifef that the sow. 9 fower lines AD ,HK,BF, andl, » ~aresn continual proportion. [ For »> by cdftrnctionyasA-DistoH K, * » foisB FtoL. And byconftruai- - = — ~ —S ee Een -—- - + - ee i — im gl oo a Se os = - at — = — —-- _ = = —— = aon = =-. = = —= ° * — - =e. - —_—--- est . > = = => ’ ? = = - me =a — — ———— —_ — _~ — — a ~ - aon = — — = - - - = Z , = = —— . - - eeneemenente . a - ee AO eee - .- ~~ --; - - = = ri = te a ee ——s = ——— = : : = a ee LS] ae =< stan —~eenwss es . - oe ——e — == = > as: = = > a — -- = — 7 —= J ¥ > Ed 4 ex a, Ge ~ eo TN > — : 7 vom 5 5 —— x< * Ss . ~ *: P's — Fo ine: = ‘= += ; == eS et ae ee SE SS SS . = ~ mm sy ——— _ - - =— ~~ - —— =~ —, — cane JS & « : : 4 2 ae og eS SS 2S eee aay Say - = — ase a ee a ee : of Euchides Elementess\ »\ © Fol.361. on,as A Disto HK, fo is H KetoB Fi wherefore H K isto BFjasB Bis to L:bythe trof che fifth,wher» cs fore the 4,lines are in:continualk proportion. }: Wherefore asthefirfbisto the third that is A D to B FE, fo isthe fquateof the firlt to the nage the fecond:thatis;the fquare of A D;torthe {quare'of H K: by thecorollary dfthe'z0.of the. fixth:And:by:the fame corollary: ;ias H'K-is to L, fo is the fquare of H K to the {quare of BE, Bue by alternate proportion the line A DistoB:Fjas K.is to’L: wherefore the fquare of ADbis to the fquare of H.K;asthe {quare of H K is tothe fquareofB F.W hereforé the iquare of H K; is middell proportionail,betwene the fquare of A D and the {quare of B F.Butas the fquares are ond to the other;fo are the'circles(whoft diameters produce the fame {quares one to thé other, by this fecod of thetwelfth: wherfore the.cirele whofe diameter isthe lime HiKzis middel propo rtionialjbetwene the circles whofe! diameters are the lines A D and B-F .. Wherefore betwene two circles cenen. ; we haue found a circle middell proportionall : which was requifite to be doone. ty q Corollary. | Hereby it4s moanifest, threelines or more being in.continuall proportion; that the circles hauing thofe tines to thesr diameters;are alfoin continuall proportion. | Asof three, our demonitration hath already proued|: fo of fower, will the proufe go forward: if youaddea fifth line in continuall proportion to the fower geuen : as we did tothe three, adde the fourth : namely, the line L. And fo, if youhaue 6,.by putting to.on¢ more, the demonitration will be ealic and plaine . And fo of asihanyasyouwill., “a ol guvtsen Sed Probleme. 8.0» Toacirele being genen, to findethree circles equal: which three circles foall be ip continuall proportion, in any proportion geuen betwene two right lines. . 3 ogy Suppofe the circle geuen to be 4 #C: and the proportion geuen to be that which is betwene the lines Xand r . I fay, that three circles are to. begetén, which three,together, fhall be equall tothe circle .4 BC : and withall in continuall proportion,in the fame proportion which is betwene the right lines Xand7,, Letthe diameterof48 C, beac. Of A C, makea {quare : by the 46.of the firft: which ~ whichletbée“4cD E . From’ the’ point D drawe a line, fufficiently long ( any way, . * without the fquare ) : which let be D9, At the point D, and from the line DO, cut aline equallto x: which letbe Dw. At thepoint ,and from the line 470, cuta line equall to r : which let be a4. At the point N,to the twolines DM and MN, feta third line proportionall; by the 12, of the fixt : whichlet be YO. From E( one ofthe angles of the fquare 4C DE, next to D.) draw arightlinetoO: making per- feéte the triangle DEO . Now fromithe pointes and N, drawe lines, to the fide D E, parallel to the fide £0: by the 31. of the firft: which let be 44 Fand N G.Wher- fore, by the 2. of the fixt, the fide D E,is proportionally cutin the pointes F and G, as D O is cut inthe pointes Mand N : ther- fore,as DM isto MN,fois DF ito FG; and as MN is to NO, fois FG to GE, Wherefore, feing{D MM N, and N O,are, by conftruction,continually proportioned, in the proportion of Xto TY: Solikewife, o are DF, FG,and’G £, in continuall propor- = +——»+_____, tion, in the’ proportion of X to T, by the 'T 11.0f the fift.. From the pointes F and G;to the oppofite fide 4C, let right linesbe —HH___ - drawen parallel to the other fides « which 302 | lines,fuppofeto be F /,and G XK : making thereby, three parallelogrammes Dz; F K;andG C, equall to the whole {quare 4¢ DE, Which three parallelogrammes,by the firft of the fixt;are one-to an other, as their bafes, D F, FG, and G,£, are . But D F, FG, and GE, were proued tobe in continuall propor- tion, in theproportion of x to r : Wherefore, the three parallelosrammes D7, XK,/and GC, by the r1.of the fifth, are alfo in continual proportion, and in the fame, which Xisin, to? . Let three {quares be made,equall to the three parallelogrammes D /,F K,and GC: by the lat of thefecond : Let the fides of thofe {quares be, orderly, 8, 7, and . Forafmuch as, it was laft concluded that the three paral- CC ean). lelogram nes, D Qo fT * Though 1 fa; without the {quare, yet you muft thinke, that st ma 2y be allo within the fquare: €5 that dinerfly. Wher fore this Pro- bleme may haue dsucrfe cafes.fo, but briefly to. aa Goyite all nay? thus be faid:cut any fide of that foétre into 3. parts sn the pre portion of X to T+ Note the maner of the drift in this demonftraa tion and con - firudtionmixt- ly: and with no determination tothe consTruc- 10248 commoly 4s i preblemets which sshere of me fo Gfed.fer an example te _young frudéetes of Garsety i ark, ee Vay — ae ns a oe = 4 ' q , | h bi | a —s oe als = Bien en nn X ann — ee © = eS —=—-— Canfiructiot» > 4%a3 Demonjiras'. tion. T he twelueth Booke iclogrammes, DU, F K,and GC (which are equal! to the fquareia' C' De) are alfolihicontifudll pro= portion,jin the proportion of Xto Ttherefore their equallspnamely; the three fquares:of 5) 7 é v are alfo equall.to the whole fquare:s4¢ D £,;and in continuall proportion, in the proportion of x to r. Wherefore the three circles,whof diameters are S$, T, andi, are éequallto the circle, whofe diameter is AC, thefide of the {quare 4 CDE; and alfo in. continuall proportionyin the proportion of xX tor: by thisfecond of the twelfth . But, by conftruction, 4C is the diameter of the circle 4B .C ) Wherefore we hane found three circles, equall- to 48 C.: namely, thecircle, whofe diameter is Sand the circle whofe diameter is 72, and the circle, whofe diameter is ¥: which three: circles, alfo, are in continual preportion, in the proportion of Xto7T . Whereforeto a circle being geuen,we haue found three: cir- cles equall in any proportion ,geuea,betwene two right lines : which was requifite to be done #9! yi quic orollary. Hereby, tis enident, that aciclegenen, Wwe may finde circles4,'5, 6, £0;:20, 100; £000, or how many foener {hall be appointed, beingsm¢ontinuall proportion, in ary proportion, ceuen berivene tive right lines : Which circles, alltagether ; hall beequallto the circle genen. 1 nails lowe f __For,euermore deuiding thé oite fide of the chiefe fquare'( which'is made of thé diameter of the circle geuen ) into fo many partes,as circles are to be made! {6 that betwene thofe partés be continued the Proportion geuen betwene two right lines * and from the pointes of thofe diuifions, drawe paral- lels, perpendiculars to the other fide of the faid:chiefe {quare;+ making fo many parallelogrammes of the chiefe {quare » as are circles to be made : and tothofe parallelogrammes ( orderly ) making equall pe sats maiitelt that the fides of thofe'lquares; are the diameters of the circles required to be made . 7 — 7 gp A Probleme: 9i | | T bree circles being genuen, to finde three equall to them: which three found circles {halle in cone tinuall proportion, in any proportion geutn-betwene two rightlinese. 0) see oxi gp sya Suppofe the three circles geuen, to be 4,B,and C, and let the proportion geuen, be, that which is betwene the right lines ¥ & Y.Tfay, three other circles areto be found, equallto 4,8, and’, & with all, in continuall proportion, inthe oe ee of. X to I. By the 2.Corollary of.my 4. Probleme,make one circle e- quall'to the three circles 4,2, and C. Which one circle fup- pofeto be D: Arid by the pro- bleme next before, let'three circles be found, equall to D, and with all, in ¢otinuall pro- ‘portion, in the fame propor- tionwhich is bétwene-Xand Y. Which thfee circles, fup- pofe to be E, F, and ‘G.I fay, that £,F,G, are equalfto 4,8, c: and with all, in continuall proportion, in the proporti6é of Xto r. For, by coftruction, the circle D is equall to the circles 4,B ;82.Czandby con- ftruction likewife, the circles E, F, and G, are:equalbtothe. fame circle Ds: Wherfore the three circles £; ¥, &G, are e- quall to the three circles4,B, & Czand by conftruction £,F, of Euclides Elementes, — Fol.362, and G, are in. continual! proportion, in the proportion of the line-X to the line 7. Wherefore E, Fi and .G,are equall to .4, B, and ¢ : and in continual] proportion, in the proportion of X to x’. Three-circles, therefore, being geuen, we haue found three circles ,equall to them,and alfo in continual] proportion, _ in any proportion geuen, betwen two right lines . Which was requifite to be done, Gj} 4 Corollary. 5. It us hereby very manifest, that unto 4-5.6.10.20. 100. or how many circles foener, hhall be £eHen We may finde 3.4.5.8.10. or how many foeuer, hall be appointed : Which all together hall be equall tothe circles geuen, bow many foeuer they are: and with all,our found circles,to be i? continu» all proportion,in any proportion affigned betwene typo right lines genen. For, enermore, by the Corollary of the4 -Probleme, reduce al! yourcircles to one: atid bythe Co- rollary of my 8.Probleme, make as many circles.as you are appointed, equall to the circles geuen, and continual! in proportion, in the fame,wherein, the two right lines geuen,are : And {fo haue you perfor. med, the thing required, Note, Whatincredible fruite in the Science of proportions may hereby grow, no mans tounge can fufhi- ciently exprefle. And fory Lam,that veterly leyfure is taken from me, lomewhat to {pecifie in particu- lar hereof. | The key of one of the chiefe treafure houfes, belonging to the State eu athematicall. Hat,which in thefe 9 -Problemes,is {aid ofcircles; is much more fayd of {quares,by whofe meanes, circles,are thus handled . And therefore feing to all-Polygonon right lined figures,equall fquares may be made, by the laft of the fecond:and contrariwife,to any fquare,a right lined figure may be made equall; and withall, like to any right lined figure geuen, by the 2¥.of the fixt . And fourthly, feing ypon the faid plaine figures,as vp6 bafes,may Prifmes, Paralleli pipedons, Pyramids, fided Columnes;Cones, and Cylinders,be reared : which being * allofone height, fhali haue thar proportion,one to the o- ther, that their bafes haue, one to the other . And fiftly, feing Spheres, Cones,and Cylinders are one to other in certaine knowen Proportions: and fo may be made; one to the other in any proportion af- figned . And fixtly,feing vnder cuery one of the kindes of figures; both plaine,and folide, infinite cafés may chaunce, by the ayde of thefe Problemes,to be foluted and executed: How infinite ( then )vpon infinite, is the number of practifes, either Mathematical, or Mechanicall, to be performed ; of compa- rifons betwene diuers kindes,of plaines to plaines,and folides to folides> Farthermore,to {peake of playne fuperficiall figuresin refpect of the contétor Area of the circle, fundry mixt line figures,Anularand Lunular-figures : and'alfo of circles to he geuen equall to the fayd vnufed figures: and in all proportions els : and evermore thinking of folides, (like high) fervpona- hy of thofe vnufed figires,(O Lord )in céfideration of al the premiffes how infinite, how ftraunge and incredible fpeculations and practifes;may (by theayde'and dire@ion of thefe few problemes), fall redie ly into the imaginations and handes of them, that will bring their miride and.intent wholy and fixedly to fuch mathematical difcourfes? In thefe Elementes., Ientend but to geuc to young beginners fome light,atde,and courage to exercife their owne witts,and talent, in this moft pleafant and profitable {ci- ence.Allthinges may not,neither yet can , in euery place be fayd . Opportunstie, and Sufficiency bef? are to be allowed. . q Lhe.3. T heoreme. The. 3. Propofition. Euery Pyramis haning a triangle to his bale: may be deuided into two Pys ramids equall and like the one to the other ,and alfo like to the whole, hase Hing alfo triangles to their bafes, and into two. equall prifmes: and thofe to prifmes are greater then the halfe of the whole P -yramis: | V ppofe that there be a Pyramis, whofe bafe let be the triangle ABC ,and his toppe the point D . Then I [495 that the Pyramis A BCD may be deuided into 100 Pyramids equall and like the one ta the other, and alfo liketothe whole,ha- uing alfa triangles to their bafes, and into two equal prilmes; and fee prilmes I ” 2? : 8 4 * Note and $+ remember one heith in 6, thefe fo- lids Py 8. — > a Lee mag. ent oe e a = = ems ee —s. = Confiruction, Demonstratt~ or of the firft partnamely, that the whole Pyra- words 38 deut- 1a The tweluetbBooke. . : prifmes are.greater then the halfe.of the whole Pyramis...Dewide (by the 10-0f, the firft) the lines. ABB G,C A;A D,BDs@>C D, into two equall partes in the pointes EF ,G,H Kana LAnihdrawe theferight inesEH,EG,GH,HK,KL,LH,EK,KF, and F G. Now forafmuch asthe line AE tsequall to the line EB, and the line AH tothe lineH D : therfore (by the 2.0f the fixt ) the line E H w a parallel to the line D B. And by the fame rea{on, the line - HK is a parallel to the lige,d B.. Wherefore ded into tive Pyramise- guall and like the one to the other, cud alfo to the whole, c& bausng trt angles to thee be &Se rm er A ; —— = = 7 7 . ES ne me ee a ee HE KB.is aparallelogramme ». Wherefore the line HK is equall to the line BB. But the line E B isequall tothe line AEs Wherefore the line A Eisequail to the line TK. And the line AH. is equal to thé line HD’. Now therfore there are two lines AE and AH, equall to two lines K H and HD, the one to the other, and theanele: EA His (by the 29.0f thefix(t)-equall to the an- gle KH Di Wherefore (by thes:of the first) the’ | bale EH is equall tothe bale K D .Wherefore the triangle A E His equall and like tothe triangle H K D.. And by the fame reafon alfothetriangle AH.G, 1 équall and like to the triangle H LD. And forafmuch as tworight lines E H cy H G touching the one the other, are parallels to two right lines K D and D L touching alfo the one the other, and not being in one and the felfe [ase plaine{uperficies with the two fir/t : thofe lines (by the r0.of the cle- senth) containcequallancles .Wherefore the angle E H Gisequall to the angle K D L.And fora{much astwo right lines EH and H G,are equallto two right lines KD ¢ DP L,the one ta the other, and the angle EH Gis ( by the 10.0f the elenenth) equall-tothe angle K D L, therefore (by the-giof the firft) thé bafe E Gis equail to the bafe L K . Wherefore the trian- gle BEG istqualland liketo the triangle K D L,, Aud by the fame reafon alfo the triangle AEG seqnall and like to the triangle HK L ..Wherefore the Pyramis,whofe bafeds the tré- angle AE Canad toppe the point H; is equall and leke to the Pyramis whofe bafe is the trian- gle HK Lyand toppe the point. D. eter er ctv *, ne sy And foralmuch as to one of the fides of the triangle AD B, namely,to the fide A Bw ’ drawen.a parallelline HK, thereforethe whole triangle A.D B is equiangle to the triangle DEK; and thar fidesare proportional ( by the Corellary 0 the 2.0f the fixt.),. Wherefore the triangle A D B is like tothe triangle .D.H.K.,.And bythe fame reafonalfo the triangle PBC us like tothe triangleD K Land the triahele ADC to the triangleDH L. And forafmuch as twovielt lines B A and AC touching the one the other are parallels to two right lines K Hand HL, touching alfo the one the other; butmot being in one and the felfe fame fuperficies with the two firft lines, therefore (by the 10.0f the eleuenth) they contaime €- quall angles .Wherefore the angle BAC is equall totheangleK HL. And as thelineBA isto the line AC, fou theline K H tothe line H Lv Wherefore the triangle A B Cu like to thetriangle K H L.Wherfore the whole Pyramis whofe bafe is the triangle AB Ce top the point'D’; 7s like to the pyrangis whole-bafe is the triangle H K L, and toppethepoimt D . But the pryamis whofe balers thetriangle H K L and toppe the point Dis proued to be like to the pyramis whofebafeis the triangle A EG and toppe the poynt H . Wherefore alfo the pyramis whofe bale is the triangle AB C,and toppe the poynt Dis like to thepyramis whofe bafe is the The conclufi- triangle AE G and toppe the poynt H (by the 21.0f the fixth) .Wherefore either of thefe py- on gad a ramias AEGHandHKL DistiketothewholepyramisABCD. . , Ete . Demonflratio of the fecond part, namely, ememe = —E arms —— op = ee RS eel - wee Se ee eens 7; any a — aa neo ss = wre weer eee =a - ~ we oe - es ae. Are — on Ses Pama oe re = ae a4 Si 2 oe = - a : - od r od . 2 2 / = a b..- - ae =< = — = See ees - ~-—_ - ee — > hes : ——————<—<—_——— a oe = : + pall ~ — —~--— —_—_------— -- ~ ——— a — 7 . = ——- _ - a — = —— ee — — apes aE = ” cg i i i TO LL AO LL A AS ~ rz = = =z : = ——=~ ~ - - = =< = = = = —_— = : x a. bof - EES — : a oes : : , ol a 9 ? - ————— 2 a See A OS. he _ 2 eet een eS a 2a; RITE ah ae se - CS : th i : ai if ay i 4o5 \ f, A ¢ P | , ou ; a> ta | a : ih TSE at ; - any Pea | ae ih 7 a] | 7 vu - ’ ’ t {is by iq ff ’ a } 1 ' ) is oo DEP ; 1 ; ; } 2 P : : ae — oP ae p i a if ce ' of he - 2s aes ie ‘ ie | 7 \ i vi i ‘ da, Ri : nt, Ve 1 a " f i : ‘ Mh ae J i ee : } : 4 ui i : ' a +, +i f P a | | t i / . F ’ bie narad it 2 ike ary i H : ee. my Gini We x the re) yy ‘ mt! : isa eH ' im t : at { vn } 4 } : j y ni } a ; ) ie : 5 ‘2 : i hia : : ; ‘ jie ; | ee | ' . b to ‘ } i | ; P “va D i ; Pare ———— = Se . [And forafmniich as the lines B E, K H,are aralleflines and equall, as it hath bene proued , there- fore the right linés:B. K and BH, which ioyne desastasether, are equall and parallels, by the 33.ofthe firit.. Againe forafmauch as thelines:B-E and: F Gyareparallellines and equall, therefore the lines E 7 ~ A 4 ; an a LT a ee Pyramids, whofe bales are the trianghs AGE, and HK of Eichides Elementes. Fol.363, and BF, which ioyne them together, are‘alfo equalland parallels’: and by the fame reafon forafnuch as the lines B G'and K H are equall parallels, the lines FK and GiH, which idyne them together,are alfo ¢yuall parallels. Wherefore BEH K;B E'G F,and KH GFyjare arallelogrammes . And forafmuch as their oppofite fides are equall; bythe 34.of the firtt : therefore the triangles EH G, & BK F,are equi- angle, by tle 8.of the firit : and therefore,by the 4.of the fame, they are equall : and moreotier, by the 15 .of the eleuenth, their fuperficiecesare parallels. Wherefore the folide BK PEH Gis a Prifme, by the 11.definition of the eleuenth.. Likewife forafmuch-as the fides of the triangle H K Lare equall and parallels to the fides of thé triangle G FC, asit hath before bene proued + It is manifeft, tharC EK L, FKHG,andCLHG, are parallelogrammes, by the 33. of the firlt. Wherefore the whole folide K L- HFC G, isa Prifme,by the. 11. definition of the eleuenth, and is contayned ynder the fayd parallelo- grammes CF KL,FKHG,and CLHG, and the two triangles H KL and GEC, which are oppofite and parallels. ] And fora{much as the line BF # eqnall to the line F C, therefore. ( by the 41. of the first ) the parallelogramme E BF G is double ta the triangle GF C. And forafmuch as if there be two Prifmes of equall altitudes, and the one haneto his bafeaparallelogranme,and the other a triangle, and if the parallelozramme be double ta the triangle, thofe Prifimes are (by the 40.0f the elenenth ) equall the one to the other : therefore the Prifine contained va- der the two triangles BK F and EH G,‘and wnder the three parallelogrammes E BF G, EBKH, and K HF G, is equallto the Prifme contained under the two triangles G F C, and H K L, and under the three parallelocrammes KF CL, LCGH,andH FFG. And it is manifeft, that both thefe Prifmes, of which the bafe of one is the parallelo- gramme EB F G, and the oppofite unto it the line K H,and the bale of the other is the tri- angle GF C, apd the oppofite fide untoit the triangle K LH, are greater then bath these 1Z, LL, and toppesthepomtes H ¢ D. For if wedrawe thee right likes BF and E K;the Prifime whofe bale is the parallelogramme E BF Gy and the oppofite vito it the right line AK, @ greater then the Pyramis whole bafe 15 the trianele E B-F ,¢ toppe the point K. But the Pyramis whofe bafeis the trianeleE BF, and toppe the point K, ws equallto the P yrame whofe bafe is the triangle AE Gand toppe the point H, for they-are contained under. equall and dike plaine fuperficieces. Wherefore alfo the Prifme whofe bafe ws the parallelogramme E BF G and the oppofite vuto it the right line HK, 0 greater then the Pyramais whofe bafeis the triangle.AE G andtoppe the point H. But the prifme whofe bafeis the parallelooramme EBF G,and the oppofite unto st the right line H E18 equal to the prifme,whofe bafe is the triangle GF C,and the oppofite fide vatoit the iriangle HK L: And the Pyramis whofe bales thetriangle AE G,and toppe the point H, is equallto the Pyramis,whofe bafe is thetrianeleH KF Liand toppe the point D. Wherefore the fore{aid two prifmes are greater then the forc{aidtmo Pyramids, whofe bales are the trt- angles AEG, HK L,and toppes the pointes Hand D». Wherefore. the whole Pyramis whofe bafe is the triangle ABC, and toppe the point D, is denided into tivo Pyramids ejuall and like the one to the other, and like allo unto the whole Pyramis, hauing allo triangles to their bafes,and into two equall prifmes, and the two pri{mes are greater then: halfe of the whole Pyramis: which was required to be demonstrated. . | If ye will with diligence reade thefe fower bookes following of Euclide, which concerne bodyes, and clearely {ee the demonftrations in them conteyned; it fhall be requifite for you when you come to any propofition, which concerneth a body or bodies;whether they be regular or not, firftto defcribe of palted paper (according as I taughtyou in the end of the definitions of the eleuenth booke)fuch a body or bodyes,as are there required,and hauing your body:, or bodyes thus defcribed , when you haue no- ted it with letters according to the figure fet forth vpo aplaine in the propofitid,follow the conitruGi- on required in the propofition.As for example,in this third propofitio ieis fayd that, Exery pyramis ha- using 4 triangle tobishale,may be deuidedintotwo pyranids.egc. Here firlt defcribe a pyramis of pafted paper haning his bafe triangled(it fkilleth not whether it be equilater,or equiangled,or not, only in this pro- pofition isrequired that the bafe be a trian gle. Fhen for that the propofition ngs one the bafe of the pytamisto be thetriangle a g.c,note the bafe of your pyramiswhich you haue:defcribed with the let- ters A B.C,and the toppe of your pyramis with the letter p.: For fois required in the propofition . And thus hgaue you your body ordered ready to the conitruction N6w in the conttruGion itis required that ye deuide thelines,a 8,8 €,¢ «.8csnamely,the fixe lines which are the fids of the fowet triangles con tayning that it ts denn ded moreoucr into two equal Prifmes. Conclufzon of the fecond pari, Demonflratio ofthe lafi part that the two Prifmes are - greater then the halfe of the whole Py~ YAMIS» Conclufion of the laft parte Conclufion of the whele propofition. The twelueth Booke> tayning the piramis, into two equall partes in the poyntet £,F, G, &¢ « Thatis,ye muft denide the line a 8 of yourpyramis into two equall partes,and note the poynt of the deuifion with theletter , and fo the line 8 c being deuided into two equall partes,note the nti of the deuifion with the lerter F. And fo the reft,and this order follow ye as touching the reft of the conftruction there put,and when ye haue finithed the con{truétionscompare your body thus defcribed with the demontftration: and it will make at very playne and eafy to be vnderitaded.. Whereas without fucha body defcribed of matter, itis hard \ _ for young beginners(vnleffe they haue a very deepe imagination fully to conceaue the demontftration by the figue:as.it is defcribed ina plaine.Here for the better declaration of that which I haue fayd,haue I {era figure, whofe forme if ye deferibe vpon pafted paper noted with the like letters,and cut the lines B A,D a,b Cand folde itaccordingly , it willmakea Pyramisdefcribed according to the conftruétion required in the propofition . And this order follow ye as touching all other propofitions which con- cerne bodyes. . aR | | ~~ = ——e x 3 = CE RS i = oS A fo 2. Oe Pe = —e — a SSS i oe — ae + — ro = eo ——— = ; —— - ——— = == : = : =e — 5 —— a en = — — — —————— en pes ESE ay ee — —_———— = : —— — — === = - ; Te ee SS - ——— id oe mein em ~ — a —— . = —— = = = SS : bs ~ = : . os - - -~ ens : 7 wa = - oon 2 ~ = o~ ee . — - 2 ; ee OE _— ~- —— ~ _————- - : - - > — ee —— te —— ; atetemnesemndtie teainn w es a ee ee ae — ———————— - - Sa mt <— == —— = = ? <5 ‘: c= : ver = . = or pas oom oe es - =~ = ———— - > = i. ee ag a = a y = am . a - ~ . ‘ . — > oe tee as ee ve . _ ~~ —— =. a as -- suse < re . ~e NSE eS i gs 3 Mi - . Rok Shee easy ee - " - ot mes —— ae — = ° . - = , A A — SN ee — - _—_- vy Another demonftration after Campane of the 3.propofition. * ear ~ ‘a - =< set ea ree es _. Suppofe that therebe a Pyramis 4 8.C D hauing to his bafe the triangle 2 C D, and let his top be the folide.angle.a:from which let there be drawne three fubtended lines 4 8,.4C, and.4Dto the three angles of the bafe, and deuide all the fides of the bafe into two equall partes in the three poyntes £,F,G: deuide alfo the three fubtéded lines 4 B,.4C, and D in two equall partes in the three points H,K,L.And draw mthe bafe theferwo lines # Fand EG: ge So fhall the bafe of the pyramis be denidedinto three fu- ‘perficieces: whereof two are the two triangles B £ F, and E G D,which are like both the one to the other,and alfo to the whole bafe;by the z part of the fecdd of the fixth,& by the definitié of like fuperficieces,8& they are equalthe one to the other, by the 8.ofthe firlt : the rhird f{uperficies isa uadrangled parallelogramme,namely, EF'GC : which is double to the triangle £ G D,by the go. and 41. of the firft. Now then agayne from the poynt # draw vato the points & and F thefe two fubtendentlines H £ and # F:drawalfo a {ubtended line from the poyntX to the poyntG . And draw thefe lines 4 X,K.L,and LAH. Wherefore the whole pyramis 4B C D is deuided into two pyramids; which are HBEF,and 4H KL, and into two prifmesof which the oneis EHFGKC,andis fetvpon the quadrangled bafe CFG E:theotherisE GDH KL, and hath to his hafe the triangle £ G D.Now as touching the two pyramids HB EF and.4 H K _L,that they are equall the one to the other, andalfo thatthey arelike both the one tothe o- ther and alfo to the whole, it is manifelt by the definition of equalland like bodyes,and by the ro.of the eleuenth,and by 2.part of the fecond of the fixth . And thar theewo Prifmes are equall it is manifeft by the laitoftheeleuenth. And now that both the prifmes taken together are greater then the halfe of the whole pyramis;hereby it is manifeft ,' for that either ofthem may be deuided into two pyramids, of which the one isa triangular pyramis equall to one of the two pyramids into which together with the two prifmes is deuided the whole yramis,and the other is a quadrangled pyramis double to the other pyramis. Wherefore itis playne Site thetwo prifeés taken together are three quarters of the whole pyramis of Euclides Elementes,. Fol.364. pyramus.dewided.Bnt if ye aredefirous to know the proportié betwen them,reade the 6.of this booke. But now here to this purpofeirthall be fufficient to know, that the two prifines taken together do ex- ceede- in quantity the two partial pyramid s,taken togcther,into which together with the two prifmes the whole pyramis was deuided. q Lhe 4. I heoreme. Lhe 4. Propofition. If there be two Pyramids ynder equal altitudes, haning triangles to their bafes , and either of thofe Pyramids be denidéd into tivo Pyramids equall the one to the other and like vnto the whole, and into two equall 'Prifmes, and againe if in either of the Pyramids made of the two firft Pyramids be fill obferued the fame order and maner : then as the bafe of the one. Pyrae mais 1s tothe bafe of the other Pyramis, fo are all the Prifmes which are in the one Pyramis to all the Prifmes which are inthe other > being equall in multitude with them, =~ (dl V ppofe that there be two Pyramids under equall altitudes, hauing triangles to ‘their bafes, namely, ABC, and DE F, and hauing to their toppes the pointes Gand H. And let either of thefe pyramids be diuided into two pyramids equall sat 3 ithe one to the other,and like unto the whole,and into two equall pri{mes (accor- ding to the methode of the former Propofition ). Anda caine, let either of thofe pyramids fo a Vi made of the two firit pyramids, be imagined to be after the [ame order denided,and [o docon- tinually .T hen! fay,that as the bafe A B Cis tothe bafeD EF, fo ave all the prifmes which are inthe pyramis ABCG,to all the pri{mes which are in the py- vamis DE F H being equallin _ multitude with them . For for- — Afmuch as the line BX ts equall _ tothe line X C, and the line-A L to thelineL C’: ( For as we [aw in the confiruction pertayning to the former Propojition, althe fixefides of the whole pyramids, arcech deuided into two equall parts, the like of which conftruc- tion is in this propofition alfo Suppofed) : therefore the line xX Lis a parallel to the line A Bc the triangle A BC, is like ta the triangle L XC, (by the Corolla-- 3 ry of the fecond of the fixthy: ~~’ and by the fame reafon the tri ~* F P : D angle D E F is like tothe trian@leRW F.And forafmuch as the line BC is double tothe line CX,and the line F E to the line F W :therefore as the line BC is to the line C X, [ois the line EF tothe line FW. And upon the lines B Cand CX ure defcribed rectiline figures ike and in like fort [ets namely the triancles ABC and LXC, and-vpon the lines EF and F Ware alfo defcribed rectilinefigutes,like and in like [ort fet ,namely,the triangles D E F é RW F But if there be fower right lines proportionall, the rectiline figures defcribed of them being like und in like fort fet, hall alfo be proportional ( by the 22.0f the fixt ) . Wherefore as the “4 DDadu. triangle 7 Ts = <= = nel — SS oS 3 Se ae eee oe = med ee Oy me - c—s “ - =—- — eg ~ - “= - - ~ in — Sema - ~~ ——| ua x re Sa eS. ~ See be —— a oma __ > : 4 7 i Re ' View re ee rf ¢ f | i ‘ Fi ro = Baths 0k TS Oh iS oo ae ee ei “- = wa eo i ~ ++ ~ ———-S —— - ier Pita ea iia SE 9 NN A Te ee SS >, RM eeer al Pe} ve - S es oe. [25 of An Affupte T he twelueth Booke triangle ABC 48 tothe triangle LXC, fo is the triatgle D E F to the triangle RW F. Wherefore alternately ( by the 16.of the fift ) as the tringle A B Cis to the triangle D E F; fois the triangle LX Cto the triangle RW F .* But asthe triangle L X Cis to the triangle RW EF, fois the prifme whofe bafe ss the triangle L X ©, and the oppofite fide unto it the tri- angle 0 MN, to the prifme whofe bafeis the triangle 3W F, and the oppofite fide unto it the triangle STV ( by the Corollary of the go.uf the eluenth). F or thefepri{mes are under one & the felfe [ame altitude, namely, vader the halfe f the altitude of the whole Pyramids, which Pyramids are{uppofed to be under one and the {fe fame altitude : this is alfo proued in the A{jumpt following ). Wherefore (by the 11 ofthe fift ) as the triangle ABC iste the triangle DEF, fo is the . prifeme whofe bafe is the triangle LX C,and the oppofite fideva- toitthe triangle-O.M N, to the prifime whofe bafe 1s the traan- gle RW F, and the oppofite fide wnte it the triangleST V. And forafmuch as there are two- pri{mes in the pyramis A BCG equal the one to the other cy two prifimes allo in the pyramis D E- FH equall the one to the other: therefore as the prifme , whofe bafeis the parallelograme B K- LX, andthe oppofite fide vate it the line M O,is to the prifr WE, whofe bafe is the triangle LX C, and the oppofite fide unto it the triangle OCMN , fo is the ; prifme, whofe bafeis the parallelogramme P E RW, and the oppofite unto tt the line ST, te me, whofe bafe is the triangle RW F Ana the opepis fide vnto it the triangle ST V. Wher fore by copofition( by the 18 .of the fift) as the primes K BX L M0,¢LXCMNO, are to the pri{me LXCMNO,foare the prif{mes PEWRST, and RWFESTF¥, tothe prifme RW F STV .Wherefore alternately (by the 15.of the fift ) as the two prifmes K B- XLMO,andLXCMNO, are tothe two prifmes?EWRST, and RWFSTV,fow _ theprifme LX C MN 0 to the prifme RW F STV .but as the prime LXC MNO isto the prifme RW F STV, fo baue we proned that the bife L X Cis to the bale RW F and the bale ABC tothe bafe D EF . Wherefore (by the 16.0f the fift ) as the triangle ABC is to, the triangle D E F , {o are both the prifmes which are in the pyramts 4 BC G, to both the pri{mes which are in the pyramis DE F H . Andinike fort if we diuide the other pyr.smids after the felfe fame maner, namely, the pyramis OM NG, and the pyramis STV H : as the bafe OM N_is to the bafe STV, fo fall boththe prifmes that are a the pyramis OM NG, be to both the prifmes which are in th pyramis STV H.. Butas the bafe O MN jstothe bafe ST ¥, fois thebafe. ABC ‘othe bafe DEF. Wherefore (by the it-of the fift ) as the bafe ABC is tothe bafe D EF, fo are the two prifmes that arein the pyramis AB CG, tothe two.prifmes that are.in the pyramis DEF H, and the two prifmes that are in the pyramis'O M NG, to the twa prifmes that are in the pyramis $ T- V H, and the fower pri{mes tothe fower prifimes.ad foalfo fhall it followe in the pri{mes wade by diuiding 5 two pyramids AK LO, and D P RS, and of all the other pyramids in gencrall, being equallin multitude. ~ men, eens in NS Wee ee pene a o> OS | | ‘\ * Flerebyit is manifest that enery pyramis is thewbird part of a prifme bax “Aid i ifthe adage sf ie bite wth it For if the bale of the prefine beany other recéiline figure the -- — — ee ? = = t « #Y i] + i fi ti ey 5. i} | Le .: fie $/ ‘me eal i ‘ f Tayi. ny 4 A . : a ; 4 ; 5 : ORE ET R ; ee Se ee eee a ears en —— 4 . tie ie cS c a} “ —— = : ” Serre 3 - SE Ss 2 —~— =: == = ah reas g a ee 2 eS ss “ +> SOS — a = 5 Sivchyon-aa rerio — a Sa ! (onfiruction. Demonftra- $10%e The twelueth Booke be deuided into three pyramids equall the one to the other; hauing.alfo triangles té. their bales. For fo do Campane and Flufias put the propofition, leauing out the former particle hauing to his bafe a triangle, which yetis red in the Greeke copye,& not left out by any other interpreters knowne abroade except by Campane and F/ufoas. Yea and the Corollary following of this propofition added by Theon or Ewclide, and améded by 44.Dee femeth to confirme this fence. Of this, 1sit made manifeft, that euery pyramis isthe third part of the prifme, haning the fame bafe with 2, and equall altuude, For, and sf the bafe of the prifme hane any other right lined figure (then atriangle) and alfo the fuperfictes oppofite to rhe bafe,the fame figure: that prifme may be de- uided into prifmes, hauing triangled bafes: and the (uperficieces tothofe bafes oppofite,alfo triangled a llike and equally, For there, as we feeare put thefe-'wordes,, For and if the bafe of the prifere be any other right lined figure. €§e. Whereofa man may wellinferre that the bafe may be any other ‘rectiline figure what- foeuer , & not only a triangle ora parallelogram me, and itis true alfo in that fence, as itis plaine to fee by the fecond corollary added out of Flxfas, which corollary, as alfo the firft of his corollaries, is in a maner all one with the Corollary added by Theon or Euchde . Farther Theon in the demonttration of the 10. propofition of this booke ( as we fhall afterward fee ) moft playnely calleth not onely fided co- Jumnes prifmes, but alfo parallelipipedons.And although the 4o.propofition of the eleuenth booke may feme hereunto:to bea let.For thatitcan be vinderfanded of thofe prifmesonely which haue triangles to their like ,equall,oppofite,and parallel fides ; or but of fome fided columnes, and not ofall + yet may. that let be thus remoued away , to fay that Exelide in that propofitié vfed genus pro /pecie, that is,the ge- nerall word for fomefpecial kinde therof: which thing alfo.is notrare,notonly with him, butalfo with other learned philofophers. Thus much I thought good by the way to note in farther defence of Euclids definition of a Prifme. | The 8:T beoveme:” Thee. Propofition. Pyramids being like ex haning triangles to their bafes are in treble propor tion the one to the ether, of that in which their fades of like proportion are. lV ppofe that thele pyramids whofe bafes are the triancles G B Cand H E F and RO hes the ey and D be i von in like fort defevibed and let AB and | SO (NDE be fides of like proportion -Thenl {9 thatthe pyramis ABCG. ts tothe Ore. pyramis D E F H in trebleproportio,of that in which the fide A B isto the fide DE. Make perfect the parallelipipedons namely, the folides B C.K L cy E F X 0.And foraf- much as the pyramis ABCG is like to the pyramis DE F H,therfore the angle ABC is equale to the angle D E F ,¢y the Be ees angle GBC tothe angle \ kate totem Te Of HE F , and .moreouer.the angle ABG to.the angle DEH, and asthedine A Bis tothe line DE, fois the line BC tothe line E F , and the line B G to the line EHS And for that as the line AB is to the line DE, fois, the line B A C tatheline EF ,and the fides about the equall an- gles , are proportional, therefore the parallelo- gramuie BM is like.to the parallelograme EP : and by the [ame reafon the pa- rallelogramme BN gstike Bo Ss 59 f% of Euclides Elementés: \ . Fol.368; to the parallelogramme E R, and the parellelogramme BK ishikevnto the pardllelocramme EX. Wherefore the three parallelogrammes BM, K Band BN arelike tothe three parals lelozrammes E P,E Xyand ER»: Bit the three parallelogrammes BM, KB; and BN are equall and like to the three oppofite parallelagrammes; and the three parallelogrammes E P, EX , and E Rareequall ana like tothe thrée oppo/ite parallelogrammes . Wherefore the pa- rallelipipedons BC K Land E F XO are comprehended under playne [uperficieces like and equallin multitude. Wherefore the (olide BC K L is like to the folide E F X O-But tke paral lelipipedons are(Ly the 33 .of the elenenth )in treble proportion the one to the other ofthat iv which fide of like proportion ts to fide of like proportion . Wherefore the folideB ORL 43 to the folide E F X O in treble proportion of that in which the fide of like proportion ABis to the fide of like proportion D E. But as the folide B C K Lis.to the folide E F X 0,fois the py ramis ABC G tothe pyramis D E F H (bythe rs: of the sifth ) for that the pyramis is the ixth part of this folide : for the prifme , being the halfe of the parallelipiped om ts treble to the vyramis, Wherefore the pyramis ABC G is to the pyramis D EF Hin treble py oportion of that in which the fide A Bis tothe fideD E.Which was required to be proved. aril Corollary. Flereby it is manife/i that like pyramids hauing to their bafes poligonon jigures arein treble proportion the one tothe other ,of that in which fide of : i + CPL vo ; like proportion ts to fide of like proportion. For if they be denided into pyramids hauing triangles to their bafes ( for like policono figures ave deuided into like triangles,and equal in multitude,and the fides ave of like propor- tion )as one of the pyramids of the one,hauing a triangle to his bafe, isto one of the pyramids of the other hauing al{o atriangle to bis bale, {0 alfo aveall the pyramids of the one pyramts hauing trianeles to their bafes to all the pyramids.of the other pyramis hauing alfo triangles to their bafes:T hat isthe pyramis hauing to bis bafe.apoligound figure, tothe pyramis haning allo to his bafe apoligono figure. Buta pyramis haninga triangle to his bafesisto a pyracwisha wing alfo a triangle to his bafe,cy being like unto it, in treble proportio uf that in which ‘fide of like proportio1s to fide of like proportio. Wherforea pyramis hauing to his bafe a paligono jigure,istoapyramis hauing al[o a poligonon figure to his bale, the fayd pyramids being like the a0 t6 the other, in treble proportion of that in which fide of like proportion isto fide of like proportion .Likewife Prifmes and fided columnes,being fet vpon'the bafes of thofe pyramids,and vnder the fame altitude (forafinuch.as they are equemultiplices ynto the pyramids ,namely,triples, by the corollary of the 7.of this booke)iha! haue the former porportion that the pyramids haue,by the r5, of the fifth,and therefore they fhall be in treble proportion of that in-which the fides of like propor- tion are. q The 9. E heoreme. Ehe.9. Propofition. In equall pyramids baning triangles to their bafes , the bafes are recipros kall to their altitudes. And pyramids hauing triangles to their bafes whofe bafes are reciprokall to their altitudes are equall the one to. the other. Neal ppofe that BC G Aand E F H D beequallpyramids,hauing to their bafes the Oe triangles BC G and E F H, and the tops the pointes A and D .ThéenI [ay that Se YS the bafes of the two pyramids BCG AandE F Hl D are reciprokall to their al- (SSS) titudes:that is,as the bale BC Gis tothe bafe EF H,foisthe altitude of the py- ramis E F H D to the altitude of the pyramis BC G A. Make perfect the parallelipipedons, 3 namely, An addition. by Campane... and F luffas. The twelueth Booke namely ,B GM Land EHS 0.And foralmuch as the pyramis BCG Ais eguall to the pyra- misEE HD; ¢ the folideB GM Lis fextuple tothe pyramis BCG A. (For the paralleli- pipedonis dupleto the Prifiaefet upon the bafe of the Pyramis, cy the Prifme is triple to the pyranus):and likewife the {olide EH P-O is fextuple tothe pyramis EF H D.Wherefore the folid@BG M Lis equal tothe folideE HP O.But in equall parallelipipedons the bafes are(by the 34.vf the eleweth )reci- prokall'te toeir altstudes. Wherfore as the bafe BIN is to. the bale E 9. ,fois the altitude of the folide EH. . P-O,te the altitude of the folide BG.ML.But as the bafe BN és to the bafe. E- 2 fois thetriangle GBC to the trianele.H EF ( by the 15.0f the fifth, for the AK triangles GEC er HE F are the baines of the paral- lelogrammzes BN and E- 2 ).. Wherfore | by the rr. of the fifth das the triangle GBCis te the triangle H~_ EF, fo isthealtitude of « | the folide EFLP Oto the B Cc. altitude of the folideBG=. ML:Bys the altitude of the[olide'E H P 0 is one and the [ame with the altitude of the pyra- miSEFH D and thealtitude of thefolide BG ML, is one and the fame with the altitude of ‘dhe pyramis BCG A. Wherefore as the bale G BC is tothe bafe HE F , fo is the altitude of the pyramisE FH D to the altitude of the pyramis BCG A. Wherefore the bafes of the two pyramids BCG Aand E F HD are reciprokall to therr altitudes. : Wieisiehirei 2 But now [uppofe that the bafes of the pyramids BCG A and E F H D,bereciprokall to sinc oe their altitudes,that tas the bafe C BC is tothe bafeH E F ,folet the altitude of the pyramis jecondpart, EP HD bétothealtitude of the pyramis BCG A:Then 1 [ay that the pyramis BCG A is e- whicbisthe quall tothe pyramis E F H D.For(the felfefame order of confiruction remaining), for that comuerleof asthe bafe GB Cis to the bafet. E F , foisthealtitude of the pyramis EF HD tothe alti- he firft. tude of the pyramis BCG A. But asthe bale GBC istothe bafe HE F,feis the parallelo- gramme GC to the parallelogramme H F Wherefore( by the 11.0f the fifth) as the parallelo. gramme G Cis to the parallegoramme H F, fo is the altitude of the pyramis E F H D tothe altitude of the pyramis BCG A. Bukthe altitude of the pyramis EF H B and of the folide EH P 0,is one and the felfefame, and the altitude of the pyramis BC G A and of the folide BG ML sis alfo one and the fame.Wherefore as the bafe G C is to the bafeH F , foisthe alti- tude of the folide EH P O to the altitude of the folide BG.M L.But parallelipipedons, whofe bafes arereciprokall to their altitudes are ( by the 34:0f the elenenth ) equall the one to the o- ther. Wherefore the parallelipipedon BGM L is equall to the purallelipipedon EHP O . But the pyramis BC G A is the fixth part of the folide BGM L,and liken the pyramis EFHD is the fixcth part of the folide E H P 0. Wherefore the pyramis BCG A is equallto the pyra- EF 2D. Wherefore in equall pyramids hauine triangles to their bales , the bafes are reci- prokallto their altitudes. And pyramids haning triangles to their bales, who i[e bafes are rech procall to their altitudes,are equall the'one to the orherewhich was required to be demonftra- ted. : Demon Bratz - ON of the first pert, F iL, Ra ! ; | | ; Hi A A i ie ay t 1 F ¥ ' 7 | } ae ee EST Sr == ~a'gr- ——— te : pce A Corollary of Euchdes Elementes. Fol.369. A Corrollary added by Campane and Fluffas. Hereby it is manife/t that equall pyramids hauing to their bafes Poligonon figures , haue their bafes reciprokall with their altitudes.eAnd Pyramids whofe bafes being poligonon figures are recipro- kall with their altitudes,are equall the one tothe other. Suppofe that vpon the poligonon figures A andB ; be ferequall pyramids .°Thén I fay that their bafes A and. B.are reciprokallwith their al- ™s wae titudes. Defcribe by the 25. of the fixth, triangles | \ 4 equal! ro the bafes A and B . Which let be C and A \ | D .Vponwhich lect there be fer pyramids equall. in altitude with the pyramids Aand B.Wherfore et Co the pyramids C and D, being fet vp6bafes equall ' with the bafes ofthe pyramids A and B, and ha- uing alfo their alticudes equall with the altitudes of the fayd. pyramids A and B, fhall be equall by theé,ofthis booke. Wherefore by thefirft part of this propofition,the bafes of the pyramids , C to D are reciprokall with the altitudes of D to C. C D Butin what proportion are the bafes C toD, in | the fame are the bafes Ato B,, forafmuch as they are equall . And in what proportion are the altitudes of D to C, in the fame are the altitudes of B to A, which altitudes are likewife equall . Wherefore by the 11. ofthe fifth, in what proportion the bafes A to B are,in the fame reciprokally are the altitudes of the pyramids B to A . In like fort by the fecond part of this propofition may be proued the conuerfe of this corollary, The fame thing followeth alfo in a Prifme,and in a fided columne,as hath before at large bene declared in the corollary of the 40. propo- fition ofthe 11.booke. For thofe folides are in proportié the one to the other, as the pyramids or paral- lelipipedons,for they are-either partes of equemultiplices or equemultiplices to partes. The 1a; Theoreme. The io. Propofttion. Enery cone is the third part of a cilinder haning one and the felfe fame bafe and one and the felfe fame altitude with it. Hy KV ppofe that there be a cone hauing to his bafe the circle ABCD, andlet there bea iC 3% cilinder hauing the felfe fame bafe,and alfa the fame altitude that the cone hath. UIZANG Then 1 fay that the cone is the third part of the cilinder, thatis, that the cilinder is in treble proportion to the cone.F or if the cilinder be not in treble proportion.to the cone, then the cilinder is either in greater proportion then triple to the cone,or els in lefve. First let it bein greater then triple. And defcribe (by the 6. of the fourth) in the. circle ABCD er {quare ABC D.Now the{quare ABCD, H : as creater then the halfe of the circle A B C- D. For if about the circle ABC D, we de- {cribe a {quare, the {quare de{cribed in the circle ABC D.isthe halfe of the {quare defcribed about the circle. And let there be Parallelipipedon pri{mes defcribed vpon Parallelipipen thofe.. quares, equall in altitude with the E dons called ~ cilinder. But prifmes are in that proporti- Prifines. on the.oneto the other, that their bafes are (by the 32. of the clexenth,and-5, Co- rollary.of the 7. of this booke) Wherefore the pri{me defcribed vpon the {quare A- BC Dis the halfe of the prifme de{cribed upon the [quare thatis de{cribed about the circle. Now the clinder ts lefle then the bet es ihe AS onl tid : wit: : ja] 4 i} + ; | ih att iS oy |b oe 4 4 dele T be twelueth Booke prifme whichis made of the {quare defcribed about the circle ABCD, being equal in altitude with it,for it contayneth it. Wherfore the pri{me defcribed vpon the fquare A B C- D and being equall in altitude with the cylinder , is greater then half the cylinder . Deuide (by the'30.0f the third) the circumferentes AB,BC,;CD and D A into two equall parts in the points E,F ,G,H, And draw the(e right lines A E,E BB F,FC,CG,GD,DH & H A. Wherfore enery one of thefe triangles A E B,BF CCG Dand DH Ais greater then halfe of that {ecment of thecircle ABC D whichis defcrtbed about it,as we haue before in the 2 3 propofition declared. Defcribe upcn euery one of thefé triangles AEB,BFC,CGD,and q D H A a prifme of equall altitude with the cylinder. Wherefore euery one of thefe prifmes fo def svibed is greater then the halfe partof the fegment of the cylinder that is fet vpon the fayd fegenents of the circle. For if by the pointes E, F, C,H, be drawen parallel lines to the lines A B,BC,C D and D A, and then be made perfect the parallelogrammes made by thofe parallell lines, and moreouer upon thofe parallelogrames be eretted parallelipipedows equalt in altitude with the cylinder , the prif{mes which are defcribed ‘upon eche of the triangles A EB,BFC,CG D,and DH Aare the halfes of éuéry one of thofe parallelipipedons.And the Segments of the cylinder are leffethen thofe parallelipipedons fodefcribed . Wherefore aifo euery one of the pri{mes which are defcrited vpon' the triangles AEB, BFC,CGDand D H Ais greater then the halfe of the [egment of the cylinder jet vponthe fayd fegment. Now therefore deuiding exery one af the circumferences remaining into two equall partes, and drawing vight lines and ray ying up-~upon enery one of thefe triangtes pri[mes equall ix altitude with the cylinder , and doing this continually we fhall at the length ( by the first of therventh \leane certaine[egments of the cylinder which fbalbe leffe then the exceffe whereby the cylinder excedeth the cone more then thrife . Let thofe fegments be A E,E B,B F,FC,C G,G D,D Handi. Whesfore,theprifme remayping whofe bafeis the poligonon figure AEBF EG DH, and altitude the felf6 5.0 ogg wf fame that the cylinder hath,is oreater then = % By shes itis the cone taken three tymes." But the pri{me manifell chat whofe bafe isthe poligonon figure AE BF Eaclate €68- © G DH dnd alutude the felfe fame that ae , hevyliniter bathyis rreble to the pyramis * aif yuderthe M0OfE bafeis the poligonon figure AE BF | gameofa CGD And altitude. the felfe fame that Peiftiee. 5 the cine hath bythe corollary of the 3. of gs is book w herfore alfo the pyramiswhofe , bafe isthe paligononfigure AEBFCGD H and toppe the felf {ame that the cone hath , is.gveater then the-conewhich hath to his bafe the circle ABCD. But it ws alfoleffe for itis contayned of it which is suapofable. Wherefore the cylinder is not in ‘Sapa eatin’) greater proportion then triple tothe cone. cing | eae polls ax = ~ I fay wsoreouer that the cylinder 1s not in leffe proportion then triple to. the cone . For if WiRETS 5t be poficble let the cylinder be in leffe proportion then triple to the cone. Wherefore by con- werfion,the cone is greater then the third part of the cylinder. Defcribe now ( by the fixth of the fourth)in the circle ABC Da {quare ABC D. Wherefore Ye ABCD wgrea- = wes === es i - a = : ba Ste Rint te _ - —— —— ee = z : : - —_— a ee _ ee Ae a , Re SSS x a pebeeae ¥ojters ——s —= = = . a0 ating oS 5 = - a . prio Fora nar teaser — ae ates Poe ng = -—s- teat a = = : _— - _ pe es) ge ee ae ee ~ - tershen the haife of the circle ABC D upon the {quare AB € D defcribe.a pyramis hauing ont cy the felfe fame altitude with the cone. Wherfore the pyramb fo decribed is greater thé balfe of the cone.{ For if aswe haue before declared we defcribe a [quare about the circle, the {quaré ABCD isthe halfe of the {quare deferibed about the circle , and if uppon the [quares bc defcribed parallelipipedons equall in altitude with the cone, which folides are a wi i calle of Euchdes Elementess »\ \ Fol.2 71, called prifimes,the pri{me or parallelipipedon defttibed vpothe|quare. A BC Dis the halfeof the pri{mee which is defcribed up the [quare defcribed about the circle for they are the one to the other in that proportia that their bafes are( by the 3 2..0f the eleueth, c 5.corgllary of the 7 -of thes booke.)Wherfore alfotheir third parts are in the felf [ame proportion (by the 1 5. of she fift) Wherfore the pyramis whofe bafe is the quare ABCD is the halfe of the pyramis [et upon the [quare de[cribed about the circle. But the pyramis fet upon the [quare. defcribed a- bout the circle 1s greater then the cone whome it comprehendeth.Wherfore the pyransis whofe bafe.is the (quare A BC D,and altitude the [elf fame that the cone hath, Greater then the halfe of the cone.) Deuide (by the 30.0f the third) every one of the circumferences AB, BC, C D,and D A into two equall partes in the pointes E, F,G,andH: and drawe thefe right lines AE, EB,BF,FC,CG,GD;DH,andH A. Wherefore enery one of thefe triangles AEB,BFC,CGD,and DH A is greater then the halfe part of the fegnient of the circle decribed about it .V ppon enery one of the[e triangles.A E B,BFC,CGD,and D HA de- [cribe a pyramis of equalhaltitude with the cone:and after the [ame maner enery one of thofe pyramids {0 de[cribed is greater then the halfe part of the feement of the tone fet Upon the Segment of the circle. Nam therefore diuiding (by the 3.0,0f the third the circumferences re- maining into.twoequall parts, cy drawing right lines cr rayfing up upon enery one of thofe triangles a pyramis of equall altitude with the cone,and doing this continually, wefhal at the. length( by thefir[t of the texth \leaue certaync fecmentes of the cone , which fhalbe'leffe then the excelfe whereby the cone excedeth the third part of the cylinder ... Let thofe feementes be AE, EB, BF,FC,CG,GD,DH, and HA.Wherefore the pyramis remayning , whofe bafe is the poligond figure A E B F CG D H and altitude the felf fame with the cone,is grea- ter then the third part of the cylinder . But the pyramis whofe bafeis the poligonon figure AEBFCGDHand ie the felf fame with the cone, is the third part-of the prif{me whofe bale is the poligono figure_A E BF CG D Handaltitude the felf [ame with the cylin- der Wherfore “the pri{meéwhofe bafe is the poligonon figure AEB F CG D H,andultitude the [elf [ame with the cylinder,is greater then tie cylinder whofe bafe is the circle ABCD. But it is alfolefe;for tt ts contayned of tt, which is impofible.Wherfore the cylinder is not in leffe proportion tothe cone then in treble proportion. And it is proved that it is not in grea- ter proportion to the cone then in treble proportion,whereforethe cone isthe third part of the cylinder Wherfore enery.cone is the third part of a cylinder ,haning one cx the felf [ame bafe, and one and the felfe fame altitude with it:which was required to be demonjtrated. q Added by M.Iohn Dee. q AT heoreme. = T he fuperficies of euery upright Cylinder, excepybis bafes,is equak to that circle whofe femidia- mseter is middell proportionall betwene the fide of the Cylinder and the diameter of his bafe. oT es , Gi Theortyhe. hs vo RS ae | pe ha > . ready ayde,and fhew the way to dilareyour'difcourfes Matheinaticalljor to inuent nd pragife thinges ‘ * A prifme bauing for his bafe a poo liconen figure as we haue often before noted ynto Owe : ’ : - b oe ; : : } te) : 1 | +e : r , i ; | t Peibi al if it! Ot) - r Wad uy uy Ls ip 7 AGE bt Ly ‘ : 7 ’ wt ' we if fr, it .. if ts Ha iz a) f i] i , %, ; Py 4] fl : a Demon #ra- tion 4s to“- \on T he twelueth Booke Mechanically..And (in deede)if moreleyforhad happened, many more ftraunge matters Mathemati- call had, ( according tomy purpofe generall ) bene prefently publithed to your knowledge : but want of due leafour cauleth you to want,that,which my good will toward you,moft hartely doth wifh you. As concerning the two Theoremeés here annexed, their veritie , is by rchimedes , in his booke of the Sphere and Cylinder maniettly demonttrated y and-at. large ; you may therefore boldly eruitto chem, and vfe them,as {uppofitions, in, any: your. purpofes:till youhaue alfo their deméftrations.. Butif you wellremeniber my inftructions vpon the firft propofition of this booke , and my other additions vpon the fecond , with the fuppofitions how a Cylinder and a Cone are Mathematically produced, you will not neede Arcbhrmedes demonttration: nor yet be ytterlyignoranntof the tolide quantities of this Cylin der and Cone here compared: (the diameter of their bafe,and heith being knowne in any meafure)nei- ther can their croked {uperficies remayne vameafured. Whereof vndoubiedly great pleafure and com- modiue may grow to the fincere ftudent,and precife practifer: The rr. I’heoreme.-*\ ~ The ¥1.Propofition. Cones and Cylinders being bnder one and the Jelfe fame altitude , are in that proportion,the one other that their bajes are. a da Be there be take cones ed cylindres under one and the felfe [ame altitude whofe he de bafes let be the circles ABCD and EF CH, and axes the lines RL cr MN, VS) apa let the diameters of their bafes be AC and EG.The 1 fay that asthe circle YS ABCD isto the cirtke EF GH, [ois the cone A L to the cone EN , and alfo Sve cylinder AL tothe cylipderE N . For if the cone AL be not to the cone E- ee Oe Zs ching Cones. WN asthe circle ABC Dis tothe crcl EF G H,then as the circle ABC D isto the circle E+ Firft cafe. t, cone E N,or greater . First let it be anto alefesmamely pte song Dalfe of che'eome: Porif medeleribe it fquare about the tir- eu . Goo ob f elasen Se “eS ~ 4 = % , eA ye SiS SI3tI3 TAR HP AGS TETAS LRE a . : ' . WW, SSR) F GH, fois the cone A L to fome folide either lefe then the the folide X . And unto that which the folide X is lefe then the cone EN let thefolede Pbe equally Wherefore the cone sx be eas E Nis equaltothefolides Xe ¥ Defcribe(by the 6.df the erp fourth)in thecircleE FGHa [quare E F G H.Wherefore | | © > the {quare is Arie then halfe the circle . Raye vpupon. |. . Hpk Si: PTE ASLO! OTS a pay MG TTS EGC Hapyramis.of- eyudll aliitude. mith she |: cone. Wherefore thepyramis ; foxayfedwpis greater then’ > / au-s ROT tleyandrpon that [quare rayfenp atyrames.of equall al. cw £% t. ety et 0 Sete ; whe ’ ~eegaat aedigenette = SER ~ of Euchides Elementes. F0l.374.. titude with the.cone , the pyramis which is fet vpon the {qnare defcrvibed with in the circle ig thé haife of the pyramis fet vpon the {quare defcribed about the circle; for they are in propor- tion the one to the other as their bafes.But the cone is lee then the pyramis whith is fet upon the [quare defcribed about the circle. Wherefore the pyramis whofe bafe is the {4uare E F- G H and top one and the fame with the cone is greater then halfe of the cone: denide (by the 30.0f the third )the circumferences E FF G,G H and H E into two equal partes in the points O,P,R,S,and drawtheferight.linesHO,0 E,E P,P FR R:RG,GS,andS H. Wherefore euery one of thefe trianglesH O E,E P F, F RG,andGS His greater then halfe the feemet of the circle defcribed about the.Rayfevp wpo exery one of the triangles HO E;E P FsF RG andG SH apyramis equallin altitude with the cone. Wherefore enery one of the pyramids fo rayfed up is greater then the halfe part of the [eement of the cone defcribed about it Now then deuiding (by the 30.0f the third the circumferences remayning into two equall partes, and drawing right lines ; and rayfing vp vppon enery one of thofe triangles a pyramis of equall altitude with the cone, and thus doing continually, we fhall at length by the first of the tenth leue certayne fegmentes of the cone whith fhall be leffe then the folide Y. Let thofe feementes beH O E, EP’ F,F RG,and GSH Wherefore the pyramis remay- wing whofe bafe ts the poligonon figure H O EP F RGS cy top the felfe {ame with the cone, 1s ereater then the folide X.Infcribein the circle A B C D a policonon ficure like and in like fort fituate to the poligonon figure HOE PF RG S,andlet the{amebe DT AV BZCY, and upon it rayfea pyramis of equall altitude with the cone AL. Now for that as the [quare of the line A C is to the {quare of the line E G, fois the poligonon figure DT AVBZCW to the poligonon figure HO E P F RGS ( by the firft of this booke). But as the [quare of the line A Cis tothe {quare of the line E G, [ois the circle ABCD tothe circle E F G H(by the fecond of this booke).Wherefore( by the 11.0f the fift)as the circle ABCD is.to the circle E- F GH, fois the poligonon figure D T AV BZ CW to the poligononfigureH 0 EP FRG- S (this foloweth al fo of the corollary.of the 2.0f this booke): moreouer as the circle ABCD is to the ciy¢le EF G H,fo is the cone A L to the folide X. And as the poligonen figure D T- AV BZCW isto the poligonon figure HOEP RGS, foisthe pyramis whofe bafe is the poligonon figure DT AV BZ CW , and toppe the poynt L , to the pyramis whofe bafe is the poligonon figure H O EP RS,and toppe the poynt N. Wherefore( by the 11. of the fift) as the cone A L1s to the folide X,fo is the pyramis whofe bafeis the poligone figure DT AV - B ZC Wand toppe the poynt L, tothe pyramis whofe bafe is the poligonon figure H E O P- F RG S,and toppe the poynt N. Wherefore alternately (by the 16. of the fift) as the cone AL is to the pyramis which is init , [oisthe folide X , tothe pyramis which isin the coneE N. But the cone AL, is greater then the pyramtis which is in it . Wherefore alfo the folide X is greater then the pyramis which is in the cone E N. But itis alfo leffe by construction Which 1s impofsible.Wherfore as the circle. B G Dyis tothe circleE F G H,foisnot the cone AL to any folide leffethen'the cone EN " : In like forte alfo may we proue,that as the circle EF GH,isto the circle ABC D,fois not the cone EN to any [olide leffethen the cone AL. NowJ {ay that as the circle ABC D, isto the circle E F GH,fots not the cone. AL to any folide greater then the cone E N.For if it-be pofGble let it be unto a greater., namely tothe folide X. Wherefore by conuerfion, as the circle E F G H,is to thecircle A B-GxDsfots the folide X to the cone AL: but as the folide X is tothe cone A L, fois the cone EN ,.to fome folide leffe then the cone A L.( as we may feeby the affumpt put after the fecond of this booke): Wherefore (by the 11.of the fift)as the circle E F G Histo the circle A BCG,fais the cone E N to [ome folideleffethen the cone A- L which we haue proued to be impofible. Wherefore as the circle A BC D,is tothe circle E- F GH, foisnotthecone AL to any folide greater then the cone EN . And it is allo proued shat tt is not to.any leffe. Wherefore as the circle ABC D 5 utothecixch EF GH fouthe cone.A L.totheconeE N. , But 25 the cone is to the cone,fois the cylinder to the cylinder, (by the 15. of the fift)for EE¢.ii. the Second cafes iz a , ae ‘ee d fet r'? cea ee. a ee 28 oh RRR, ae he A rhe aA Peete Notte Demon tra- tien touching cylinders, Fis cafe, Conftrutl ton. _intreble proportion of that in which the diameter BD is altitude with thé cone.Wherfore the pyramis{o raifed up,ts Thetwelueth Booke = the one isin treble proportion tothe other Wherefore by the rr - the fift)as the circle A BC D istothe circleE F GH, foarethe cylinders which are fet vpon them the one to the other,the faid cylinders being Under equall altitudes with the cones.Cones therefore and cy- linders being under one & the felf fame altitude,are in that proportion the one to the other, that their bafes are:which was required to be demonftrated. q The 12: T heoreme. Ihe 12. Propofition. Like Cones and Cylinders are in treble proportion of that in which the die ameters of their bafes are. ; : / BDistothe diameter F H,the cone ABCDL hall be to the diameter F H,cither to fome folide leffe then the cone EF GHN, or to fome folide greater. Firstletit be untoa leffe,mamely,to the folide X.Defcribe( by the 6.of the fourth) ththe circle EF GH, a fquare EF GH . Wherefore the Square EF GH is ereater then the halfe of the circle’EF- GH. Raifevp from the [quare E FG H a pyramis of equall greater then the halfe part of the tore . Diuide now (by the 30. of the third) the circumferences EF ,F G,GH,& HE, into two equall partes in the pointes 0, P,R; S; and drawe ce ae thefevight lines E-0,0 F, F P,P G,GR,RH,HS, andS E. Wherefore enery one of thefe See | ae, ae Sob is weet 4 i. ~ . . . : : (ee btaty ~ %& . £% of Euchides Elementes. Fol.3°72, triangles EO F,F PG,G RH, and HS E, is greater then the halfe of the Segment of the circle E F G H, defcribed about ech of them. Erecte upon euery one of thefetriangles E o F, FPG,GRH, and HSE,a pyramis , hauing one and the felfe fame altitude with the cone. Wherefore euery one of the pyramids fo raifed vp, is greater then the halfe of the Segment of the cone defcribed about them. Now therefore diuiding the circumferences remayning into two equall partes, and drawing right lines, and raifing Up Upon enery one of the triangles a pyramis, hauing one and the felfe [ame altitude with the cone, ana,thus doing continually, we fhall at the length (by the 1. of the tenth) leaue certaine Seementes of the cone,which fhall be lefve then the exceffe whereby the cone E F G H N excedeth the folide X . Let thofe feementes left be EO,0 F,F P,PG,GR,RH »>HS,andSE. Wherefore the pyramis remayuing,whofe bafe is the Poligonon figureEOFPRGRHS,and toppe the point NV, is greater then the folide X . Defiribe ( by the 18.0f the lixt ) inthe circle ABC D,vn- to the Poligonon figureEOF PGRHS,a Poligonon figure like and in like fort fituate,and let the famebe AT BV CZDW. , and from it raife up a pyramis, hauing one andthe felfe fame altitude with the cone ABC L. And let one of the triangles comprehending the pyra- mis, whofe bafers the Poligonon figure AT BV CZ D Wand toppe the point L, be LBT, and let alfo one of the triangles comprehending the pyramis, whofe bafe is the Poligonon fi- gure EOF PGRHS, and toppe the pornt N, be NF O, and drawe thefe right lines K T and MO. And foralmuch as the cone ABCD L is like to the cone EF GH N, therefore (by the 20.definition of the eleuenth ) as the diameter b D is tothe diameter F Hf fo ws the axe K L to theaxe i XN: But as the diameter B D isto the diameter F H [0 ( by the rs. of the fift ) ts the femidiameter B K to the lemidiameter F M. Wherefore (by the rr. of the Sift) mB KstoF M,foisK LtoMN. Wherefore alternately alfa ( by the 16. of the fifty aBK stoK L, fois F MtoM N.Wherefore the fides about the equall angles b K Land F M N (wluch angles are equall, for that they ave right angles ( by the 18. definition of the eleuenth ) are proportionall. Wherefore ( by the 1.definition of the fixt) the triangle BK L uw like to the triangle FMN. Againe for that as BK isto KT, fois FM toMO, and they comprehend equall angles, namely, BK T and FMo, for what part the angle B KT 45 of thofe fower right angles which are made at the centre K. the felfe fame part isthe angle F M O of the fower yight angles which are made at the centre At - fova{much therefore as the fides about the equallaneles are proportional, the triangle-B KT is like to the triangle F MO. Againe, fora{much as it was proued, that asB K isto K L, fois FM to NM, but B Kis equall toKT,andF MtoM 0, therefore aT Kustok L, foiso MtoM N.Wher- fore the fides about the equall angles T K Land OMN (which angles are equall, for that they are right angles ) are proportionall .Whereforethetriangle L KT is lake tothe trian- gle MNO. And for that (by the 6 of the fixt ) and by reafon of the likenes of the triangles LK BandNM F,as L BistoB K; fou NF toF M, and againe by reafon of the likenes of the triangles B K T and F M0, as. K Bis to BT, fois MF to F O, therefore of equalitie (by the 22.0f the fift) a L BistoB T, fois NFtoFo. Againe for that by reafon of the likenes of the triangles LKT anmdNOM, a LT istoT K, fois NO to OM, and by rea- Sow of the likenes of the trianglesT K By OM Fas KT is toT B, foisMO100 F : ther- fore of equalitie (by the 22.0f the fift) as LT is toT B {01s N Oto OF . Andit was proued that asT Bis to B L, foisOFtoFN. Wherefore againe of equalitie,asT Lis to L B, fois ONt0 NF .Wherefore the fides of the triangles LT B ce NO Fare proportionall. Wher- fore ( by the s.of the fixt) the triangles LT B and NO F, are equiangle . Wherefore alfo theyare like . Wherefore the pyramis, whofe bafeis the triangle BK F, and toppc the point L, is like unto the pyramis whofe bafe is the triangle F M O,and toppe the point N..For they are comprehended under like plaine fuperficieces, and equalli» multitude. But pyramids be- ing like, and hauing triangles to their bafes, are ( by the 8.of the twelfth ) in treble proporti- on the oneto the other of that in which fides of like proportion are. Wherefore the pyramis EEe.iys eos ee & L Demonfirati- On teading to an impofirbi~ bitie. ee - SE “=> - ms ct a es ee eee ee SS re ax_tat— FP ~ =. xt oS - 2 —-- ——— r= —= a ee = SS _——= = = - - —~- = = = ow = ee —— — _ ; : = 3 a , —-— — = Re py OT es ae lt PT — mas oN ee ae Ree ane r : ; ser erere - . —~ + — + —v se - oa A a a a — es Ae Pe hae Fahl Pet PAT os EAS wT Es SE nn ts a ee <= ~ — nae “ —-—:- aed : ina ; ; : ey iar ee * * # * : : =A5 sata “= : es c = = : - : 3 : ; a ; =e : = ee pe rooms: = ee a = a = - . . eae een eee nk nna iate a ~ ‘ ee SA T he twelueth Booke BET L isto the pyramis F MO N, iniveble proportion of that in which the line BK és te the line FM. And in like fort if we draw right lines from the pointes A,W,D,Z,C,and¥; ' tothe point K, and likewi[e from the pointes E,S,H,R,G and P, to the point -M, and raife up upon the triangles pyramids haning the felfe fame altitudes with the cones, we may proue zhat enery ove of thofe pyramids of one o the Jelfe fame order, is to euery one of the pyramids of the felfe fame order, in treble proportion of that in which the fide of like proportion B K ts to fide of leke proportion F M that is, of that which the line B D hath to the line F H. But as one of the antecedentes ts to one of the confequentes, foare all the antecedentes to all the con- fequetes (by the 12.0f the fift) Wherfore as the pyramisB KT Listothe pyramis F MON, {ois thewhole pyramis, whofe bafeis the Poligonon figure AT BV CZ D W, and toppe the point L, to the whole pyramis,whofe bafe is the Poligonon figure E 0 F PG RH S,and toppe the point N . Wherefore the pyramis, whofe bafe is the Poligonon figure ATBYCZDW, and toppe the point L, ts to the pyramis,whofe bafe is the PoligononfigureEOF PGR HS, and tappe the point N, in treble proportion of that in which the line B D isto the line F H. Cc And it is fuppofed alfo that the cone, whofe bafess the circle ABC D, and toppe the point L, is tothe folide X, in treble proportion of that in which theline B D is tothe line F H. Wherefore as the cone, whofe bafeis the circle ABCD; ana toppe the point L, isto the folide X, fois the pyramis, whofe bafe is the Poligonon figure AT BV CZ DW, cy toppe the point L, tothe pyramis, whofe bafe is the Poligonon figure EOFPGRHS, and toppe the point N. Wherefore alter- nately ( by the 16.0f thefift ) as the cone, whofe bafe is the circle ABC D,c toppe the point L, isto the pyramis which is in it, whofe bafe is the Poligonofigure AT BV CZ DW, cy toppe the point L, fois the folide X,to the pyramis, —_ bafeis the Poligonon ficureE O F PGRH S, and toppethe point N_. But the forefaid cone is greater then the pyramis which is in it, forit containeth it . Wherefore the folide X is greater then the pyramis, whofe bafe isthe Poligonon figure EOF PGRHS, and toppe the point N.But it was Speed to be leffe:which is impofsible. Wherfore thecone ABC DL is not to any [olide leffe then the come EF GH N,in treble proportion of thatin which the diameter £% fa vy ” of Euclides Elementes. — Fol.3'73. diameter B D is to the diameter F H.In like fort alfo may we proue,that the cone EF GH N is not to any folide leffe then the cone 4 BC D L, in treble proportion of that in which F H stoBD. Now alfo I fay, that the cone ABCD L isnot to any folide greater then the cone EF GHN, intreble proportion of that in which the diameter B D is to the diameter F H. For if it be pofsible, let it be to a greater,namely, to the folide X. Wherefore by conuerfion( by the Corollary of the 4.of the fift) the folide X isto the cone ABC D L, intreble proportion of that in which the diameter F H is to the diameter BD. But asthe folide X is to the cone ABCDL, fois the cone EF GHN to fomefolidelefe then the cone ABCD L ( as itis eafie to fee by the Afswmmpt put after the 2. Propofition) . Wherefore alfo the cone EF GHN is vinto fome folide leffe then the cone ABC D L, in treble ipeepctins of that in which the diameter F His tothe diameter BD : which is proued to be zmpofsible. Wherefore the cone ABCD L is not to any folide greater then the cone E F GH N intreble proportion of that in which the diameter B D is tothe diameter F H, and it is alfo proued that it is not to any lefve. Wherefore the cone ABC D istotheconeEFGHN in treble proportion of that in which the diameter B D is tothe diameter F H. But as cone is to cone, {a1s cylinder to cylinder (by the 15.0f the fift ) for the cylinder is triple to the cone which is de[cribed on the one and felfe fame bafe, and hauing one and the felfe fame altitude with the cone . F or it is proued ( by the 10.0f thetwelfth ) that euery cone is the third part of a cylinder, bauing one and the [elfe fame bafewith it, and one Gx the felfe fame altitude . Wherefore the cylinder is unto the cylinder in treble proportion of that in which the diameter B D is to the diameter F H.Wherfore like cones ¢ cylinders are in treble proportion of that in which the diameters of their bafes are: which was required to be proued. The 13.I heoreme. The 13.Propofition, Ifa Cylinder be dinided by a playne fuperficies being a parallel! to the two oppofite playne fupers frcieces: then as the one Cylinder 1s to the other Cylinder, fois the axe of the one to the axeof $| the other. me V ppofe that there be acylinder AD , whofe axelet Oi be EF , and let the oppofite bajes be the circles A- B VE B,andC F D, andlet A D be dinided by the fu- pon thofe circles imagine thefe cylinders P R, RB,DT,T W to EE¢.ity. be Second cafes Second pare which concere neth Cilltn< ders. Confirutlzon. T he twvelueth Booke ne Demonfira- be fet. Now forafmuch as theaxesL.N,NE, and EXKare a equall the one to the other : Therefore the cylinders PR, RB, p L 0 and BG are( by the 11.0f the twelueth in proportion the oneto | the other as their bafes are.But the bafes are equall. Wherefore alfo the cylinders P R, R B, and BG are equall the one to the o- ther. And forafmuch as the axes L N,N E, and E K are equal er the one to the other,and the cylinders P R,R B, and BG are al- > i) fo equall the one tothe other , and the multitude of the axes L- N,N E,and E K 1s equall to the multitude of the cylinders P- C2ORe R, R Band B G: therefore how multiplex the whole axe K Lis to the axe EK , fe multiplex is the whole cylinder P G tothe cylinder BG. And( by the fame reafon) alo how multiplex the whole axe MK isthe axe K F fo multiplex isthe whole cylin- der WG to the cylinder GD. Wherfore if the axe K L beequal ee | the axe K M,thecylinder P G i equall to cilinder GW. And A |G if theaxe K L be greater then the axe K M ; thecylinder P Gis greater then the cylinder GW. And if it be leffe zt is lef[e. Now therefore there are foure magnitudes , namely , the two axes D ra E K,and K F and the two cylinders BG , and G D , and vuto theaxe EK and tothe cylinder B G,namely,tothe firft and the — third ave taken equemultiplices, namely, the axe K L,.and the cylinder PG. And likewife unto the axe G F, and vuto thecy- VY F Sedat) T linder GD, namely, the fecond and the fourth, ave taken other equemultiplices namely the axe K M1, and the cylinder GW. Andit is proued that if theaxe KL }excede the axe KM, the yy | WR. cylinder P Gexcedeth the cylinder GW-and that if it beequall - aM at is equall,and tf it be leffe stisleffeWherfore (by the 6.defini- tion of the fift) as the axe E Kistothe axe K F [ois the cylinder BG to the cylinder G D. If therfore a cylinder be diuided by a plaine [uperficies being a parallel ta the two oppofite plaine Superficieces:then as the one cylinder isto the other cilinder, {ois the axe of the one to the axe of the other ; which was required to be proued. ) ye gq Ihe 14.T heoreme. Lhe 14.Propofition. ~ Cones and Cylinders confisting ‘vpon equall bafes are in proportion the one to the other as their altitudes. | V ppofe that the cylinders F D ,and EB , and the cones AGB, andC K D,do nt anal : of Euclides Elementes. Fol.374.. a parallell to exther of the oppofite plaine fax | perficieces: therefore ( by the 13. of the twel- A ueth)as the cylinder CM is to the cylinder’ ° | /\\ F D,fois the axeL Ntotheaxe LK. But the cylinder CM ts equal to the cylinder E B, and the axe LN to the axeG H . Wherefore as the cylinder EB is tothe cylinder F D , fo ttheaxeG Htotheaxe K L. But as the cylinder E B is to the cylinder F D,fo(by the 15.0f the fift)is the cone AB, G tothe cone C DK , for the cylinders are in treble proportion to the cones ( by the 10. of the twelueth) . Wherefore ( by the 11. of the fift)as the axe G H isto theaxe K L, fo isthe cone ABG to the coneC D K,cy the cylinder E B to the cylinder F D.Wherfore cones cy cy linders confifting upon equal bafes are in pro- portion the one to the other as their altitudes: which was required to be demonftrated. Demonfira- tion touthing Co NESe . | q Ube rs.T heoreme. The 15.Propo/ition. En equal Cones and Cylinders , the bafes are reciprokall to their altie tudes.And cones and Cylinders whofe bafes are, reciprokall totheir. altie tudes,are.equall the one to the other. Se SS SONY ppafe that thefe cones AC L,E GN gr thefe cylinders AX,E 0, whofe bafes r¢6 . : | QOS are the.circls ABCD; EFGH; ee KAS OX)] 7d axes. K Land M N (which axes. KISS ‘i are alfa the altitudes of the cones & | cylinders). bé equall the one totheother. The Lfay |. that the bafes of the.cylinders X A @ EO. are reti prokaltotheir.altitudes,that is that asthe bafe-A- BC DistoshebafeE F GH;fo the altitudéM KX to the altitude K L. For the altitude K L is either equall to the altitude MN or not.:Kirst let it be e- guall.But the cylinder AX ws equal tothe cylinder \V E0.But cones and cylinders confiftine vader one a i and the felfe fame altitude, are tn proportion the | Fir part of the propofitts demonfiraed touching Cones, Iwo cafes in this prepofia= £207, The rf cafe By one te the other us their bafes are(by the rr of the twelueth) Wherfore the bafe ABC D is equall to the bafe EF G .Wherefore alfo they arerecipro- kal: 46 the bafe ABC Dis tothe bafeE F GH,fo } ws the altitude MR tothealtitude K L. rs “° But now fuppofe that the altitude LK be not ye equall tothe altitude M N', but let the altitude “SS ~~ Second cafes Conftratiwn. MN ( eT a a eee eee nee lok ‘Thetwelueth Booke. M Nhe ereater. And (by the 3.of the firft ) from thealtitude M N take away P M equall to the al- titude K L’, {0 that let the line PM, be put -eqnal to the line K L. And by the point P let there be extended 4 playne fuperficies T V S which les cut the cylinder E O , and be a parallell to the twa\ oppofite playne [uperficieces , that is , to the circles | Dewonftre. EF GH,and RO. And making the bafe the circle teon touching EF GH, er the altitude M Pimagine a cylinder cylinders. ES. And for that the cylinder AX ts equall to they. | cylinder E O, and there wan other cylinder ES, therfore(by the 7 .of the fift)as the cylinder A Xys |. to the cylinder E S,fo w the cylinder E 0 tothe cy: linder E § But asthe cylinder AX us tothe cylig=\ |} der E S,fois the bale: ABCD to the bafe E Fs: GH. For the cylinders AX ,and ES are under: one and the felfe fame altitude . And as the, ‘ fs = cylinder E O , isto the cylinder ES, fois the altio Pgs) Ps ze M N tothe altitude M P.For cylinders cofi- Sica, SS 9h 64" fing upo equall bafes are in proportion the one to . the other as their altitudes. Wherfore as the bafe A BC D is 0 the bafe E F GH,fo us the ab titude M N to the altitude M P-But the altitude P «Mis equallto the altitude K L.Where- fore as the bale ABC Dustothe Lafe€ F G H, fois the alutade'M No the altitude K L. Whereforein the equall chlinders AX, and E O the bafes are reciprokall to their altitudes. But now {nppofe thatthe bales of the cylinders AX,and E O-bereciprokat ta their altitudes, Sceond part that is,as the bale ABC Dis tothe bafeE F GH, {01s thealtitude MN to the altitude KL. cemonfiratede Then 1 fay that the cylinder ‘AX is equall to the cylinder EO. For the felfe fame order of conftructio remayning, for that as the bafe ABC D isto the bafeE F GH3fo is the altitude MN tothe altitude K L, but the altitude K L is equall to the altitude P M. Wherefore as thebafeAB.OD is-to-the bafeE CM, fo is thealtitude M N to'the altitude PM. But as the bafe A BC Dis tothe bafe E F GH fois the cylinder AX tothe cylinder ES for they are under equall altitudes: and as the altitude MN 1s to the altitude P M,fo-is the cylinder E 0 to the cylinder E S(by the 14. of the twelueth). Wherefore alfo a3 the cylinder A X is to ot) the cylinder ES , 0 isthe cylinder E 0 tothe cylinder ES . Wherefore the cylinder A X is mee eauall to the cylinder E O (by the 9.0f the fift) . And fo alfo is itin the cones which haue the oS felfe fame bales and altitudes with the cylinders. Wherefore in equall cones and cylinders, the bafes are reciprokall to their altitudes Gc. which was required to be demonfirated. ee ee ee Zz : } | ' wi 4 i ’ ie 7 i] 1 4 ui 4 : | Lt i i, 1% | ie nt 6 “4 se ozs A Corrollary added by Campanéand Fluffas. Hitherto hath.bene fhewed the paffions and proprieties of coues and cylinders whofealtitudes fa perpendicularly vpon the bafes.N ow will we declare that cones and eslinders whofe altitudes fall ob. liquely vpontheir bafes haue alfothe felfe fame paffions and. proprietieswhichthefore[ayd conesand ctlinders bane. oe ee <7 _Forafmuch as in the tenth of this booke it was fayd,that euery Cone is the third part.of a cilinder hauing one,and the felfe fame bafe,8 one & the felfe {amealtitude with it,which thing was deméftra- ted by a cilinder geuen, whofe bafeis cut bya {quare infcribed in it, and vpon the fidesof the {quare are defcribed Ifofceles triangles , making a poligonon figure, and againe ypon the fides of this poligonon 7 figure are infinitely after the fame maner defetibed other Mfofceles triangles taking away more théthe o\eo Seosel halfe,as hath oftétimes bene declared:therfore icis manifelt;chat the folides fet vpon thefe bafes, being ynder of EuclidesEleméntese\ © Fol.373. vnder the famealtitude that the cilinderinclined is sand beingalfo included im the fame cilinder , de take away more then the halfe of the cilinder, and alfo more thé the halfe of the refidue,as it hath bene proued in erected cylinders.Fot thefe inclined folides being vnder equallaltitudes and vpon equall ba» fes with the erected folides are equall to the,ereGted folides by the, corollary of the 40. of the eleuenth, Wherforetheyalfo in likefortas the ere&ted, take away more then the halfe. Iftherfore we cépare the inclined cilinder,to a cone fet vpon the felfe fame bafe. and hauing his altitude erected , and reafon by an argument leading to an impofiibilitic by the demonftration of the tenth of this booke , we may proue that the fided folidé included in“thelinclined cylinderisigreater then the triple of his pyramis, and itis alfo equall to the fame which is impoffible.And this is the frit cafe,wherein it was proued that the.cilinder not being equall to the triple ofthe cone is nor greater then the triple of the fame. And as touching the fcod‘eafe, we may afte: chefkne manerconclude that that fided’ folide contayned in the cylinder is greacer\then the cylinder : whichis very abfurd | Wherefore if the cylinder be neither greater then thetriple.of the cone,nor lefleatmuft nedes be equall to the fame. The demonitration of thefe inclined cylinders moft playvely followerh the demioniration of the ¢reSted cylinders: for it hath already bene proued,that pyramids,and fided folides ( which are alfo called generally Prifmes ) being {er vpon equall bafes.and.vnder one. and the felfe fame altitude , whether the altitude be ereéted or ine clined’, areequali the one to the other, namely , are in proportion ‘a8 their bafesare by the ¢. of this booke: Wherefore a cylinder inclined ‘thall ‘be triple to euery cone(although alfo thé’ cone be efected) {et.vpon one and the fame bafe with,it,and being vnderthe fame altitude . Butthe cilinder ereSted wac the triple of the fame cone by the tenth of this booke. Wherefore the cilinder inclined is équall to the cilinder erected being both fet vpoh one andthe felfe fame bafe; and hauing one andthe felfefime al. titude. The fame alfo cometh to paffe'iin onésjwhich are the third partes of equall cilinders, & there- fore are equall'the one to the other, | Wherefore according to the eleuenth of this bookeir followeth ,that cylinders and cones incli- ned or erected , being vnder oné and the felfe fame altitude , are in proportion the one to the other as their bafesare.For forafmuch as the erected are in proportion as theit bates are 5 and to the erected ci- linders the inclined are @quall: thereforethey alfothall:be in proportion as their bafes are. And therefore by the 12.0f this booke like cones and cylinders beinginclined are in triple propor- tion of thatin which the diameters of the bafes are’. For forafmuch as they are equall tothe ere&ted which haue the proportion by the r2,0f this booke,and their bafes alfo are equall with the bafes of the erected therefore they alfo fhalt-haue the fame proportions Wherefore it followeth by the'13.of this booke,thata eylinderinelined, being cut by. a.playne fu- perficies parallelto the oppofite playne fupefficieces therofihall be cut according to the proportions of the axes. For fuppofethat vpon one and the felfe fame bafe he fer an ere&ted ceylinderand an inclined cylinder, being both yndeér one and the félfe fame altitude; which let be cut by aplayne fuperficies pa- rallel to the oppofite bafes. Now itis manifett thar the fections of the one cylinder are equallto the fec- tion of the othercylinder, for they are fet vpon equall bafes , and ynder oneand the felfe fame altitude, namely ,betwene the parallel playne fuperficieces. And their-axes alfo are by thofe parallel playne fu- perhicieceseutproportichally bythe 17\ of theeleuenth. Wherefore the inclined cylinders(beimg equall tothe erected. cyhoders hall hhauetha proportion of their axes, as alfo have.the ereéted.Forin ech,the proportion of the axes'is one and the fame. Wherefore inelined Cones and Cylinders being fer vpon équall bafes, thall by the r4.0f this booke béinproportion.as theiraltitusdes ard For fora (much as the iaclined:are equall to the ereéted which haue the felfe fame bafes and altitude,and the.eretéd are in prpporian as their altitudes; thérfore the inclined fhall ‘be improportion the one to the other as the felfe fame altitudes which are common to ech; hamelyjto eShitelined and.to the ereéted. And therefore in equall cones and cylinders whether they be inclined or ereéted , the bafes fhall be reciprokally proportionall with the altitudes, atid cofitragiwife by the 15. of this booke . For foraf- much as we hauc oftentimes fhewed that the ihelined cones and cylinders are equall to the ereéted ,ha- uing the felf fame bafes and altitudes with them,and the ere@ted ynto whome the inclined are equall, have.theinbafes reciprokall proportionally with their altitudes, therefore it followeth , that the in cli- ned( being equall to theerected )haue alfo their hafes and altitudes ( which are common to eche ) reci- prokally proportionall .Likewife if theifaltitudés 8c\bafes be'tecipt okally proportidnall they théfelues alfo fhall be equall, for that they are equall ro the ereéted cylinders and cones fet vpon the fame bafes and being vnder the fame altitudesavhich ereéted cylinders aresequall the one to the other by the fame 15.0f this booke. Wherefore we tay conclude,that thofe paffions & proprieties which in this twelfth booke hauc bene proued to be incones and cylinders whofe altitudes are ereéted perpendicularly to the bafes;arevalfo mthofecones ahdicylinders whofe altitudes are fet obliquely vpon their bafes, How- beit this is to be noted that fuchinclined conesor cylinders are not perfect roundas are netaesna er yeaa By i Hh i a> PT, a a a er See ts ne er > Bie rte gin — —_ eS ee i ; i] - 4 ; be ) fet te ‘ t © ; A | V4 ' . f i Ny ‘. ae eter sere 4... pS ee ~ + ete — ahtinentc te LO = .___ _.___—. T herwelueth Booke > char 'ifchey be eut by a playne: fuperficies paffing at rightangles with their altitude, this fection is 2. Conicall feGtion which is called E&p/fs;and fhall not deferibe in their fuperficies 2 circle as it doth in e- rected cylinders & cones, but a céttaine figure, whofe leffe diameter is in cylinders equall to the dime- tient ‘of the bafe: that is,is one andthe {ame with it. And the fame thing happeneth alfo in cones incli- ned, being cut after the fame maner. , Lhe 1. Probleme. The 16. Propofition. T wo circles hauing. both one and the felfe. fame. centre being geuen.,to ins feribein the greater circle a poligonon figure, which fhall ‘confit of equall and euen fides and fhall not touch the Juperficies of the leffecircle.. f~| fame centre, namely, K At isrequived inthe greater circlewhich letbe ABC D | td 10 infcribe a policonon fieure which fhalbe of equal and euen fides and not touch Confirnffion, Weel 5 the circle E F G A. Drawe by the centre K aright line BD. And (by the 11. of the firft) from the point G rayfevp vate the right line B.D.a perpendicular line A.G,and exe: tend it to the peint C . Wherefore the line A C toucheth the circle E F G H (by the 15.of the third). Now therfore if( by the 30.0f the third) we dinide the circumference B A D into two equall partes , and againe the'halfe of thatinto — | two equal partes,and thus do cotinually,we fhall : = y ( by the corollary of ther. ofthe tenth ) at the H length leaue a certayne circumference leffe then the circumference AD . Let the circumference | | left be LD. .Avd fromthe point L, Drawe ( by | ge10gord eH iy the 12.0f the firft) vntothe line B D a perpendi-6. * ) i.Dee, culare line™ L M,andextendeittothe point N. ‘ | * norethis b fy bse of aie is ‘ Suppofe , V ppofe that there be two tircls ABCD, andE F G H haning one c thefelfe. if therefore there be applied-right linesequall tothe line L D continually into the circle Ax’ BC D(by ther of the fourth) there [bathe de[cribed in thecwele ABC D 2 poligonon figure madebythe which fhalbe of équall,and * ewen fides and fhall not touch the leffe circle, namely, EF G H:. ff many fo euer,namely ,E Aand EB. And draw thefe | angles DE Aand DEB are equall (for the line DE i rected perpendicularly to the playne fuperficies).And the right lines D A & DB which fubtend thofe angles are by the 12. definirion of the eleuenth equall: which right lines moreouer(by the 47.0f the firft)do contayne in power the {quares of the lines DE,E A,and DE,EB: if therfore from the {quares of the lines D E,E A,and D E,EByetake away the fquare of the line D E which is c6mon vato them,the refidue namely the fquares of the linesE A & EB fhallbe equal. Wherfore alfo the lines EA and EB are equall. And by the fame reafon may we proué, that all the right lines drawne from the poynt E to theline which isthe comon fection of the fuperficies of aah and of the playne fu- | perficies,are equall . Wherefore that line fhall be the cir. cumference of a circle, by the 15. definition of the firft. c *Butifit eo the plaine {uperficies which cutteth the {phere,to pafie by the centre of the pte Demonftrae tion. * The circles fs t ¢sthe right lines drawne from the centre ofthe {phere to their made:or fo cona common fection, thal! be equall by the 12 .def fuperficies of the phere. Wherefore of neceff tie th of the common feGion fhall bea circle, and his centre thal {phere. lohn Dee. Exchdehath among the definition of folides omitted certa kinde of Analogic. As a feement ofa {phere,a fe@or of a {phere {phere: with fvch like. But that(ifnede be , derfland a fegment of the fphere AB C to bethat part of the {ph (whofe center is E)and the fphericall fuperficies A FB. To whi A DB ( whofe bafe is the former circle : and to iter fector of a fphere,or folide fe&tor(as I call it). D E extended to F ment,to be the poynt F:and E Fis the altitude of the fe ter thé the halfe {phere,fome are lefle.As before A B Fis lefle,the re then the halfe {phere. QA Corollary added by the fame Flufas. By the forelayd affumpt it is manifelt . thar if from the centre of y Iphere the pevdicularly’ wnto the circles which cutee the /phere, be equal: thofe perpendtcular lines fo dravwne fall upon the centres of the fame circles. lines drawne per- circles are equal, ead the For the line which is drawne fré the centre of the {phere to the circumference,containeth in pow er,the power of the perpendicular line,and the power of the line which ioyneth together the endes of thofe lines. Wherfore fr6 thar fq the li : © center of the {phere to the circum. ference or cémon featio drawne,which is the femidiamete: of the fphere, taking away the power of the it followeth, that the refidues how many fo ever they be , be equall powers,and therefore the lines are e uall the one tothe other, Wherefore they will defcribé equall circles , by the firft definition of the ehird. atid vpon their centers fall the perpendicular lines by the 9.of the third . "And thofe citcles ypon which falleth the ersofthe lines drawne from the centre o quall,to thepowers of the perpendicular lines and alfo ir ci , the greater that the powers of the perpeadicul away from the power contayning them both are, the leffe a mayning., whic idi be greater or leffe,the circles thould be vnequall as it is perpendiculars to be equall.* Alf the : that are drawne from the centre of the {phere :f FFF j. from before manifeft.. But we fuppofe the vpon thofe bafes are the leaft of all, nition of the eleuenth.For that common fection isin the dered in the ¢ playne fuperficies comprehended ynder that line fohere, arecal- I be one and the fame with the centre of the ed the greatest circles: All o- ther, net ha- wing thei center of the {phere, to be their center yne, which were eafy toconceane bya al/s,are called athe vertex,or toppe of the fegment ofa lefSe csrcles, geué,in this figure next before,yn- Notethefe den > feriptions, gment {phericall.Of fegmentes, {ome are greae = ConfirnGlions T he twelueth Booke from the centré of the phere to the circumference of the circles, are in power equall both to the pow= ers of the perpendiculars and to the powers of the lines ioyning thefe perpendiculars and thefe fubten- dent linestogether : making triangles reCiangleround about : as moit eafily you may conceaue of the figure here annexed. A ___ the Center of the Sphere, AB the lines from the Center of the Sphere to the Circumte- rence of the Circles made by - B the Section, BCB the Diameters of Circles made se ag et by the Sections. AC the perpendiculars from the Center of the Sphere to the A, Circles:whofe diameters B C- : Bare one both fides or in any , | fituation els, \ P CB theSemidiametrs of the Cir- cles made by the Sections. AQ a perpendicular longe then 5 Mata 5) — WA = =e AC: and therefore the Semi- diameter O Bis leffe. 3 / + ACB, & AO Btriangles rectangle. B | qUbe2.Probleme. ~ The17.Propofition. T wo [pheres confifting both about one ex the felfe fame cetre sbeing gene, to in{cribe in the greater {phere a folide of many fides ( whichis called a Polyhedron )which fhali not touch the fuperficies of the leffe [phere. SFT V ppofe that there be two [pheres about one cy the felfe fame cttre namely, about AVS A.It is required in the greater {phere to in{cribe a Polyhedron,or a folide of ma- LN V/A ny fides, which fhal not with his fuperfictes touch the {uperficies of the leffe phere. IIE Let the fpheres be cut by fome 2g 52 fuperficies paling by the center A.T hen Conftructione [ball their fettias be circles." (For( by the 12.definttion of 1 he elenenth) the Diameter remat- * This is alfo ning fixed, and the femicircle being turned round about maketh a (phere. Wherefore in wr — “, what pojitio [0 ener you imagine the femicircle to be,the playne {uperfictes which paffeth by it fare added z hal make in the (uperficies of y phere a circle. And it is manifeft that is alfo.a greater circle, out of Fluf- for the diameter of the [phere which is alfo the diameter of the femicircle , and therefore alfo fas. of the circle t( by the 15. of the third ) greater then all the right lines drawne in the circle or Nate what 4 fpbere! which circles {hall haue both one center being both alfoin that one playne uperficies, eget ee ; by which the [pberes were cut . Suppofe that that fection or circle in the greater [puere be B C- By par: D E,andin the leffe,bethecircleF GH. Drawe the diameters of thofetwo circles in fuch Firkt part of forte that they make right angles,and let thofe diameters be BD , and CE. And let the line she Conflruc~ AG, being part of the line AB, be the femidiameter of the lefse [phere and circle ,as ABis ‘pom. the femidiameter of the greater [phere and greater circle: both the fpheres and circles hauing one and the fame center. Now two circles thatis, BC DE and F GH conjifting both about onc,and the felfe fame centre being geuen , let there be deferibed (by the propofition nexte going Lefore)in the greater circleBC DEa poligonon figure conjfifting of equall and euen fidessnot touching the leffe circle F G H. And let the fides of that figure in the fourth part af ae she circle, namely, inBE,beBK,KL,LM, and ME. And draw aright line from the point St Nae - ne ee eee 3 te = ~ aot ep epee dv o~ . =~ ian ree * of Euchdes Elementes. Fol.379. point K tothe point A,and extende it to the point N. And (by the t2.0f the eleienth ) from the Arayfeup to the fuperficies of the circle B CD Ea perpedicular line.A X,and let it light Nore; uppon the fuperficies of the greater {phere in the point X. And ly the line AX, and by either + rox kinow of thefe lines B D,and K N extend playne fuperficteces.Now by that which was before [poke, full wel, thofe plaine {uperficieces fhalin the* Superfictes of ¥ phere make two greater circles.Let their **## Sa oe femicireles confifting upon the diameters B Dand K NbeBX D.and K X Nw And foraf- Essig oo a much as the line X Ais cretted perpendicularly to the playne [u- perfictes of the circle BCD E.Therfore al the plaine [uperficicces which are drawne by the line XA are erected perpendscularly to the {uperficies of the circleB C D E ( bythe 18. of the eleuenth). Wherefore the femicircles BX D,and KX N are erected perpen: dicularly to the playne (uperficies of the circle BC D E : And for- afmuch as the Tenth BED,BX D ,andKX N are equall, i for they confiff vpon equall diameters B D, and K N': therefore alfo the fourth parts or quarters of thofe circles mamely,B E,BX circnmferences of thé cir cles ure: but by thefe csr- cutnferences the limstatsd andafsigning of circles is Cled:und fo,the circumfe. rence of a csrele, G/wally called u circle, which in this place can not offend, This figure is restored by M.Dee his diligence. For sn the greeke and Latiné Exclides,the line G Lthe lime A G,and the line K= Z, (in which three lynes the chiefe pinch of both the demonftrations doth fund), are Gntruely drawen:as by comparing, the fiudsonus may per- ceawe . Note. Ton mult smagine the right line AX, tobe perpedscular Cron the dé- ameters BD andC Ei: though here AC the fe- midiater, ferme to be part of AX. And fo sn other posntes sn this figure, and many other firengthen Jour timagihation accer= ding to the tenér of cone Strustions: though sa the delsneatio in plasne, fenfé be net fatiffied, and K X are equal the one tathe other. Whercfore how man’ fides of a poligonon figure there arein the fourth parte or quarter'B E ,fomany alfoare therein the other fourth partes or quarters B X,ana- KX ,equall to the right lines BK, KL,L Mj and ME. Let thofe fides be defcribed,and let them be B 0, 0'P;P R)RX,KS,ST, TV; andV X: and drawethele 20 Note. éguall ré right lines SOT Pyand V R. And from the pointes O and S.Drawe to the playne uperfities BK, in refed of thecircleB CD E perpendicular lines , which perpendicular lines will fall upon the com- of M. Dee his demonfi ration mon fectious of the plaine {uperficteces,namely upon the lines BD Cr KN, (by the 38.0f the’ roteing. FFf i. eleuenth) t..Were these poset Z that pas BBY the Letter Grs-. der atsh MM. Dive hes demg- fistien, Thetwelneth Booke elenenth ) for that the playne fiperficieces of the femicirclésB X D , K X N areeretled pera pendicalarly to the playne {uperqcies of thecircleBC DE . Let thofe perpendicular lines be GZ;tandS W.And drawe arght line from the point Ztothe point W. And forafmuch as athe equall femicircles BX Dand KX N the right lines BO, and K S are equall,from the ends wher of are drawne perpedcular lines 0 Z,andS W therfore (by the corollary of the 35. of the eleweth )the line 0 Z is equall to the line S W,cy the line BZ is equal tothe line K W. { Fiaflas proueth this an other way hus: Forafmuch as in the trianglesS W K , and O Z B, the two an- glesS W K,and O ZB are equal, fo that by coftrudtion they are right angles,and by the 27.0f third the angies 'W K S,and ZB O are equallfor they fubtend cqual circumfer ences SX Nand OX D, and the fide S K ss equall to the fide O B as t hath before ben proued. Wherefore (by the 26.0f the firft) the o- ther fides & angles are equall, namely, the line O Zto the line S W,and theline B Z to the line K W, J Bat the wholeline B Ais equalliothewholeline K A:bythedef- nition of a civtle. Wherfore the nfidueZ A ts equall. tothe refidue W A Wherfore the line ZW is. aparallel to the line BK (by thez. of the fixt). And forafmuch asether of thefe lines O Z,6 SWis ereited perpendicularly to the phyne fuperficies of thécitcle BCD. eee A E, therefore the line O Z is aparilleli.to the line SW;( by, the 6.0f .\. the cleuzth)and it is proued thatit is al{oeqisl untoit Wherfore rye’ NR the lines W.Z and S-O are al{o equalland parallels (hy the 7 sof the. « . _ eleucath,and the 33 of the firft saad by the 3 of the first) Ana for- . aftauch asWZis a parallelite 0: But ZW is a parallell.#s.t0.K B : Wherefore S.0 is alfo a parallell to K B,(by the 9.of the eleuenth)..And the lines BO, and K S do knit them toge- Sher WV herefore the fower fided foure BOK S isin one andthe {elfe ame playne fuperfictes: ~ of Euctides Elementes. Fol.378. For(by the 7 .of the elewenth if there be any two parallel right lines,and if in either of thems: be taken a point at allauentures aright line drawn by the{e points isin one, & the felfe fame playne fuperficies with the parallels. And by the fame reafon alfo euery one of the fomer fided fguresS OPT andT P RV is imone and the felfe fame playne ge pone the triangle RX is alfoin one and thefelfe fame plaine fuperficies(by the 2.of the elenenth)..Now if we smagine right lines drawne fro the pointes O,S,P,T, R3¥ ,tothe point A,there [halbe defcri-. bed a Polyhedro or a folide figure of many fides ,betwene the circs ferences B X and K X compofed of pyramids, whofe male are the Sower fided figuresB KOS,SOPLF,T.P R¥ andthe triangle V RX,and toppe the point A. And if inewery one of the fides K. L,L Mand-M E we vfe the felfe [ame confiruction that we did in BK, and moreouer in the other. three quadrants or quarters, and alfoin the other halfe of the{phere; there fhall then be madey a Polyhedron or [olide figure confifting of many ides de[cribed in the phere, which Polyhedron is made of the pyramids whofe bafes are the forefayd fower fided figures, and the triangle V RX , and others which arein the [elfefame-order with them : and common toppe to them allin the point A. Now I fay that the for[ayd polihedron folide of many fides toucheth not the fuperficies of the le(Je phere in which is the circle F G H. Draw (by the 11.0f the eleuenth) \fro the poynt Ate Setond part the playne {uperficies of the fower fided figure K BOS-a perpendicular line AT , and letit °[*h# e0n- fall upon the playne{uperficies in the point Y.And drawe the(e right lines BY Gt K. And — forafmuch as the line AY is eretted perpendicularly to the pleyne [uper ficies of BKOS, FF fay. therefore S « * 4 —— ‘ ==> —— esas a ———— - 7) CN «a ee. a r- SSS SS Oe ile ee ce a : ee eee oe aoe nan - ~~ - - = - — ~ : — . : - : — “= 4 -= —= < =- oer - ——* =r — As = — = * — 4 — a ad — — — = . a i —_ i 4 > : ~~ = Sc ee eee =2-0*- Oo — = - ras a. SSS: ' - : ii Vi Us ' uh vit. ae 4g Se ay. ae) if 4) Ye! | thee uaa ¢ uev Hi ' nt f] y | ? 4 } Hu) f + Wa! AH : i f ! 4) at) tbh 4 Ne a ’ in aiibt th fe 1 Gertie | titept } / } : Hy aT ot oe AM " : ‘ 4 ee ate Bh h ; » ae, i ia . Nee & : iy ih 5 : } 4 “ [4 4 b ‘¢ ie | 1 Uebel i ' th 4° y a4] Ari = el yh (Be a th a “4 4 rt eS eae = rye SS SS — a — ied \ } { { j ' - : i : +o { { Mi! ‘ i : HI ' ; fi ; ' 1 - i Mh nt 4 { v1 } = U i H oy ee L | | | - ( | in : th ; i t Te : : ead 1h el +t} t { > ue ae) ' t - f - ie f te i} i A \ 4 4 ry » | Oia , mae i } t Bat a | \ * Wi ! : hha) } | 1) aoe d er} io { i" ] wl ; ae ; ig itn miro ue 1 oa) NEES ; ee in , ayer i t [ert a jot - > ; i . Hi - te Prey Ti eel me i ‘ P| itt zt PE SS RAST hee = . heal mo Jeng Barba t- 3+ +-—= = a M J 4 ee s - bd -_ ae a Os ree eee ae 7 = Sy ARF ish pa ma al — are ~ ee Sees = Ls SEMA: Thetwelueth Booke . ohereforethe fame A Vis erected perpandicularly to.all the right lines that touch it, anid are Second part —snahe Plate ¢ fuper ficies of the fowerfided figure( by the.2.definition of the eleventh) . Where- of the demon- oe agg eae ge : age: | fration fore theline AY ts erected perpendicularlyto either of thefelinesBY , and YK .And foraf- mchast by the rs.definition of thefirft)theline AB isequall tothe line AK, therfore the {qnareof the line wx B is equalttothe fquare of the line-A K.Aud tothe fquare of theline AB are equall the [quaresof the lines AT,And YB bythe ap.ofthe firft) forsheangle Ba 2 ef ° “ r oe | 7 ~ én ” Pack oth o 4 - > ‘ ats Sith Yeenars in 1h 46 05k" _ ; Y Aisa@richt angle. And to the {4 : quave ofthe line AK areegual °°" the fayaresof the lines AY and 1K: Wherefore'the{quares eo SS the lines.AY;and Y Bare equalto the [quares of thelines AT, oh Ana K: takeamay the {quare of thé line’ A Y whith ts common / to ther both Wherefore the yefiduenamely,the Tanare of ¢ e B- Y is Gitall-tothe-refidne snameély ; to the pian of thelineT K. Wherefore the-tint BY is equal othe line K: In like fort slo K may we proue that right lines drawne from the point T to the > Pap! porntes 0, andS are equall tocither of the lines BY andT RK. OO Kdefiiibedeircle , und it hal palfe by the poyntes O , dnd S, and the fower fided eure K- So \ ZC the 2\of the fix becaije'A K isepeater then AW)sbiit the line WZ is equall tothe line's O.Wherfore the line B K is greater there WHeS'O: Bui the line B Kis equal to either . “LSS LONGUS RPS ST GE PARIS TG AIS bh. SMRS Th Stas “4 4 % a4 i ~ "ke ' Wi herefereanitre the center the poynt Y , and the fpace either theliné BY or the lineT 4 fr of Eviclidles Elementes. F0l.372, of the lines K $,and BO by confirdchion. Wherefore either of thé lines K Sand B Ois reas ter then the lineS Ov.And forafmuch asinthecircleisa former fided figureK BO S , andthe fides B.K,B.O.and K-S.are equat ard the fide O'S is leffe then anyone of them yand the line BY is drawnefromthe centre of the.circle: ther efoxathe {quare of the line K then the double of thefauare of the line BX ( by there anangle greater thena right anglecontayned of the Bis greater of the fecond) (for that it fubtendeth twaequaldlines BY , and K,which angle BY Kis. an obtufeangle.Rorthe giduelesat the céterY are equalto 4.xight aneles: of which threesnamely the anglesBEK 3K ES , and BY O-are equall by the g.of the firft,and the fourth nawmely,y angle SL Ousleffethen any of thofe three angles, by the 25.0f the "fir ft.) Drawe (by the 12.0f the first) from thepoint K ta thé line BZ; 2 perpendicular line WH K Z. And forafmuchastheline B Dis lefethen the double tothe line DZ ( for théeline BD is double tothe line D.Aswhich is lee then theline D\Z) sbut.ds the line BD i to the line D Z,f0 isthe parallelogramme contained. under the lines. DB and BZ;to the parallelo- gramme contained under the lines D.Z and ZB . B (by the 1. of the fixt ) : therefore if ye defcribe up- < onthe line BZ a [quare, and making perfecte the | parallelograimme contained under the-lides-Z D D as and ZB, that which is contained under the lines | D.B cy BZ; fhal! be leffe then the doubleto that whichis contdined wander the lines D Z and ZB... Anaif ye draivea richt line fromthe point K tothe point D, that which is contained vader the lines D Band BZ, is equall tothe {quare of the line BK (by the C orollary of the S.of the jixt.) forthe angle BK D tsaright angle, by the.31.0f the third, for it is in the fem micitle BED») : and that which is.contained under the lines.D Zand Zz B, is equall to the [quaxe of the line K Z, (bythe fagnec orollary ).. Wherefore the fquare of the line K B, islefethen thegoubleto thefquareaftheline-K Z But the (quare of the line K Bis greater then the double to the {guare.of the line BX,as before hath.bene proucd. Wher fore the {quare of the line K Z, is greater then.the {yuareof theline BX (by the s0.0f the fift ) And ‘foraf- much as.( by thers .definttion of the fir theline B Ads equalltothe line K A.therefore al- fothe [quare of the line BA is equallto she fquareof theline.K A But( by the 7.0f the first) vutothe[quareof the lined B, are-eqnall the {quares ofthelines BY & LAs forthe an- gle BY Aas by construction, a right.angle.)...And ( bythe fame reafon).to thefguare of the line K A, arecquall the fquares of thelines K Zand Zs for theangle KZA is alfo by confiruction, aright angle). Wherefarethefquares of the-linésBY and Y A, are equall to the {quares of the lines K Zand 24s Of whichthe {quare-of the line.K Z isgreater ther the {quare of the line BY, as bath beforehene prowed.. Wherefore the refidue, namely, the {quare of the line ZA; is lefvethep the Square of theline X-t..Wherfore the line Y Ais erea- ter then the line AZ..* Wherefore the line AX is much ereater then the line AG. But the line AY falleth upon.one of the bafesof the Relihedrow }and the line. AG falleth vpon the fi ote, of the le(je [phere Wherefure the Polihedron toucheth not the {uperficies of the An other and more ready dernonfiration te prone that.theline AY is greater then the line AG Raifevp ( by the 11,0 thefrft) fromthe poynt Gta.the line AG a perpendicular line G L.Anddrawa right line {rothepaint A tothe poyitL.Now thé deniding (by the 30. of the third) the circumference E Binto-halues, Cragaynethat halfe into halues,¢y thus do- ing continually,we fhall at the length by the corollanypof the firft of the tenth, leaue a certayne circumference which fhal be lelfe then the circumference of the circle BC D which is {ubten- ded of aline equall to the line G L.Let the circumference left be K B. Wherfore alfo the right “ine K B is lefe then shévight lineG L . And forafmuch as the fower fided figure B K O Sis FFf.tiy. £73 ° rf. W hich of necefs. ity foall fallGpon 2, a3 M.Deeprouerh it: and bis profe ss fet afe ter at this marke whe following, I.Dee. * But AZ is greater the A- G,as in the for- mer propofitsa, KM was eus- dent to be grea- ter then KG: fe may it alfo be made manio fel that KZ doth neyther touch nor cut the circle F Ge H, Another proue that the line A Y ss greater thé the line AG, SSS aS 9 Pit 1} ct Hoy : ii} ean \ rin i? } Hall I t aie ae ’ ih : til : - ’ ia} ‘et - 7 iat i} im) i > He | i He ; i i. | \' , 1 | i a) i 4 | oF | A a {tt 7 1! i | it) +) ie fg : ; f i ! ‘ } The twelueto Booke in acircle,and the lines O B,B K and KS areequall;and the line 0 S is lefe,therefore the an gle BY-K is an obtufe angle.Wherefore the line B K is greater then the line BY . But the line G Lis greater then the line K B.'Wherefore the line G Lis much greater then the line BY. Wherefore alfo the {quare of theline GL 4s greater then {quare of the lineBY . And foraf- much as (by the 15 definition of the firft)the line A Les equall to the line AB, therefore the fquare of the line A L is equail to the {quare of theline AB. But unto the {quare of the line ‘4 Lare equall the {quares of the lines AG andGL, and tothe {quare of the line A B are e- \guall the {quares of the lines BY and A .Wherefore the {quares of the lines AG and GL are eqwallto the {auares of the lines BY and¥ A, of which the {quare of the line BY is lefve then the [quare of theline G.L.Wherefore the refidue,uamely , the (quare of the lineY Ais greater th? the [quare of theline AG.Wherfore alfothe line AY is greater the the line AG. Wher fore tivo [pheres confisting both about oneand the felfe [ame center,being genen, there isinfcribed in the greater {phere a polihedron or {olide of many fides which toucheth not the {uperficies of the leffe [phere:which was required to be done. q Corollary. And if in the other [phere,namely,in the leffe [phere be in{cribed a Polihedron or fo- lide of vzany fides like to the polibedron in{cribed in the {phere B C D E, then the polihedron inferibedinthe {phere BC DE isto the polihedron inftribed inthe other {phere in treble proportion of that in which the diameter of the {phere B C D E isto the diameter of the other |phere.For thofe olides being denided into pyramids equall in number and equallin order, the pyramids {hall be like.But like pyramids are (by the 8. of the twelfth) the one to the other in treble proportion of thatin which fide of lke proportion is to fide of like proportion. Wher- fore the pyramis whofe bafeisthe fower fided figure K BOS and toppe the poynt A is to that pyramis which is of like order in the other (phere,in treble proportion of that in which fide of like proportio1s to fide of like proportid, that is of that inwhich the fide A B which is drawne frothectter of the {phere which ss about the céter A,ts to the fide which is drawn fro the ceter of the other [phere. And in like fort alfo euery one of 9 jaca which is in y [phere which is — about the cétre A isto euery one of the pyramids of the felfe fame order in the other {phere in treble proportio of that in which the fide AB isto the fide which isdrawne from the center of the other [phere.But as one of the antecedentes is to one of the confequentes’, fo are all the antecedents to all the confe- quentes by the 12. 0f thefifth. Wherefore the whole polihe- 0 dron folide of many fides which isin the sti which is about the center A,is to the whole polihedron or folide of many fides which isin the other [phere,in treble proportion of thatin which the fide A B is to the fide which 1s drawne from the center of the other phere, that is,of that which the diameter, B Dis to the diameter of the other fphere,by the 15.0f the fifth: which wasrequiredtobe ~~ demonftrated. q Majter seg ee ’ . LL LLL LA LLL LL eesti aiaaiai ta . Fol.386, M.Dee his deuiife, to helpe the imagina- tion to young ftudéts in Geometry: and to make his demonftration more evident as concerning the errors by hym correéted itt Euclides figure, by the ignorant, miflined: I.Dee, This figure is anfwerable ‘to the firfte plaine: which, citting the two Spheres by theifcommon center A, made two concerns tricall circles (hauing the fame center with the two Spheres)namely B C D E,and EF G- H.Vppon which, if you aptly reare perpen. dicularly , the fecond figure contayning two concentricall circles, (to the firft e- quall) and make the pointes noted with like letters to agree.and afterward vppon the fea cod figure, fet.on the third figure being here for the betterhandling made a femicircle: which vppon the firft figure mutt alfp be e- rected perpendicularly : And laftly if you take the little quadrangled figure BO KS, and make euery point to fouch,his like: & then reade the conftrudtion & wey the de- monitratio (twife or thrife being red oner) fhall you in this delineatié in apt paftborde, or like matter framed, finde al things in this probleme very evident. I neede not warne you,that the line AY may eafely be imagined, or with a fine thred fupplyed: or.ofthe right lines imaginable betwene P and T, and betwene R andV, I neede fay nothing, trufting that the ereat exercife paft, by that tyme you are orderly come to this place, will haue made you {uf- ficient perfect to fupply any farther thinge herein to be confidered. | ~The little fowercornérd peeces remay . ning to the femicircle, are'to be let through the firft ground playne: therby to ftay this fe micircle the better in hisapt place and fitu- ation: which it willthe more aptly doo, if ye do abate flauntitigly,the contrary araffes of the flict of it,and of the flitt of the fecond figure,into which itis to be let: abating thé alike much:.a litle will ferue. Experience, by aduife, will teach fufficiently, o Pica Se = = Se SS A — = - = : —- ~~ —-= : 5 2 = —~ - - : : - ——_ = = =a Sw SSS = ie. ho ee = SS aS < ae se ‘ > ae pat Lge Se - pate oS es SS SS SSS See: he i nee — = 3 RSI ATS a. ™ = => =~ : : aS : ; 7 2 —S—.—- . SS ee SS - - Se 3 — a =r = - = ——————oo —— — poke Te a r = = aalietel- a —— =—s — — a ~ ee Se) = =—=- = 2 tiene = a roo a os c ee nes < = eT his as4w 5; afumpt,ts 5; prefenrly >> pron ed, bs The twelueta Booke q Majter Dee his aduife and demonftration, reforming 4 great errour in the deficnation of the former figure of Euclides fecond Probleme: with rio Corokaries ( by him inferred ) upon his {asd demonstration. x. AT heoreme. %% T bara right line drawen from the point K, perpendicularly upon the line BX, doth fall vpon the poimt X : Wwe Will,thus,make enident. Y the premiffes,it is manifeft, that the point Z is that poine where a right line from the point O,be- ing perpendicularly let fall co the circle BC D E, doth touch the fame circle. Which point Z alfo is proued to be in the right line B A D, the common fection of two circles cutting eche other: being one to the other perpendicularly erected . Thefe thinges,with other, before demonitrated, I here make my fuppofitions . Confider now the two triangles reGangles OZ B and K ZB: OF which, the angle O ZBisequallto theangle KZB. For, byconftruction, they are both right angles: * and the angle ZB O is equallto,the angle Z B K . For, if from D to K you imagine a right line : and the like from D to O : you haué two triangles in equall femicircles, rectangles,namely, D K Band DOB : which haue the diameter BD common: andB K, the Chord, equall to B O the Chord, by conftruétion. Where- fore (by the 47.of the firit) the third fide,namely, DK, is equall to the third, namely,D O : Wherefore (by the s.of the fixt) the angle ZB O,which is DB O,1s.equall to Z B K, which is DB K . (For the line Z DB, by conftruction, is part of DB) . And feing two angles of-O.Z B, are proued equall,to two angles of KZB, of neceflitie the third,namely, Z O B,is equall to the third,namely,Z K B,by the 32.0f the firtt. Wherefore the two triangles rectangles O Z B and K ZB, are proued equiangled. By the fourth, there- fore,of the fixe, their fides are proportionall-: therefore by the premifies, proued, as B O isto BK, fo is O Zto KZ, andthe third line, which fubtendeth the angle Z OB, to the third line which fubten- deth the angle ZKB. But, by conftruction,B O is equallto B K : therefore O Zis equallto KZ: And the third alfo is equall te the third , Wherefore the point Z,in refpecte of the two triangles reCtangles, O ZBand K ZB, determineth one and the fame magnitude, in the line BZ. Which can not be: ifany other point, in the line B Z, were afligned nearer, or farther of, from the point Bs. One onely poynt therefore, is that, at which the two perpendiculars K Zand OZ fall : But, by conttruction, O Z fal- leth at Z the point, and therefore at the fame Z, doth the perpendicular, drawen from K, fall likewyfe: Which was required to be demonitrated. Although a briefe monition, mought herein haue ferued for the pregnant or the humble learner, yet for them that are Well pleafed to haue thinges made plaine, with many wordes,and for theftiffenecked bufie body,it was neteffary, with my controlment of other, to annexe the caufe & reafon therof,both inuincible and alfo euidenr. o 4 Corollary. 1. _ ” - Hereby itis manifest, that two equall circles cutting onethe other by the Whole diameter if froms one andthe fame end of their common.diameter,equall portions of their circumferences be taken : and from the pomtes ending thofe equall portions, twe perpendsculars be let downe to their common diame- ter, thofe perpendiculars fhall fall vpon one and the fame point of their common diameter. a. Secondly st followeth that thofe perpendiculars are equal. q Note. eg es seule ad, fuppofition eche to other perpendicularly ereéted, we procede and inferre now thefe Corollaries, whether they be perpendicularly ereéted or no : by reafon the demonftration hath a like force , ypon our {uppofitions here vfed. q Ube 16. Theoreme. T he 18. Propofition. Spheres are in treble proportion the one to the other of that in which their diameters are, Suppofe of Euchdes Elementes. . Fol.381, il V ppofe that there betwo [pheres ABC and DEF, andlet their diameters be |B Cand E F .Then I fay that the [phere A B Cis to the [phere D E F in treble proportion of that in which the diameter B C is to the diameter E F. For if not, Si then the [phere A BC ts in treble proportion of that in which B C is to E F , ei- ther to.fome {phere lef then the {phere D E F , or to fome {phere greater . Firf let it be unto a le(Se, namely, toG HK .\ Andimagine that the [pheres D E F andGH K be both about one and the felfe fame centre. And(by the propofition next going before)defcribe im the grea- ter (phere D E F a polihedron or a foltde of many fides not touching the {uperficies of the lefe SphereG HK . And {uppofe alfo that in the {phere ABC be infcribed a polihedron like to the polihedron which is in the [phere D E F Wherefore(by the corollary of the fame) the po- libedron which isin the {phere AB C , isto the polthedron which isin the {phere D E F in treble proportion of that in which the diameter B C is to the diameter E F But by fuppofition the {phere A BC is tothe [phere G H K in treble proportion of that in which the diameter B- Cis to the diameter E F Wherefore as the [phere AB Cis to the [phereG H K , fois the poli. hedro which is de{cribed in the {phere A B C to the polihedro which is defcribed in the [phere DE F by the 11.0f the fift.Wherfore alternately ( by the 16. of the fift)as the [phere A B Cis to she polthedron which is de{cribed in ito is the {phere G H K to the polihedron which is in D the [phere D EF .But the [phere ABC is greater then the polihedro which is de[cribed init. Wherfore al{o the [phere G H_K is greater then the polihedro which is in the [phere DEF by the 1 4.0f the fift.But it is alfo lefe,for it is contayned in it, which impoffible. Wherefore the {phere ABC is not in treble proportzo of that in which the diameter BC, is to » diameter EF, to any [phere le(Se then the [phere D E-F In like fort alfo may we prone that the {phere D E F és not in treble proportion of that in which the diameter E F is to the diameter BC, to any [phere lefe then the [phere ABC. Now I fay that the [phere .A B Cis not in treble proportio of that in which the diameter B Cis to the diameter E F to any [phere greater thé the {phere D E F. For if tt be poffible , let it beto.a greater,namely, to L M N.Wherfore by conuerfion the Sphere L M N isto the {phere A BC in treble proportion of that in which the diameter E F 1s to the diameter B C.But as the {phere LM Nis to the [phere A BC, fois the [phere DE F to fomse {phere leffe thé the {phere A B C,*as it hath before bene proned,for the [phere L MN is greater then the [phere D E F.Wherfore the [phere D E F is in treble proportio of ae in whic Two cafes in this propoft= £701, The firft cafes Denzonftra- tion leading te an smpofirbs~ bitte = Second cafes * As it $s eafie to gather by the afsumpt pur after the feconh of this bookg. Note: a genes gall rule, Couftratiton. The twelueth Booke which the diameter EF is to the diameter BC to fome Sphere lefce thé the [phere ABC, which 45 prouedto be impoffible Wherefore the {phere ABC is not in treble proportion of that in which BE isto E F toany {phere greater thé the {phere D E F.And itis alfo proued that it is not to any lefse Wherefore the [phere A BC is to the phere DE F in treble proportion of that in which the diameter BC is to the diameter E F :which was required to be demonftrated. A Corrollary added by Fluffas. Hereby it is manifeft that {pheres are the one to the other as ke Polibedrons and in like fort defm svribedin then aresnamely,eche are in triple proportion of that in which the diameters. A Corollary added by M.ADee. It isthen enidext, how to gene two right lines, baning that proportion betwene them, which aup two Spheres genen, haue the one tothe other, | ‘nats For, ifto their diameters, as to the firft and fecond lines (of fower in-continuall proportion ) you adioyne a third anda fourth line in continual proportion (as I haue taught before ) : The firftand fourth lines, fhall aunfwere the Probleme. How generall this rule is,in any two like folides, with their correfpondent (or Omologall) lines, Ineede not, with more wordes, declare. q Certaine T’heoremes and Problemes ( whofe vfe is manifolde, in Spheres,Cones Cylinders and other folides) added by Ioh.Dee. x AT heoreme. 1. T he whole Juperficies of any Sphere, us quaarupla, tothe greateft circle, in the fame Shere con tayned . It is needeles to bring Archsmedes demonftration hereof, into this place: feing his boke of the Sphere and Cylinder, with other his workes,are euery where to be had, and the deméitration therof,eafie. se ATheoreme. 2. Exery Sphere, is quadrupla, to that Cone, whofe bafe is the greateft circle, height,the femidiae meter of the fame [phere. es This is the 32.Propofition of Archimedes firft booke of the Sphere and Cylinder. A Probleme, I. eA Sphere being geuen,to make an vpright Cone,equalltothe fame : or in any other proportion, geen beiwenetwo right lines, nie! saci = Suppofe the Sphere geuen, to be 4,his diameter being 8 C, and center D: with a line equall to the femidiameter D (-which let be NO _) deferibe a citcle W RP: whofe diameter let be WP, and center O, itis evident, that N R P is equall to the greateft circlein.4, contayned. Arthecenter 9, leta perpen» dicular be reared equall to B D ( the femidiameter of 4 ) which {uppofe to be O @ ; It is now plaine that to the Cone, whofe bafe is the circle wRP, and height @ , the Sphere 4, is quadrupla : by the z.Theoreme here, and by conftruétion « Take aline equall to.w /, which let be F E : and with the fe- _ midiameter F E (making the point F center ) defcribe a circle : which fuppofe to be £.X.G, and dia- _ mererEG. Atthe center F, reare a line perpendicular to £ KG, by the 12.0f the cleuenth ; and make Demonstrc- Siete . icequall to'O Q » Lerthat line be FZ: I fay thatthe Cone, whole bafe is the circle £X.G, and heighe the line F L, is equallto.4. Forfeing F Z, the femidiameter of £ X.G,is equallto N” ( the diameter of N &P ) by conftruction : £ G, the diameter of £ KG, fhall be double to w rv . Wherfore the {quare of £ G, isquadrupla to the {quareof WP: by the 4.ofthe fecond . But as the {quare of £G, is . the quare of Exuchides Elementes.” Fo/.382, {quare of N’P, fois the circle E K.G to the circle w RP: bythe» ofthis ewelueths Wherefore the circlé E KG, is quadruple tothecircle NAP. And FL, the height is (by conftruction)equall too @ the height . Wherefore the cone, whofe bafeisthe circle é.K Gand height-F D;1s qivadruple'to the cone; whole bafe is WR P,and theheightO Q; bythe rt.of thistwelfth. But vato the fame cone whofe bafeis NR/?,and height O 2, the Sphere 4 1s likewife proued } K quadrupla!. Wherefore the cone whofe bafeis EXG and heighe FL, 1s equall to the Sphere -4: by the 7.ofthe fift. Toa Sphere being geuen therefore, we haue made an vpright cone equall. And as concerning the other partofthis Probleme, it is now eafie to execute ,and that two wayes : Imeane to 4 the {phere geuen,to make an vpright cone Jn any proportion geuen betwen two rightlines. For, lecthe pro- portion geuen, be that which is betwene Xand r. By the order of my additions, vpon thea. of this twelith booke : to the circle EKG make an other circle in that proportion that X is tor: which let- be Z. Vpon the center of Z, reare a line perpendicular and equallto FL. I fay thatthe cone, whofe bafe is Z,and the height equall to FL,is to 4,ia the proportion of Yto r.For the cone vppon Z, by conitruction, hath height equall to thesheight. ofthe cone LEKG: and OO See ) Loe tins contruction, isto KG, ds Xisté T? Wherefore, by the'rr.of this twelfth, the cone vpon 2, is to the cone 4 £KG,asXisto Yr. Butthecone L £ K Gis proued equal! to the {phere 4. Wherfore the cone vpon Z, 18 to 4, as Xis to r, by the 7.0f the fift. To a Sphere gcuen therefore, we haue made a cone in any proportion geuen, betwene two right lines . Secondly, As-<%5 (6 F}fo'to FL, let there bea fourth line, by the 12.0f thefixt : and fuppofe itto be #77. fay that ‘acone, whofe bafe is equallto EG, and height the line 1¥,,is to «4,as x 1s to Y .For,by the a#iorthistWwelfth, Cones being fet onequall bafes, are one to the other, as their heightes.are : Bur, by-conitradtion; the height W, is to the height F 2, as % X isto Y:. Whercfore the cone whieh hael his bafeequallte EG; and height the line W, is tothe cone L EXG,as XistoY . And itis prouéd, chat rote cote DE ICG) the Sphere 4 is equall: Wher fore, by the 7.of the fift, the cone, who bale is eqdalbto BKiG, and height the line W; is to 4, as Yis tor. Theretore a Sphere heing geuen;~wehaue madean vplight cone, inany ‘proportion geuer be- twenetwo rightlines , And before, we made an vpright cone, equail to the Sphere geuen . Wherfore a Sphere being fSeuen, We haue nadean vpright cone, ¢ quall to the fame, or in any other proportion, gcuen betwenetwo right lines. Icall that an vpright cone,whofe axe is perpendicularto his bafe. @] 4 Corollary. Of the fizft part of the demonstration, itis enident: A Sphere being propounded, that'a Cone, Whofe bale hath his (:midrameter, equall to the diameter therof,and heicht equallto the Jemidiame- ter of the fame Sphere, is equall to that [phere propaunded. | q-A Probleme. 20 ef Sphere being genen and a civcle,tovenre an TPR C one,upon that circle ( asa bafe,) equak fotn at e Sphere genen : or in.an 1 proportion betwene two right lines affigned, Suppofe the Sphere geuen, to be Q :: and the circle geneti tobe C+ By the firtt Probleme make an vpright cone equall to Q the Sphere geuen : which cone fuppofe to be A’+ and (by the 2.Probleme of my additions vpon the fecond of this awelfth booke) as C the circle geuen,is to the bale of A, fo let GGg,j. the Theferond partof the Probleme tivo Wwayce$s executeds i. as An vpright C one, rd’ , The tweluetb Booke tlie heightofA, be to aline found t,whichletbe D. Themitas evident, that the cone, which hath for his bafe C, the circle géuen, and height, thedine D, laft found, fhall be equall ro Q the Sphere geuen: which conelet be F .. For, by conitruction, F hath his bafe and height in reciprokall proportion with the cone A, made equall to Q- the Sphere geuen : Wherfore 1 by the rs.of chis twelfth, and 7.0f the fifth; this vp- right cone F, reared vpon C, the circle geuen, 1s e- ze quail ro Q_, the Sphere geuen : which thing the Probleme firit required. And the fecond part of this Probleme is thus per- formed . Suppofe the proportion geuen to be that Thefecond whichis betwene X& Y.Then,as Xis to Y,fo letan part of the other right line found, be to the height of F : which Probleme. line let be G. For this G, the found height (by con- ftruétion) being to the height of F,as X 1s to Y, doth caufe this cone (which let be M) vpon C,the circle geué (or an other to itequall)duely reared ,to-be vn- » to the cone F}asX is to Y, by thet4.of this twelfth. ~*~ But F is proued eqiiall to the Sphere geuen: Wher- » fore M, 1s to the Sphere geuen,as Xis to Y. And M, ov is reared vpon the circle geuen:or his equall. Wher~ fore, a Sphere being geuen,& a circle, we-hauerea- “ sos red an vpright cone, vpon that geuen circlé (asa G Ai | bafe}.equall to the Sphere.geuen : orinanypropor- © ———-~__ = tion, bet Wene twerightlinesaffigned : which was’ ‘., | required to be done. ? & me) OF , 4 fit | 14 tives | : he 7 ui { Hy 4 : ) ‘ | I iJ 7 | Roky ie bi : + ; } { H 15 ,. Hue SR oct + ie i} : : a i, bs oe } 4 + ee 1} oe , i 1 7 r \ by . |. 4 ai - 3 - | t ; { ef Sphere being genen,anda right line, to make an upright cone; equall to the Sphere. enen, oF in any other proportion geen. betwene two right lines : Which made cone,fhall hane bis beaks e couall a * (793 roche right lime peugn. | te OR? sSappofe the Sphere geuenstobe R: andthe. rightlinegeuen,to be S. To'R the Sphere geuen, make an'vpright cone,equall: by the firlt Probleme. : whichcone fuppofe to be A. Then-as S, the line ge- nem, isto the height of A, folet the bafeof A; bero an other citele, which let be K, by my additions, vp- on thefecond Propofition of this twelfth bocke.I>2\ fay.thatan.vpright cone, hauing his, height, equalles «: to S, the right line geuen, and his bafe K,isequall tor» ; the Sphere geuen . Let this cone be noted by L: foro» by confiruction., thys cone 1,.and As haue theyr: heightes and bafes reciprokall in proporti6: Wher- fore this cone L,andthe cone A, are equall,'by. the .. 15. of the twelfth). But A is equall to the Sphere ge> | uen by conftruétion. Wherefore L is equall to the Sphere geuen « And the height of L,is equall to the. right line geuen, by conttruction <_ which ought to Pa ds G-ccnd ve Plate acirce, LIN TIM For the fecond part : findeacircle, which tha The fe = haue to thie bafe of L, any proportion appointed?in part of the right lines: as the proportion of X to ¥.+ which;by Probleme, my additions, vp6 the fecond of this booke, ye haue learned to do. Then, with the height, equall to the heigth of L rearéd vpon this laft found circle,which let be T, asa bafe, you fhall f{atiffie the Probleme. Let chat Cone be V . For thislaftcone V, isto L,as his bafe is tothe bafe of L, by the 11.0f this twelfth.. <>’ But & is proucd equall to the Sphere geuen: Wher- « : | fore by the 7. of the fift, this lalt cone V hath to - is : the #7 Sn ee eS = - e - = > ee ‘agli Se = = "ET oa = — nr ‘” ‘2s ; of Euchides Elementesa\ \ F0l.383. ché’Sphere geuen, that proportion whichis: betwehe X and! ¥; dtigned :-andforafmuch asthe height of this cone’ Vs 19equall to cheheight of Li and the. heighrofksequallto Ss the wight line, geuen ¢ by conftru@tion:) pitis euident, thata Sphere being geneo; Kamight lineswehavemadeanypright cone, equalkto theSphere.geuen; oriiti any:other proportion geuen:betwene two right-ines»iwhich made cones, haue their height equallto the right line geuen : which ought to be done. q Vuwilling Iam co vie thus many wordes, in matters fo plaine:and.eafie .| But this, (J thinke ) can not hinder them, that by natur@are not {fo quicke of inuention, as tolead euery, thing, generally {po- ken, to a particular execution. | @ ATheoréme. 3. Euery Cylinder w hich hath his bafe,the greateft Circleina Sphere, ¢ heith equallto the diame- ter of that Sphere, is Sefquialterato that Sphere.Alfothe fuperficies of that Cylinder, with his two: bafes is Se(quilaterato the fuperficies of the Sphere:and wathout bis two bafés ; 1s equal to the fuperfi- cies of that Sphere ies? Suppole,a {phere to be fignified by A, and an vpright cylinder hauing to his bafe a circle equall to the greateft circle in A contayned , and his heith equall to the diameter of A , let be fignified by FG .T fay that F G,is fefquialter to A: Secondly I fay that the croked cylindricall fuperficies of FG; together with the fuperficieces of his two oppofiebafes, is fefquialtera to the whole fuperficies, {phericall of A. ThirdJy I fay that the cylindricall fuperficiesof F G, omitting his two oppofite bafes', ts equall to the fuperficies of rhe fpere A. Let the bafe of FG, bethe circle FLB ‘whofe center, fupyofeM, and diaméter F B.And the axe ofthe fame F G,let be,.M H. Which is his heith (for we fuppofe the cy+ linder to be vpright) :and fuppofe H,to be his toppe.or vertex. Forafmuch as , by fuppofition M H ts equall.to the diameter of A.Let MH be deuided into two equall partes in the point N , by a playne fu- perficiés pafling by the point N, and being parallell to the oppofit bafes of F G, By the thirtenth of this twelfth booke,itthen foloweth,that the cylinder F G , is alfo deuvided into two equall parts :being cy linders: which two equall cylinders let be 1 G, and F K: the axe of | G fuppofe to be H N : andof FK the axeto be N M.And for that,F G,is an vpright cylinder ,and at the poynt N,cut by a playne Superficies parallel to his oppofite bafes , the common fe- étion of that playne fuperficies and the cylindérF Gmultbe a* circle’ equal, to his bafe ELB , and haue his center, the point N. Which circle,let be 1 O-, K:And feing that FL Bis, by fuppof- tion, equall to the greateft circie in A, 10 K,alfo, fhall be equall to the grea- te(t circle,in A,contained‘Alfo, by rea- fon M H,is by fuppofitionyequal-to- the diameter of A: and NH, byconttru- &id half of M H, it is manifeft chat N- His equallto the femidiameter of A. 3 Ifitherefare,you fuppofe a cone to’ haue the circle OKs his bales and NH to his heith,the fphete A, fhall be to that Cone, quadrupla , by the2. Theoreme.Let tharcone be HIOK. Wherefore A;1s Guadriplate- HO K.And the Cylinder'l G hauing the fame bafe; with H TO K(the circle O K)and the fame heith,(the right line’ N’H)is triple to the cone H LOX “ by the ro.of this twelfth booke. But to I G,the whole cylinder F G,is double,as is proued: Wherefore F G,is triple and triple , to the cone HI OK, that is, fextuple. AndA is proved quadrupja tothefame HIOK. Wherefore F G is to H1O K,as6.to r:and A,is to H 1 O K,as 4.to 1: * Therfote F G is to A,as 6,to 4: which in the leaft termes, is,as 3 to2.but3 ta2z;is the termes of fefquialcera proportion. Wherefore the cylinder FG, isto'A’fefquialtéra in proportion... Secondly, forafinuchasithe {uperficies of a:cylinder(histwo oppo- fite bafes excepted ) is equall to that circle whofe femidiameter is middell proportionall betwene the fideofthe:cylinder, and the diameter,ofhisbafe:.Cas vato the 10,0f this booke ,lhaue added.) But of F-GsthefideB G,being parallelland equallto the axe M H, muttalfo be equall to the diameter of A. Andthé@bafeE L:Bsbeing(by {uppofition)equall to the greateft circle in A contained ,muft haue his di- ametér(EB)equal.to the fayd diameterofA,The middle proportional therfore betwene BG and FB, being equalleché to.orher,thali bea ling, equall toveither of them. . Lais by osiifeeB Gand FB together, as oneline,jandyvpor that ling-compofed , as. a.diameter make a fem icincde:andifiom the center ; tothe circumference yf aline perpendicular to the fayd diameter: by-the-tzof the fixeh,thatperpendicular,is middel proportional betwene F Band B G,the femidiame- ters :and hehim felfe alfo.a fernidiameter:and thertore by the definition of a circle, equallto FB, and likewiie;to: BGs} Andacircle,haning his femidiameter,equall to the diameter F B,1s quadruple to the citcleF LB. {Forthe fquareofeuery whole line ,is quadcuple tothe {quare of his halfe line , as gel GGg.j). ¢ ot 2 + This 772A Ca« fely be demon- ftrated,asin the 17 .propofitton the fection of a [phere was pro ued to bea cir= cle. + For taking ae way all doubt, this, asa Leme« ma afterward ss demoftrated, A Lemma (asitwere) 4g, prefenrly de- monflrated 5 Meee tas : : en + = = * + - —=-*+ ~ > ear ————— : 7 = ——— = - — —— SSS —— = > ~ ar. = 3 a= a ene oe eee = ee == Se a - ~~ a - = — ne ee oe <= z : = =a p= Se =S = = Si ies ; —=—= Conftrauction, Tremou|ftra- $iGH, The twelueth Booke be proued by the 4.of the fecond:and by the fecond of this twelfth, circles are one to the other, asthe {quares oftheir diameters,are.] Whertore the fuperficies cylindrical] of FG, alone, is quadrupla to his bafe F LB. But ifa certayne quantity be dupla to one thing , and an other , quadrupla to the fame one thing, thofe two quantities together are fextupla to the fame one thing. Therefore feing the bafe,oppo- fiteto FL B,( being equall toto FLB) | added to F L B, maketh that cépound, double to F LB : that double added to the cylindrical {uperficies of F G,doth make a fuperficies fextupla to FLB. And the fuperficies of A, is quadrupla to the fame FLB, by the frit Theo- reme. Therefore the cylindricall fuper- ficies of F G, with the fuperficieces of his two bafesis to the fuperficies FLB, as 6 tor. and the fuperficies of Ato F- LB,is as 4to 1. Wherfore the cylindri- call {uperficies oPF G,& his two bafes, together, are to the fuperficies of A,as 6 to 4: that is, in the {malleft cermes,as 3 t02. Which isproper to fefquialtera , proportion.Thirdly,itis already made evident that the fuperficies cylindrical,of P G*fonely by it felf) is quadrupla to FL B. And aifo it is proued , that the fupérficies of the {phere A , is quadrupla to the fame F L B, Wherefore by the 7.of the fifth, the cylindrical] fuperficies of F G, is equall to the fuperfi- cies of A. Therfore,euery cylinder,which hath his bafe the greateft circle in a fphere,and heith equal to the diameter of that fphere,is fefquialtera to that {pere: Alfo the fuperficies of that cylinder with his two bafes, is fefquialtera,to the fuperficies of the fphere:and without his two bafes is equall to the fu- perficies of the {phere: which was to be demonftrated. The Lemma. If AbetoC,as 6,to t:and B,to Cas 4t0 1:4,istoB,as 6,to 4. For,feing Bis to C,as 4 to 1,by fuppofition: therefore backward,by the 4.0f the fifth,C is toB,as r,to 4. Imagine now two orders of qnantities,the firft, A C B : A,C,and B the fecond,¢,1,and 4.Forafmuch as,A,is to : : 9 \ C,as 6, to 1, by fuppofition: and Cis toB, as 1, to 4,as ; we haue proued: wherfore,A isto B,as6 to 4,by the 22 Se ERENCUR ES of the fift.Therfore, if A be to Cas 6 to 1,and B tojC,as 6 I 4 4to 1:A is to B,as 6,to 4.which was to be proued. ELE | COMOSTL SS Si Ncichcunmbumeumemmmecdl Notw. Sleight things(fome times) lacking euidét proufe,brede doubt or ignorance.And,I nede not warne you,how gen:rall,this demonttration is : for ifyou putin the place of 6 and 4,any other numbers, the like manner of conclufion will follow.So likewife,in place of r.any other one number may be, as, if A be to Cas6 tos :and B ynto C,be as 7 tos: A,fhall be to B,as 6 to 7. &c. cA Probleme. 4. T aa Sphere geuen,tomake a cylinder equall, or in any proportion genen betwene two right lines. Suppofe the geuen Sphere to be A : and the proportion geuen to be that betwene X and Y. I'fay that acylinder isto be made, equall to A* or els in the fame proportion to A, that is betwene X to Y. Let a cylinder be made (fuch one as the Theoreme next before fuppofed)chatihall haue his bafe equal to the greateft circle in A, and height equall to the diameter of A : Lerthatcylinder be the vpright - linder B C.Let the one fide of B C, be the right line Q C.Deuide Q C into three equal parts: of which, let Q Econtaine two, and letthe third part be C E. By the point E fuppofe 2 plaine ( parallel co the bafes of BC )to paffe through the cylinder BC; cutting the fame by thecircle DE .1 fay that the cy. JinderB E is equall to the Sphere A. For feing BC, being an vpright cylinder, is cur bya plaine, pa- rallel to his bafes, by conftru@tion : therefore as the cylinder DC, is tothe cylinder BE, fo is theaxe of DC, to the axe of B E, by the 13.0f this twelfth . Wherefore as the axe is to thelaxe, fo is ‘cylinder to cylinder. But axe is to axe, asfide to fide,namely, C Eto QE, becaufe the axe is parallel to any: 8 | ofan ras =e i. ee of Euchdes Elementes. Fol.384., ofan vpright cylinder: by the definition of 2 cylin- der. And the.circle of the fe&ion, is parallel to the bafes, by conftruction. Wherefore in the parallelo- gramme (made of the axe, and of two femidiame- ters, on one fide parallels, one to the other, being coupled together by aline drawen betwene their endes in their circumferences, which lineis the fide QC )itis euident, that the axe of B.C is cutin like proportion, that the fide Q Cis cut. Wherfore the cylinder DC, is to the cylinder BE, as EC isto QE . Wherefore, by compofition, the cylinders D CandB E, that is, wholeBC, areto the cylinder P yg ‘ : bee sande te ae bLlS S. atl ; | BE,asC Eand Q E (the whole right line Q C)are Re ee eT a TT Ruth ; to Q E.But by céttruction, Q Cis of 3-fuch partes, 3 | G————__ yy Or atte ad as Q Econtaineth 2. Wherefore the cylinder BC, Bc ENR eS ee \ is of 3 .fuch partes, as B E contayneth 2. Wherefore BC the cylinder, is to B E, as 3:to 2 : which is fefquialtera proportion, But by the former Theoreme, B Cis fefquialtera to the Sphere A : Wherefore, by the 7. of the fift, BE is equall to A. Therefore toa Sphere geuen, we haue made a cylinder equall, ‘ Or thus more briefely omitting all cutting of the Cylinder, Forafmuch as B C is an vpright Cylinder: his fides are equall to his axe or heith: therefore the two cylinders’, whereof one hath the heith QC and the other the heith QE, hauing both their bafes,the greateft circle in the Sphere A,are one to the other as Q Cis to Q E, by the 14.0f this twelfth,buc Q C is to Q E,as 3.to 2,by conftru@tion:and 3.tO 2,is in Sefquialtera proportion: therefore the cylinder B C hauing his heith Q C,& his bafe the greateft circle in A coteyned,is Sefquialtera to the cylinder which hath his bafe the greateft circle in A conteyned,‘and heith the line QE. But by the former Theoreme, B C,isalfo Sefquialtera to A: wherfore the cylinder hauing the bafe B Q (which by fuppofition,is equal to the greateftcirclein A conteyned )and heith, QE, is equall tothe fphere A, by the7.of the fift . And the fide Q E,be to Q P,as Y is to xX, by the 12.0f the fixt. Therefore backeward 9 QP isto QEasX to Y. Wherefore the cylinder hauing the bafe the greateft circle in A and heith the line QP , isto. the cy- der haning the heith Q E,& his bafeithe Sreateft circle in'A; conteyned, is proued equall tothe Sphere Fors by the 7.of the fift,the eylinder whofe heith is QP and bafe the greatett circle in A,con- teyned,is.to the fphere A,as X to ¥: Therefore toa {phere geen: we haue made a cylinder, in any pro- portion geuen betwene tworight littesand alfo , before we hauetoa {phere genen,madea cylinder e- quall.Therefore to a {phere geuen;we hatie madea cylinder equall,or in any proportion geuen betwene AProbleme. .5, cf Sphere being genen, anda circle upon that circle asa bale, torerea cylinder, equall tothe Sphere LeHen: or in ary proportion wenen bétwene two right lines, AProbleme. G. AA Sphere being geuen , anda right line;tomake a cylinder , equall tothe phere gewen,orin ANY Ga ther proportion, betwene two right lines Leuen. In this s.and 6.probleme,firft make’a cylinder equall to the {phere uen,by the4.probleme:and then by the order of the 2.and 3-problemes,in cones,execute thefe accord; gly in cylinders. A Probleme. © 7: T wo vilike Cones or Cylinders .b cing geuen,to finde two right lines , which hauethe Same proper tion one to the other,that the two £euen cones or cylinders haue one tothe other, Vpon one of their bafes rere a cone(ifcones be compared jora eylinder/if cylinders be compared) equall to the others ; by the order of the fecond and third problemes : and the heith of the cone or cy- The fecond part of the Probleme; * GGg.il. linder . b mets as FSS ee Ss : : 7. | ae te >| Wee i} LT) UF yt | te i t - | woe } ’ als : } +} bt Hy ahey + : et : + eel Hf Bit | af 4 ;t lei ta ie Ne re oP! ie, itr | eee : 7 7 Tt { ' +14 alt i | ih} - a ia it” ul ha q eH 4yi | : te ie re x I. f : a Conftrutiton. Demon flra- 20%. ; " ~ a 2 a 1 An other way of executing this probleme. Belo The rwelueth Booke linder,on whofe bafe you rered an equall cone or cylitider ; with theinew heith-found , haue that pro- portion, which the cones or cylinders haué;one to the other,by the 14,0f this wwelfth booke. ¢ A Probleme. rig id en vpright Cone , and Cylinder being genen : t0 finde two right lines hauing that proportion, the one to the other, which the Cone and Cylinder hauejone to the ether. Suppofe , 2 EK an vpright cone and 4 Ban vpright cylinder genen . I fay two right lines are to be geué which thall haue that pro- portion one to the other, which 2 EK and .4& haute one to the o- ther.Vpon the bafe B A, erectea Conejequall 0S XK; by the'or- ”” der ofithe fecéd préblethe: which letbe 0 B H,and his heithlet be oO C,and let the heith of 4 8, (the cylinder) , be CSyptoduce cs;to P: fotharc’, be triplatoC $,8 make perfe¢t the cone 2 BA. I. fay that PC and OC haue that pro portionswhich 4’B hathto GFK, For} by“conftrction, O BH 15 .¢-* quallte BER and ? BH is-equal to 4 B as we will proue,(Afium pr ~~ wife} DAnd BW) and oO B Haren ~ oe | vpurrond' bale tamcly B.A Wher py pit oa t4 oF this twelfth, 2 Be Gini hore Hatit'o Bi atetirero the other «og er say lo as theirbelches' rc and 6 Coates one tothe other) Wherefore the yo ai yor BO 9 ib tis: cylinideand cone eqnall to 7 FH and-0.8 Mare as 2 Cis to.0.C:by the 7.0f the fifth. But4 8 thecylin- der Q°P K the cone; are équall to PBH.and OB H:by conitraction: wherefore 4 2 thie cylindersis to 2 ER ule Cones PCA to OC.” Wherefore.we hane found ewo sight lines:haning that proportion that atconednd'a eyifiider ceuen Haile onc to thé other. Which thing.we may,exeeutevpon the bafe of the coneinenensaswe did vpon the bafe of rhe cylinder genen,,.on.ghis maner... Vpon:the bafe.of the cone QE swinehwbatelet be £K, erect'a cylinder , equall to 42, by the order of my fecondiprobleme: Which cylinder let be £ D,and GT his heith,and let the heith ofthe cone 2 E K,be 2 G.Take the line G R,the third part of 2 G,(by the 9.0f the fixth ): and with playne paffing by &,parallel to ZX, cut of the cylinder £ F:which fhall be equall to the coh@@ SX jby-the affumpt following: I fay now, that -4- B,the cylinder,is to 2 E K the cone,as G T,isto G R.For the cylinder £-D is to thecylinder E F,asGT is to G R,by the 14. of the twelfth, and to & Dis thecylinder4.B equall : by,conftruction :-and to EF, weiltaie ouch he ca ie'S PKs equal; Wherefore by the7, of the fifth ,.4 Bist. OEK,asGT isto GR. Wherfore an vpright cone,& a cylinder being gétien, we hauefound tworight lines hauing the fame proportion betwene them , which the cone and the cylinder, haue one to the other : which was requifite to be done. . mae B nw i= +> t. 94 Nicans . “bt dn affiamptr ’ aS YOR AVG IBIS THT SATS ~ “ If a cone and a cylinder , being both on one bafe,are equal one to the other : the heith of the cone is tripla te the heith.of the.chlinder'..c-dudsfacone and acylinder being borh on one bafe, the heith of the cone be tipla tothe heith'of thecylandersthe cone and the cylinder are eghall, ~~ We will vie the cylinder A B & the cone P BHinthe folmer probleme: with their bafe & heithes fo noted as before.] fay if P B H be equall to A B,that C P the heith of PB H, is triplato CS theheith of the cylinder A B. Suppofe vpon the bafe B H, aconeto be rered ofthe heith of C'S, which let be $- B Hoxtrsunanifeftthac'A B is eripla'to that cone S B H, by the 10.0f this twelfth. Wherforea.cone.equal to AB the cylinder is tripla'to $B HH thé one; ‘by the 7, ofthe fifth ; but P BH is fuppofed equall to AB,.Thereiore P B H is triplato S B H,therefore the heith of P B H fhall be tripla to the heith of SB H by the 14.0fthe twelfth But the heith.of P.BH,is C P:and of S B Hjthe heithis CS: wherefore C P is tripla to°C S.And C Sis the heith of the cylinder\A B by fuppofition . Therefore'a coneand’a cylinder, ing . at being SS ae es) F e- c fe of Euchides Elementesi:s Fol.385. being both on.one bafe,and equall, the heithof the cone is triplatothe heith of the cylinder . And the fecond part,as ealeiy may be confirmed.ForifAB a cylinder, and PB H acone , haue one bafe , as the circle about B H: and the heith of P BH be tripla to the heith of A B;I fay that P BH,and A B are equal. The conuerfe The heith of AB let be(asafore)C S$: and of PBH, the heith, let be C P: of the heithC S, imaginea of the afsiipt. Cone vpon the fame bafe BH: by the ro.of this twelfth, A B fhall be triple to that cone. And the cone PB H hauing heith C P,(by fuppofition)triplatoC S, fhall alfo be tripla to that cone $ BH: by the 14. of this twelfth. Wherefore by the 7.0f the fifth A B and P B H are equall. Therefore, ifa cone and a cy- linder being both’on one bafe,the heith-of the cone be tripla to the heith of the cylinder, the’cone and the cylinderare equall.So haue wedemonitrated both partes:as was required. => z=. eed a7 { Wit i i Ait ' LY ia / , vid iui u qj 4T beoreme.. 4. ( T he fuperficies of the fegment or protion of any [phere,is equall to the circle, whofe femidiameter, és equall to that right line which is drawne from the toppe of that fegment to the circumference of the sircle, Which is the bafe of that portion or fegment. | As in the Sphere A, aSegment being cut of by thecircle, whofe diameteris C E - & the fame circle being the v ms Y bafe of the Segment, ZB Q whofe top alfo is D:the GE Z, croked fuperficies {fphe- HyA ZZ ricall of the fame Seg- YY = ment, is equall toa cir- ig = cle “whofe Semidiame- SHA = if ‘sir. : E teris equall to the right AW Wis line D © . As is the cir~ \ AWS SS cle B. This hath-s4rehimedes demonftrated in this firft booke of the Sphere and Cylinder,in his,4o.and 4% propofitions:and I remitte thenrthether , that will herein demonttratiuely be certified: would with all Mathematiciens, as well of verities eafy,as of verities rare and obfcure , to feeke the caufes demon+ ftratiue,the final] fruite thereof,is perfeCtion in this art. | | sf Note. Befides all other-vfes and commodities , that are of the Croked fuperficieces, of the,Cone, Cylin- der,and Sphere, fo eafely and certaynely,of vs to bé dealt with all: this is not the leait,thara notable Er- ror, which among Sophifticalhbrablers,and vngeometricall Mafters and Doétors;hath a longtime bene vpholden : may moft evidently, hereby. be confuted, and vtterly rooted out of all mens fantafies for.c- uer.The Error is this, Curus,ad rectum,nulla eft preporrso,thatis: Betwenecroked and {traight,,isnopro- 4 great error portions This error, in lines, fuperficieces,and folides, may with more truedemonftrations be ouer- eo mmonly throwne, then the fauourers of that fond fantafie are able, with argument, either probable or Sophi- uiaintained fticall to. make fhew or pretence to the contrary. In lines,I omitte ( as now ) Archimedes two wayes,for the finding of the proportion of the cizcles circumference to aftraight line . [meane, by the inf{cription and-circum({cription of like poligonon figures, and that other,by fpirall lines.And I omitte likewile (as now)in folides ,of a parallelipipedon,equall to a Sphere,Cone,or Cylinder: or any fegment or feétor of the faydfolides . And onely , here require you to confider in this twelfth booke, the wayes brought to your knowledge,how to the croked fuperficies of a cone and cylinder, and of a fphere, (the whole,any fegment or fector thereof ) a playne and ftraight fuperfictes may be geuen equall: Namely, aCircleto ” oh be geuen equall , to any of the fayd croked fuperficieces afligned , and geuen. And then fartherbymy » Siraight Additions vpon the fecond propofition,you haue meanes to proceede inall proportions,thatany man » andcroked eat ititight lines geue,or afligne.Therfore,Curmsad rettum,proportio omnimoda poteft dars.One thing itis, » all maner to demonttrate,that betwenea croked line and a ftraight “ora croked fuperficies and aplayneorftraight », ¢f propor- {uperficies, &c. there is proportion. And an other cite itis ,to demontftrate a particularand f{peciall 5, #20 may be kinde of proportion, being berwenea croked fuperficiesand a ftraight or playne fupérficies, For this al- fo confirmeth the firlt. This fhort warning will caufe you to auoyde the fayd error, and make you alle hable to cure them,which are infected therewith. HAI Hh We Betwene 3 LeKene x ATheoreme. $. eA ny tivo Spheres being genen , as the Sphericall Superficies of the one,is ta the Sphiricall Super- ficses Confiruction, Demouftras tBOte wrth te The afin ,.° tio of acir~ 45 clecoaptcd 4, ai a {phere. Cousirnition. - Aor The twelueth Booke ‘Yicies of the other?So is the reatepre ircle conteyned in that one,to the greate/t Circle conteyned in that other And as greatest Circle,1sto greatest Circle, fois 8 phericall fuperficies to Sphericall fu perfictes. For thefuperficies of every {phereis quadrupla to his greateit.circle,by my firft Theoreme:wher- fore, of two.geucn fpheres ,.as the {phericall {uperficies.of the.one, isto his greateft circle, fo is the {phericall fuperficies of the other,to his greateti circle whetfore by alternate proportion,as {pherical.{u- perficies.is co {pherical fuperficies,fois gucatett circle to, greatelt circle.And therfore alfo as greateft cir- cle,is to greatelt circle, fois {phericalfuperiicies to fphemeall faperficies: which was to be deméitrated. x AProblemes» os _ A Sphere being genen,to.geue an os her Sphere ,towhofe Spherivall fuperficies, the fiperficies Sphe- vical of the Sphere geutn {hall haue any proportwn,sberwenesoright lines geuen- 5 avant .. Stppofe A,to be a {phere geuen,and the proportion geuen,to be that,which is betwene the righe lines X and Y.I fay that a {phere is to be geven to whofe {phericall {uperficies, the fuperficies {phericall of A,fhall haue that proportion which X hath to Y.Let thegreateft circle,conteyned in A.the {phere be the circle B C D.And by the probleme of my additions,vpon the fecond propofition of this booke , as Xis to Y, fo let the circle BC D be to an other cirelé found. let tharother circle he EE Gs and his dia- meter EG, 1 fay that , the {phericall fuperfici- , | " es of the {phere A, hath | to the iphericall fuper-_ ficies of, the fphere, whofe gréateft circle is EFG,(orWNus equall) that proportidn, which X hath to Y.Por(by con {truction)B C Dis to E- FG,asXisto Y:and by_ | 7 "44¢ p- the theorumé@ néxtbel OMS s12kge 93 26 Gn00G 8 fére,as Be Disto E‘F2 it pes er te Sh T9130 Ls Gp fois deetphericalifg! 02 . owalde bes opi perficies of A ( whofe greatett circle is BC D, by fuppofition ) tothe a ae Oe fphericall fupe: ficies,of the {fphere, whofe grea- tet ircle -1s° B Fe G23" yt bah 21 9 te saded: wherefore By thee 3 i; ie end ics , ; ‘ rh 4 ofthe Afmayxas t6°¥ FO" tia tess 55d:5 tse s So as the “fpheridall’ figuio ue boro01 ylteiw bas perficies of Aljto the Means ss te pees Pee aot ea “afeeiniay es. {phericall fuperficies of the {phere; whofelatéatett circle is EF G: wherefore, the {phere whofe diame: téris E Gf thé diameter alfo of EF F'G 49 is'the fphere’, to whofe fphericall fuperficies , the {phericall fue perficies of the {phere Ayharh that propition which X hath to ¥."A {phere being geuen therefore, we have géeuen-an orher fphere,to' whofe fier all {uperficres, the fuperticies fphericall of the {phere geué Hiath arly proportion geuen berwene two right lines: which ought to be done. rrrrr ar : 7 - ' ~* : . . + yt) : : ss v4 aha é 7 Ls ‘4 ** : . - > [< - uf > 3 ae | A Probleme. TO. : ef Sphere being geuen anda Circle leflethent begr eateft Circle, in the fame Sphere conteyned, . bocoapt in the Sphere genena Cacle equaltto the Circle geuen. & veoleen a iy 4geke Suppofe-A:,to:beithe (phere reuett tarid-the circle genen lefle then thé greateft circle in A contey- ned,to be FaCGuibfaygthacio the Sphere Ava citcle,equall to the circle F K G,is to be coapted. Firft yn= deritand,whae we medne here,by coapting of acircleina Sphere. We fay, that circle to be coapted ina Sphere, whofe whole circumferéce is in the fuperficies of the fame Sphere. Let the greatett circle in the Sphere A conteyned,be the circle B CD W hofe dianteter fuppofe to be BD , and of the circle F K G, Jet F G be the diameter By the 1. of che fourth, lec a line equall to F G, be coapred in thecircleB C D. Which line coapted,let be B E.And by the line B E,fuppofe a playne to pafle, cutting the Sphere a Bitie cD wht Ot 1: Neee4 Pelee ai . tobe _ fe + aan of Euchides Elementes: co be perpendicularly erected to the fuperficies of BCD. Seing that the portion of the playne remayning in the-fphere,is called their common fection: the faydfeétion fhall-bea-circle , as be- foreis proued . And the common [ectionof the fayd playne,and the greateft circle BC D , (which is BE by fuppofition ) fhaltbe the diameter of the fame ctr- cle ,as,we will proue . For, let that oircle be B L- EM. Let the center of the {phere Abe the point H: which H,is alfo the céter ofthe tircle B C Dy becaufe B C D is the greatett circle in Avcontey- ned. From H, the center ofthe fpheré A, leta line perpendicularly be let fall to the’circle B L- EM. Letthatline be H O: andic#s evident that H O hall fall vpon the commion fection BE, by the 38.of the eleuenth.And it deuideth B E,into ewo equall parts, by the fecond part of the third propofition of the third bookes, by which poynt O , all other lines drawne in the'circleB LEM, are, at the fame pointe:O , deuided into two e- quall parts. As if from the poynt M;bythe point O, aright line be drawne one the other fidecom ming to the circumference , at the poynt N:itis manifeft that N.O M is deuided ihto two equal partes at the poyntO: by reafon,if from the cen= terH , to the poyntes Nand M, rightlines be drawne,H N and HM , the fquares of H M, and H N are equall: for thar all the femidiameters of. the {phere are equal: and therefore their {quares are equall one to the other: andthe {quare of the perpendicular H O,iscommon: wherefore the fquare of the third line MO fs equall tothe f{guare of the third line N O : andtherefore'the | line M O to the line N O.So therefore is N-M equally denided at the poynt O . And fo may be proued of all other right lines, drawne in the cirqle B L. BM, pafing bythe poynt O,to the ciréumference one both fides. Wherefore O is the center of thecircleBLE Mw : and therefore B E paffing by the poynt O is the diameter of the circle BLE M.Which circle(I fay)is equal to F K G: for by conitruction B E is & quall toF G:and BE ts proued the diameter of BL EMs\and PG is by fuppofition the diameter of the circle F K G: wherefore B L E Mis equalltoF. KGthe circle genen:and.B LE M isin Athe {phere geué, Wherfore we haue in a {phere geuen coapted a circle equall to a circle geuen: which was to be done. sf Corollary. Befides our principall purpofe, in this Probleme ,euidently demonstrated, this is alle made mani- fest : that if the greateft circle in a Sphere, becut by an other.circleseretted vpouhimat right angles, that the other circle is cut by the center,ana that their conimon fektion is the diameter of that other circle; andsherefore that other circle deutded is into two equall partes. x. A Probleme. rk. < et Sphere being geuen,and acircle,lefethen double the greatest circle in the: fame. Sphere con- tained, to cut of , a fegment of the fame Sphere, whofe Spherical juperficies, fhall be equall to thecirs cle genen . Suppofe K to be a Sphere geuen, whofe gréateft circlelet be AB € s and; thecircle geuen,fuppofe tobe DEF . 1 fay,that a fegmenr6f the Sphere'K,is to. be cut of, fo-great, that his Sphericall fuperfi- cies, fhall be equall to the circle DEF . Let thediameérer ofthe circle.A B.C, be theline AB. Atthe point A, in the circle AB C, coapta right lineequall to the femidiamerer of the circle DEF ( by the firft of the fourth) . Which line fuppofe to be A'H). From-the point. H, to the diameter AB, leta per- pendiculardine be drawen : which fuppofe to b¢HI : Produce-Hi.tothe-other fide of the circumfe- rence, and fet it come'to the circumference atthe point L..By therighrline HIL ¢ perpendicular to AB ) fuppofea plaine fuperficiesto pafle, perpendicularly erected vpon the circle ABC’: and by this Bil z2 HHh;): playne Demin rae 30% This is mani feft:if you con fider thetwo triangles relt- angles, OM auadHoNn: andthen with all, vfe the 4.7. of the frst of Exlide, Conflructions == ——— —— : = = —_—? & 6S = = = ae a = ak Be = SE <= Sas Sees = - < = —— r= —ee-- - a —_ = 2 a fe amet a z = ~== ~~. — ‘ ~ — a <—- —— - Pe ea avers a = 2 = ee - = it — - = SS" = ——— c . —— - ms —-— = =. - - 3 ~ = —_ - it , ——<= Ae PR Co eh Se ee Demonftratio » . Conftruétion. Lhe twelueth Booke riaine fuperficies, the Sphere to becutinto two feg- . mnentes : oneleffe then the hale Sphere, namely, H A- LI: and the other greater ther the halfe Sphere, name- ly,HBLI. Tfay, that the Spherical fuperficies of the feoment of the Sphere K, in which the fegment (of the greateft circle) H A LI, is conayned,( whofe bafe is the circle pailing by HI L, and topre the point\A ) is equall to the circle DEF. For the:irele,whofe femidiame- ter is equall to the line A H,is qquall to the Sphericall fu- perficies of the fegment HAL, y the 4.Theoreme here added.And (by conftruétion )A H is equall to the femi- diameter of the circle DEF: therefore the Sphericall fuperficies of the fegment of tle Sphere K (cut of by the circle paffing by HIL ) who toppeis the wom, A, is equall to thecircle DEF. Wherefore,a Sphere being. , geuen, and a circle leffe then louble the greateft circle - in the fame Sphere, we haw cut of, a fegment of the fame Sphere, whofe Sphericdl fuperficies, is equall to the circle geuen : which was equifite to be don:. q An aluife. In noting or fignifying f Spheres, fometimes we vfe by one and the ie circle, in plaine defigned,to re~ prefent a Sphere andalfo thegreatelt circle in the fame contained : and likewife,by afegment of that circle, fig~ : nifie a fezment of the fame Sjhere, as by a ftraight line, we often fignifie the circle, which is the bafe of a fegment of a Sphere,Cone,r Cylinder : and fo in fuch like. Wherin, confider our fuppofitions: and rake heede when we fhift fron one fignification to an other, in one. and) the fame defignation : and withall remember the princiyall intent of our drift : and fuch light thinges, can not either trouble or offend thee , Compendioufnrs and artificial cuftome , procureth fuch meanes : fufficient,to ftirre vp # magination Mathematical : »r to informe the prattifer Mechanicall. | Ee er ee, Probleme. “#2. Tocut a Sphere geuen jnto.two fuch fegmentes,that the Sphericall fuperficseces of the fegnsentos hall bane one to the other, ary proportion geuen berwenetworight lines. — | : Suppofe F to bea Sphere geuen: and the proportion geuen,to be that, which is betwene GHand HI. I fay, chat the Sphere f, is to be cut into. two {uch feementes ; that the Sphericall {uperficies of thofe fegmentes, fhal} haue hat proportion, one to the other, which, the right line G H, hath to the right line HI. Suppofe ABCE to be agreateft circle, inthe Sphere F, contained : and his diameter, to be A B. Deuide A B into wo fuch partes, as G I is di» 3" | uided into,in the point H(by the ro.ofthe fixt)Let thofe « partes be AD, and DB. Sothat; as G Histo HI, fous. ADtoDB.Bythe point D,letaplaine fuperficies paffe, cutting the Sphere F, and he diameter AB: So, that ‘ynto that cutting plaine, thediameter A B, be perpendi- . “cular : and the Sphere alfo tlereby deuided into two feg-* ~- mentes, whofe cémon bafe uppofe to be the circle C E, hauing the centerjthe point): andthe roppe of the one _to be the point A, aad the oppe of the other to be the point B : and the fegmentesthem felues, to be noted by EA C,andEBC: Drawe fromthe two toppes, A and B, to C (a point in the circumference of their common bale) tworight lines AC aid BC. Pfaynow, thatthe Sphericall fuperficies ofthefegment EAC, hath tothe Spherical fuperficies of the fegment EB C, the fame’ Pemonflvatio proportion,which G H hahto HI. For, forafinuchas: This in maner of 4 Lemma,4s prefently pre- wed. circles haue that proportior, one to the other, that. the =) 0:3 {quares of their diameters hiue one to the other ( by the: Gc 2. of this twelfth ) s And th fauares of theyr femsdsame- _ gers | have the fame proportios one to theother , which the fquares af theyr diameters haue . EF For like partes haue a _—_-- =, ae of Euchdes Elementes: Fol.387. that proportion one to the other , that the whole magnitudes, whofe lke partes they-are., have the one to the other ; by the 15.0f the fift. Bur the fquare of euery diameteris quadruple to the {quare of > his femidiameter : as hath often before, bene proued : therefore, circle haue one to.an other, that proportion, that the {quares of their femidiameters haue one to the othe ] . Wherefore, feing AG and BT are femidiameters of two circles, whereof eche is equall to the Sjhericall {uperficies of the feg- mentes, betwene whofe toppes and circumference of their bafe, they aredrawen: by the 4.Theoreme of thefe additions : it followeth that both thofe circles, : whofe femidiameers they are: andalfo thofe Sphericall fuperficieces, which are equall to thofe circles, haue the one tcthe other, the fame propor- tion, which the {quare of A C hath to the fquare of BC . But A C is drawaa betwene the circumference of the bafe, and toppe of the fegment Sphericall, EA C, by conftruction and likewife BC is drawen betwene the toppe, andicircumference of the bafe, of the Spitericallfegnent EBC by conttruaion: Wherefore the Spherical fuperficies of the fegment EAC, isto the Sphericall fuperficiés. of the fegs ment E BC, as the {quare of A C is to the iquare of B C’. Bur the {quate oA C is to the {quare of BC, as'A Dis to DB : by the Corollary of the Probleme of my additions vpoithe fecond of this twelfth: And A DistoDB,asGHistoHI : by conftruétion. Wherefore the Splericall fuperficies of the feg- ment E AC, is to the Sphericall fuperficies of the fegmentEB C,as GHiitoHI. Wehaue therfore, cut the Sphere geuen, into two {uch fegmentes, that the Sphericall f uperftieces ofthe fegmentres,haue one to the other any proportion geueg berwene two right lines : which vas to be done. : @ ACorollary. 1. — _ Flere it appeareth demonstrated, that, circles are one to the other, athe {quares of their femidi- ameters are,one to the other. Wherby (as occafion fhall ferue) you may, by force of the former argument, vie other like partes of the diameter, as wellas halues. AC orollary. 2. * It is alfo enident, that the Spherical fuperficieces of the tio feomertes of any Sphere ,t0 Wwhofe common bafe,the diameter ( palfing to their two toppes ) 35 perpendicular, bane. that proportion the one to the orber, that the portious of the fayd diameter pane the one to theerber - that faperficies and that portion of the diameter onthe one fide of the common bafe,being compred to that Juperficies,and that portion of the diameter ,on the other fide of the common bafe, ~* | ; - . : © ACorollary. 3. It lhewife enidently followeth that the two Sphericall fuperficieces of wo feementes of a Spheres Which two fegmentes are equall to the Sphere,are tn that proportion the ontto the other that their axes (perpendicularly erected to thesr bafes.) are in, one to the other : where fener inthe Sphere thofe feg- mentes betaken, | I fay thatthe Sphericall fuperficies ofthe fegment C AE, and the Sphericall fuperficies of the fegmene F GH, hauing their axes A D and G I ( perpendicular to their bafes ) : are in proportion one to the other, as AD isto GI: if the fegment, of the Sphere contai- ning CAE with (the fegment of the fame Sphere) F GH, be equall to the whole Sphere. For feing the diameter ( or axe ) A D, extended to tlfe other pole or toppe , oppofiteto A ( which oppofite toppe, let be Q_) doth make'with the fegment @ A E, the comple- ment of the whole Sphere: and by fuppofition, the fegment F G H, with the fegmenrC AE, are equall to 33 2? Note here of bafe the whole Sphere : Wherefore from equall taking | Iolidene ne CAE (the fegment common ) remayneth the feg- oleae Z ae to ment C Q E, equallto the fegment F GH. And ther- * bri by, Axe, Bafe s Solitie, at fuperficies Sphericall of ring any far the fegment F G H, mutt (of neceflitic ) be equall to ther proofe fore Note *1 fay hatfe a circular reuo- kutioz: for that fuffifeth in the whole diameter ST,to defcribe « circle by? sf st be moned about bis center Q, Se. ‘Lib.2.prop.2. de Sphera €5 Cylindro. *. culation and pra@tife,thal bea great ayde and direction. ° | of thie fame othtr fegment remayning. , a T be twelueth Booke the Axe,Bafe:Soliditie and fuperficies Spherical of the fegment CQ._E: Wherefore, by the fecond Corollary here, and the 7-of the fift, obr conclufion is inferred, the {ulperficies Spherical], of the feg- ment 'C AF; to be, to the fuperficies Sphericall ofthe fegment F GH, asADistoGL. wo 4) - - - 4 y - “ 8 AT peoreme. é. Toany falide Jallor off aSphere;thatv pright Cone és equall whofe bafe is equallto the corinex Spherical fuperfioset'of that Jettoryand heith equall to the femidiameter of the fanse Sphere. © Hereofthe demonfiration in refpeét of the premi- fes : and che common argument of infcription andcir- cumfcription of figures 1s eafy: and nevertheleffe, if your owne wirte will not helpe you fuficiently : you may take helpe at archimedes hand , in his firtt booke & lait propofition of the fphere and cylinder. Whether if ye haue recourfe, you thal! perceaue how your Theo- reme here amendeth the common tranflation there: and alfo ‘ourdelineation geueth ‘more Luely thew of the chiefe circumftances neceflary to theconftruction, then there you fhall finde. Of the {phere here imagined to,be A,wenotea folide fector-by the letters P:Q RO, So that P Q R doth fignifie the fphericall {uperficies, to that folide feGtor belonging + (whsch ss alfocommon to the [cement of the [ame phere’ R Q ) and therefore a line drawnefrom the toppe of that fegment, ( which toppe fuppofé to be -Q_, ) isthe femidiameter of the circle, whichis equall co. thefphericall fuperficies of the fayd _- folidé feor,or fegment * as beforeis taught. Let that line be QP. By Q draw a lint contingent: which let be SQ T.At the poynt Q from theline QS, cut a lineequall to PQ ‘which letbe'S Q-And vntoS Q._, make QT equall,then draw the rightlines O $ © T.and:O:Q:, About which O Q (as anaxe faltened) if you imagine the triangle O $ T, to make an *halfe circular renolution,you fhall hauethe vpright cone O'S T: (whofe-heith is O Q , the femidia- meter of the {phere,and bafe the cifcle, whofewdiameter is $ T, equal to the folide fector P QRO. ’ - + ATheoreme. 7. T oany fegment or portion of a Sphere, that cone is égitall , Which hath that circle to hisbafe , which és the bafe of We fog mer aud heith, a right line, which unto the beith of the feguset bath that preportra which the femidsameter of the Sphere , together with the heith of the other [egement remayning hath to theheith 4: This is well demonttrated by Archimedes 8¢ there- fore nedeth no inuentién of myne, to confirme the |. fame; and for that the fayd demonftration is ouer long here to be added jf willrefere-you thether for the de-_ monftration: and here fupply that whichto Archimedes demonitration fhall geue light,and to your farther {pe- Suppofe K to bea {phere : & the greatett circle K incd- - = teynedslerbe'A BC E,and his diameter BE,& céterD. Let the {phere K,be cutte by a playne fuperficies, per- .. , say pendicularly ereéted Vpon the fayd greateftcircleAB- ft C E:zlet the fection be the circleabourAC:Andletthe = 2 ae of Euchdes Elementes. feginentes of the {phefe be the one that wherein is A- Fol.388. B C,whofetoppe is B:and the other let be that where- inis AE Cand his toppe , lét be E: I fay that acone which hath his bafe the circle about AC, & heith a line which to B F(the heith of the fegment, whofe toppe is B, ) hath that proportion that a line compoted of DE, the femidiameter of the {phere , and E F ( the heith of the ether remayning fegmenc, whofe toppe is E ) hath to EF, (the heith ot that other fegment remayning) , is equall to the fegment of the {phere K, whofe toppe is B.To make this cone , take my eafy orderthus . Frame your worke for the finding of the fourth proportionall line: by making E F the firtt:and a line compoled of D- E and EF,the fecond:and the third, Jet be B F: then by the 1z. of the fixth, ler the fourth proportionall line be found: which let be F G : vpon F the center of the bafe of the fegment,whofe toppe is B, ereCtaline perpendi- ~ cular equall to F G found: and drawe the lines G A and G C: and fo make perfect the cone GAC. Lfay, chat thecone GA C ,1s equall to the fegment (of the {phere K)whofe toppe is B. In like maner,for the orher fegmét whofe toppe is E,to finde the heith due fora cone equal to it: by the order of the Theoreme you mutt thus frame your lines:let the firft be B F: the fecond DB and B F, compofed in one right line,.and the third muft be E F: where by the 12,0f the fixth , finding the fourth, it fhall be the heith ,'to rere vpon the bafe , ( the circle a~ bout A C , ) to make an vpright cone, equall to the feg- ment, whofe toppeis E. @ Logistical. . The Logifticall finding hereof is moft eafy: the diameter o’ the {phere being geuen , and the'pora tions of the diameter in the fegmentes conteyned ( or axes of the fegmentes ) being knowne.Then or= der your numbers in the rule of popes 2 asI here haue made moft playne, in ordring of the lines: for che fought heith will be the produéte. | aes Corollary. 1. | | Hereby and other the premifes it is enident that te any fegment of a Sphere, whofe whole diame. ter is kuowne and the Axe of the figment geuen, An upright cone may be made equall ; or in ‘any pro- portion , betwene two right Iinesa jigned and therefore wlfoacylinde: may tothe [44 fegment of the Sphere ,be made equall or in any proportion geuen,betwene two right limes, — . | * ACorollary. 2. eM anifeltly alfo,of the former theoreme,it may beinferrid that a Sphere, and his diametegben ing denided ,by one and the fame playne fuperficies , to which the fayd diameter is perpendicular - the two fegmentes of the Sphere,are one to the other inthat proportion, in which a reclungle parallelipipes don hauing for his bafe the [quare of the greater part of the diameter, and bis heith a line compofed of the leffe portion of the diameter and the fenmdiameter:tothe relaugle parallelipipedonbauing for his bafe the fquare of the leffe portion of the diameter ,e> his heith aline compofed of the femidiameter @& the greater part of the diameter. | ATheoreme 8%. Exery Sphere,to the cube,made of his diameter, is (in maner J milter. ; HHb.i. As Nore. 2 > —- ie aan co > na = = SES s Pe. oe >= — = = mare = = oe. > => ape =: - » — . 7 = s = a >. = — —e ~ ——— —— << — ie Soe Se 2 oy ee Ss -* — 7 -. — ee > iit, == = = 7 - <<. ~ ~ een — = — a . — r i -—— ss cmerreet - : > ; - = rs as 3 a a > ; SSS ~ ; oS EP : » - “75 Seal - = = - —- = = > = a =f ——— - to — = — = - 3a ea ee = — = e = eee Lot gag eee. 1... Sie pis a Fen sk 2 % = +” Se 4-= S385 — . ~~ 4S ese’ = ¢ ~ . > > ae Oe -- —a > > - a ee 2 - —— =< — e SS Sree A ee | - S . = Sa | ~ ; : oe * —— = + ~ = hom hentt ar r St -5——_ Ses SS o = 3 2 = See OPE a ae ‘< —— = =o —- a 2 Sater a . C << Soe = — = Sea Se = = rn Z . es Oe = = a 7 > eP-8 -<- =e 2 ae ee txts x +—= = = ee ss > Sie ee i Fa =: Ses See ee ee ——— —. = ee = = Se, = == a= ==: aes i ee = = = = ——— a = = eS = ~ a ee See ee — Fine Se ee . —s ee eee ee = = as a - ; = = rramie= lintel = i = aeeel : , — — = - = <2 ‘aoa = vf = 2 a a RSet + See SF AE eee i= » OZW,as 4 bale, and ofthe . circle , in the os A ,con- wife {eing the cylinder ZM, © T he twelueth Booke Asvpon the firft and fecond propofitiés of this booke, I began my additions with the circle (be- ing the chiefe among playne figures)and therein brought manifold confiderations, about circles : as of the proportion betwene their circumferences and their diameters: of the content or Area of circles: of the proportion of circles to the {quares defcribed of their diameters: & of circles:to be geuen in al pro _ portions,to other circles: with dinerfe other moft necefiary problemes ( whofe vfe is partly there {peci- fied): So havej in the end of this booke,added fome fuch Problemes & Theoremeés, about the {phere (being among folides the chiefe ) as of the fame , either in it felfeconfidered , orto cone and cylinder, compared (by reafon of fupefficies,or foliditie,in the hole,or in part: )fuch certaine knowledge demon ftratiue may arife,and {uch mechanical exercife thereby be detifed,that(fure Iam )to thefincere & true {tudent great light,ayde,and comfortable courage (farther to wade)will enter into his hart: and to the Mechanicall, wittysand induitrous deuifersnew maner of inuentions, & executions in his workes will (with {mall trauayle for fete application)come to his perceiseraunce and vnderftanding. Therefore, e~_ uen as manifolde fpeculations & prattifes may be had with the circle , his quantitie being not knowne: in any kinde of fmalleft cerrayne meafure: So likewife of the {phere many Problemes may be executed and his precife quantitie,in certaine meafure,not determined,or knowne: yet,becaufe, both one of che firi¥(humiane) occafids of inuenting and flablifhing this Arte,was meafuring of the earth(and therfore. called Geometria , thatis , Eatthmeafuring) , and alfo the chiefe and generall end (in deede ) is mea- fure: and meafure requireth a determination of quantitie in a certayne meafure by nuber expreffed: Ie was nedefull for Mechanicall @arthmeafures , not to be ignorant. of the meafure and contents of the circle , neither of the {phere his meafureand quantitie,as néere as fenfe can imagine or wiih . And (in very deede)the quantitie and meafure of the circle, being knowne, maketh not onely,the cone and ty- linder, but allo the {phere his quantitie to be as precifely knowne, andcertayne, Therefore feing in re- {pect of the circles quan ritie( by Archimedes fpecified) this Theoreme Is noted ynto yous! wil, by order, vpon that(as a fuppofition )inferre the conclufion of this our-Theoremes. Suppofe a {phere tobe — fignified by As whofe diame- ter letbe RS.To RS,lera line equal] be taken , which let be T V: of T V, by the 46, ofthe firlt, defcribe a fquare. Let that fquare be T Y. With in TY let acircle be infcri- bed : by the2.of the fourth, which circle fuppofe to be OZW. That OZ W is e- ea to the greateft circle in. the fphere A conteyned, itis euident by the diameter , e- gual.to T:V.dfvp6 the {quate od T Y,asabale, be ere@ed,a pa: : rallelipiped6 re€tagle , whofe eith 1s equallto T V, itis e- uident that that parallelipi- pedonis acube. Which let be done : and that cube pro- duced , let be noted by T X. Likewife, if ypon the circle «>, heith equall to the line TV, . a cylinder be erected, itis ma nifefehat the cylinder hath , his bafe equall to the greateft teined: & heith,a line equall to the diameterof the fame _. {phere A. Which cylinderlet ¢ produced and noted by Z- M.. Efay now thatthe fphere - A,ts tothe cube T X,(in ma- ner) asthe number 11. isto ©». the. number.21. For feing the: . cube T X, was produced of his bafe,(the {quare T Y),be- ing brought into the heith of aline equallto TV: &like- Tr \ + Ee of Euchides Elementes. Fol.389. is produced ,ofhis bafe(che circle O.Z W)being brought intoa linesequal tothe faid T V:ic followeth feing their heithes is all one,thae the cube T X thall be to the cylinder Z.M,asthe bafe.of T X,( which iz the iquare TY ) is to the bafe of 2M, thatis the circle O Z W. But the {quare T Y;is to the circle OZ W..as the number 14.15 to the nfiber.r.(inmaner), by Archimedes demottratio: wherfore,the cube T X is tothe cylinder Z.M,as thenumber t4.,1s to the number 11.(wellnere). And by my third:Theoreme Chere added jthe cylinder Z M,is tothe Sphere A,in fefquialrera proportion: thatis,as3, to2. Where- fore the cylinder Z M,hauing the fame 11.equall partes (which he conteyneth in refpect of the cube T ~ X, being 14,0f the fame partes) deuided into 3.equall portions ,euery one of thofe portionsis 3 . And allowing to the Sphere A,two of thofe portiéns*icis euident,that the Sphere A fhall be 7—fuch partes . is ; 3 as are 14.in the cube T X:and rr.in the cylinder Z M . Wherefore the Sphere A, is tothe cube TX,as os to 14. The fraétion being reduced,maketh ~ :.and the number 14. being brought to the fame name, . . . ae . and denomination of thirds, maketh —.Put away now theyr common denominator : and then remay- neth,for the Sphere A,22.fuch partes,as the cube T X hath 42. And then depreifing them , to the fmal- left rermes:for the Sphere A, you fhall haue r1.fuch partesas the cube T Xconteyneth 21 . Wherefore euery Sphere,to the cube made of his diameter is,as 11.to 21. which was requifite to be demonitrated. Note. Fs Wherfore ifyoudeuide the one fide (as T-Q_) of the cube T X into 21 equall partes, and where 11,partes do end,recke~ ning from T, fuppofe the point P :and by that point P,imagine a plaine (paffing parallel to the oppofite bafes) to cut the cube | T X: and therby,the cube T X, 7 . to be deuided into two rectan- | N gle parallelipipedons , namely, T N,and PX: It is manifeit, * TN, to be equall to the Q Sphere A, by conftruétion:and 21. the 7.of the fift. Note, 2, Secondly, the whole quan-11. 7 titie,of the Sphere A, being cé- tayned in the reCtangle paralle- lipipedon T N, you may eafilie tranfforme the fame quantitie, into other parallelipipedons reCtangles, of what height,and of what parallelogramme bafe T you lift: by my firitand fecond Problemes vpon the 34.of this booke . And the like may you do,to any affiened part of the Sphere A: by the like meanes deuiding the parallelipipedon T N : as the part afligned doth require.As if a third, fourth, fifth, or fixth,part of the Sphere A , were to be hadina parallelipipedon,of any parallelogramme bafe affigned,or ofany heith afligned: then deuiding T P, in- to fo many partes(as into 4.ifa fourth part be, to be tranfformed: or into fiue, ifa fifth part , be to be tranfformed &c.)and then proceede,as you did with cutting of T N,from TX. And thatI fay of paral- lelipipedons,may in like fort(by my fayd two problemes,added to the 34 of this booke) be done in any fided columnes,pyramids,and prifmes: fo that in pyramids and fome prifmes you vfe the cautions ne- ceffary,in refpett of their quantities, compared to the quantities of bodyes hauing parallel, equall, and oppofite bafes : whofe partes thofe pyramids or prifmes are: as before in their propofitions,is by Ew- e/sde demonttrated.And finally, feing, in thefe prefent additions , you haue the wayes and orders how to geue to a Sphere,or any fegment of the fame , Cones,orC linders equall, er in any proportion be- twene two right lines, geuen : with many other mott neceflary {peculations and prattifes about the Sphere: I truitthat I haue fufficiently fraughted your imagination, for your honeft and profitable ftu- die herein, and alfo geuen you ready matter, wherewith to ftop the mouthes of the malycious , igno- HHh.ilij. rant, * A reflangle Paralletspspe~ On 2eucN,e= Guak toa Sphere geuen< To a Sphere , o? fo any part of & Sphere affig- ted: asathird,' fourth fifth ége 0 gene a paral lelspipedon e- guall, Sided Columes Pyramids, and prifmes to be ge- nen equallte a Sphere, or te Any certayne partthereof, Toa Sphere or any fegment, oF [efor of the fame ,to genea cone or cylinder equall or snany prepertion af- figned : SS a —— = ——— — —— — -—— ~ T as > - = yee VAR NS + — - _ ee = eee ra eee = = Se Ss ee —— is el —— = vampire Hoe ah eos seerrtge ome - —— —_— Se ae eins - aS ees SIO or : So Se = ~< nen 4 ar nig SS a ee —— coe ee Rene as ale : a a ak, — » = -- - oe EN ee Farther vfe of Sphericall Geometrie. T he twelueth Booke rant,and arrogant, def; pifers of the moft excellent difcourfes, trauayles, and inuentions mathematicall. Seingafwelthe heanenly {phéeres,& fterres their {phericall foliditie,with their conuexe {pherical fuper- ficies3to'thé earth at all rimies refpecting ,and their diftances ‘from the earth ; as alfo the whole earthly Sphere aud globe it felfe,and infinite other cafes , concerning Splieres or globes , may hereby with as mucheafe and certainety be determined of, as of the quantitie of any bowle, ball,or bullet, which we may'gripe in our handes(reafonyand experience, being our witnefles) :and without thefe aydes >fuch thinges of importance neuer hable of ys,certainely to be‘knowne,or attayned ynto. Here ende M.John Deehis additions vpon the lait propofition of the swelfth booke. - A propofition added by Flufas. | If a Sphere touche a playne fuperficies:a right line drawne from the center tothe touche, {hal be erected perpendicularly to the playne faperficiess “4 | Suppofe that there bea Sphere B C DL : whofe centrelet be the poynt A. And let the playne fu- perficies G C I touch the Spere in the poynt C , and-extend a right line from the centre A to the poynt C. ThenI fay thattheline A Cis ereéted per- L pendicularly to the playne G IC.Let the {phere be cutte by playne fuperficieces pafling by the right line L A C: which playnes let be A B C D- Land ACEL, which letcuttheplayne GCI by the right lines GCH and KCL. Nowitis manifeft (by the aflumpt put before the 17. of this booke ) that the two fections of the {phere B fhall be circles , hauingto their diameter the line LAC, which is alfo the diameter of the fphere; Whereforetheright linesGCH and x K CI which are drawnein the playne G C Ijdo at the poyntC fallwithont thecirclesBCDL ¢ and ECL. Wherefore they touch the circles __ in the poyntC, by the fecond definition of the _ K/-— Pe third. Wherefore the rightline LAC maketh | rightangles with thelines GC Hand Kk CI by . the 164of the third. Wherefore by the 4.of the eleuenth the rightline A C is ereCted perpendicularly to to the playnte fuperficies G C I wherein are drawne the lines GCH andKk C1; If thereforea Sphere touch a playde fuperficies,a right line drawne from the centre to the touche, fhall be erected perpendie cularly to the playne fuperficies: which was required to be proued. 7 oly eit Sx The ende ofthe twelfth booke of Euclides Elementes. ee nas we Fol.390. | The thirtenth booke of > sang Evuclides Elementes. | won IN ha. S THIRTENTH BO 0K E are fet forth The argument -|jcertayne moft wonderfull and excellent pafSions of pr sh¢ shir iia lyne deuided by an extreme and meane proportl= ‘senzh booke. on:a inatter vndoubtedly of grearand infinite vfe in ~ ) = ¥-\4 Geoinetry,as ye fhall both in thys booke,and in the ¥s4tother bookes following moft euidently perceaue. It =¥ |fteacheth moreouer the compofition of the fiue re- -Z34 gular folides,and how to infcribe them in aSphere ra 4}! WY oat 1 geuen, and alfo ferteth forth certayne comparifons LENG AEE PSS of the fayd bodyes boththe onetothe other, ands PETS alfo to the Sphere, wherein they ate defcribed? The 1: Theoreme. The 1. Propofition. y | i rN Ba , di . “die —— a phig Ve, Wi) TRieeeee oe B | If aright line be deuided by an extreme and meane proportion , and to the greater fegment, be added the halfe ofthembole line: the fquare made of ‘oh phofe'two lines added ‘topether fhalbe quintuple to. the fqnare made of the SO balfe of the-wholelpiens > « pin | Vppofe that the right — us line AB be deuided by an + extreme.and meane pro- EAMG) portio in the point C.And let the greater fegment therof, bé AC. And unto A C,adde directly a ryght line AD, and let A D beequall tothe halfe of the line AB. Then I fay that the fquare of the line C D is quintuple to the {quare of the line D A.Defcribe (by the 46. of the firft) upon the lines. AB and DC [qnares,namely, AE @ > D F. And in the {quare D-F ,defcribe and make complete the figure. And ex~ tend the line F C, to the point G. And foralmuch as the line AB is deuided by an extreme and meane proportion in the point C,therefore that which is contayned under the lines_A Band B Cis equall tothe fquare of the line AC. But that which is contayned vn- der the lines: AB and BC, is the pa- rallelogramme C E,and the{quare of : Consirnallion. E ——— — ee t= =~ = > — : = ie aod : aie Z So - = 3-5 Se a —~ = — Seo - — =< be — ss . = - be 5 a ro = oe - —— — = 2 ee r= —— ——= Pe TT ial Sark T be thirtenth Booke che line AC is the [quare H F.Wherefore the parallelogramme C E is equall to the {quare H F. And forafmuch as the line B A, is double to the line A'D,by confiruction: butithe lyne B Ais eqnall tothe line K A,and the line A D, to thelyne AH: therefore allo,the lyne K A, is double to the line A H. But as thelyne.K Ads to the ine-A-H, [ois the parallelogramme C- K to the par allelogramme CH Wherefore the parallelogramme C K is double tothe paralle- logramme CH, And the parallelogrammes LH and. CH are double to the parallelo- gramme CH (for fupplementes of parallelogrammes are by the 43-0f the firft equal the one to the other) Wherefore the parallelagramme C Kis equallto the parallelogrammes L H ¢ CH. And itzs proned, that gu pe ste CE is equall tothe {quare F H . Wherefore the whole {quare A E ts equal so the gnomon MX N. And forafimuch as the line B A,is dou- ble tothe line 4D, jy pa tt of the line B Ass, by the '20.0f the fixth quaaruple to the {amare of the line D A, that ts, the [quare AE ta the [quare DH. But the{quare AE isequall tathe cnomo M X N, wherefore the gnomo.-M X Nis alfo quadruple to the {quare DH .Whereforethe whole fquare DF is quintuple to the [quare D H. But the {quareD F, » isthefqudre of the ine C D, and the [quare DH is the |¢mare of the line D A .-Wherefore the {quane of the line CD, os. quintuple to the {quare of theline DA. If thereforearight line . be deusded by an extreame and meane proportion, and to the- greater {egment, be added the halfe of the whole line: the{quare made of thofe two lines added together fhalbe quintuple to the {quare made of the halfe of the wholeline: Which was required to be demonftrated. Thys propofition isan other way demonftrated after the fiueth propofition of this booke. . J : . ¥ : i. If avighe line be inpowwer quintuple to afegment of the fame line:the dous ' ble of the fayd fegment is deuided by an extreame and meane proportion, and the greater Jegment thereof is the other part of the line geuen at the be oiining. ; 2 > Lame Pr V ppofe that the right line i This propofi- 1D C be in power quin- f fton ts the con , | nerfe of the tuple to a fegmentof the . former. || fame line, namely, to.A- ar 3 | a D, and let the double of Bi the line A D be the line AB. Then I hi 3 [ay that the line ABisdenided by an extreme and meane proportion > ana. | the greater fegment thereof is the lyne Conftruttion, 4 C.Defcribe on either of the lines AB and C D {quares namely AE and D- Dewontrae . F- And in the {quare D F make per- tion, _ felt thefieure,and extendthe lineFC — tothe point G. And forafmuch as the [quare D Fis quintuple to the {quare D H, by {uppo(Sition, therfore the eno- mon M N X is qhadruple to the {quare D H. Andfora{much as the line AB is double to the line A D,therefore the {quaré of the line A Bis quadruple to the fauare of the line A D (by the 20. of : er - po ee a ee a aie es ~- pa — in Sie cet eee a eet ae sh tas lee <2) : Pt. oO oP ~~ ae ~~ _ °% te Pe a — Y et a teal — — 5 na 7 4 -* 2 TO ee ed - 0 awe See eee S as xt > . ee oe 2 . 7 - - te a * -- ~ ’ -- ” :

— - | Sa i pS a a SSS — i = — \ =~ = = a aor = = == — — - -— = — - iN = oe —- — = = Sees Se ee = Ee Ee iS ee =— =~ : — a ~— owmoereet as ie . - are di Sn ow Dy . ace = ¢ ow — eth - o> - _~ ¥ * _ , - ~ ete = » kA a RS Aer Tee Ue ‘ ni = et ~ ===e . 4% = = a ee ee ee pt Se eee —— ror ae oR ae wy —— 7 > — 5 il ae f. 61) eg ee eee. phe: edi a + ts Pty } | et are 7 } cin ie ' wy >} i) ee. | it : ¢ : , - 2 ae : ub * ue u 7 F } Ld es « i ) aa Fewis t By} f ' Th ¢é : Leia f ; | aaa : 4 wad . 7 a She I> ; * et ee : [ay af ab a 1 A i wa ; ri i | ‘ y } wv , Bis i, ieee — — inte — ne ia nen - li Fo eal “ * Sere eee 2 T he thirtenth Booke andCB:by the firft of the fixth (for at is their equall cach -) WH vi Bek ore,the {quare of A D, thalbe grea~ felis ter then the fquareof AC: by the Ai": c B 14. of the fifth . But the line AD, | D is lefle then the line AC,by fuppofi- tion: wherefore the fquare of AD | | is lefle then the {quare of AC . Andit ts concluded alfo to be greater then the fquare of A C: Where- fore the (quare of A D, is both greater,then the {quare of AC : and alfo leffe. Which isa thing impof- fible.The {quare therefore of A D,is not equal to the parallelogramme vnder AB, and DB. And there- fore by the third definition of the fixth , AB is not deuided by an extreameand ficane proportion , in the point D:as our aduerfary imagined . And ( Secondly ) in like fore will the inconueniency fall out: if we afligne A D,our aduerfaries greater ferment , to be greater then our A C. Therefore {cing neither onthe one fide of our pointC : neither on the other fide of the fame point C,any point can behad , at which theline A B can be denided by. an extreame and meane proportion, it followeth of neceflitie, that A Bcan be deuided by an extreame and meane proportion in the point C,onely .Therefore,a right line can be deuided by an extreame and meane proportion , but in one, onely point: which was requi- fite to be demonitrated. ATheoreme. 2. What right line fo ener being deuided inte two partes, hath thofe hi: two partes, proportional , to she two fegmentes of a line deuided by extreame and meane proportion: ss alfo tt felfe deusded by an ex- treame and meane proportion:and thofe his two partes ave his two fegmsents of the fayd proportion. Suppofe,A B,to be aline deuided by an extreame and meane proportion in the pointC,and AC to be the greater fegment.Suppofe alfo the right line D E, to be deuidedinto two partes , in the point E:and that che part D F, is to F E,as the fegment AC, isto C B:or DF, 10 be, to A C,asF Eis toC B. Eor fo thefe partes are proportionall , to the fayd fegmentes . I fay now, that D E is alfo deuided byan extreame and meane proportion in the pointF . And that D F, FE, are his fegmentes of the fayd pe portion.For,feing,as A C,is to CB:{fo isD F,to FE: (by fuppofition).Therfore,as A C,and C B(which is AB)are to C B: fois D F,and F E; (which is D E Jto F E:by the 18.0f the fifth. Wherefore(alternate- ly)asABistoDE: foisCB toRE.And therefore,the refidue AC ; ts tothe refidue A G 5 D F,as A Bisto D E, by the Gifth of the fife. And thenalternately,ACistoAB,asDE, F E isto D E. Now therefore backward , AB is EE DE Te nt ae toAC,as DEistoDF. Butas ABistoA- C,fois AC toCB : by the third definition ofthe fixth booke. Wherefore D Eis to DF,as A Cisto CB: by the 11. of the fifth.And by fuppofiti- on,as A Cisto C B,fois D Fto F E: wherefore by the 11.0f the fifth , asD Eis to DF: fois DF toF E. Wherefore by the 3.definition of the fixth ;D Eis deuided by an extreame and meane proportion , in the point F. Wherefore D F,and F Eare the fegmentes of the fayd proportion . Therefore , whatright line fo ever , being deuided into two partes , hath thofe his two partes, proportionall to the two feg- mentes of a line deuided by extreame and meane proportion:is alfo it fae deuided by an extreme and meane proportion, and thofe his two partes are his two fegmentes,of thefayd proportion: which was - requifite to be demonftrated. Note. “* - Many wayes,thefe two Theoremes,may be demonftrated : which I leaue to theexercife of young ftudentes.But vtterly to want thefe two Theoremes,and their demonftrations:in fo pore aline,or Thechiefe line rather the chiefe piller of Euctsdes Geometricall pallace,was hetherto, (and fo would remayne ) 2 great nat Eucides Aigrace. Alfolthinke it good to note vnto you, what we meane,by one onely poynt . We meane , that che quantities of the two fegmentes,can not bealtered,the whole line being once geuen .And ae . e Geometric. What isment _ fromeither end of the whole line,the greater fegment may begin : And fo asit were the point of {ecti bere by,A fedi- ON MAY feeme to be altered : yet with vs , thatis no alteration : forafinuch as the quantities of the feg- omineneonely mentes,remayne all one. meane,the quantitie ef the greater fegment,is all one : at which end fo cuer posit. it be taken: And therefore, likewife the quantitie of the leffe fegmentis all one.&c. The like confidera~ tion may be had in.Ewelsdes tenth booke,in the Binomiall lines, &c. Jobn Dee. 1569. Decemb.78, i T he 3. ° _ of Evclides Elementes. > F0l.39%. The3, Theoreme. The 3.Propofition. If a right line be desided by an extreme and meane proportion, and to the le/Se fegment be added the balfe of the gerater fegment: the {quare made of | sbofe two lines added together is quintuple to the [quare made of the half line of the greater fegment. Rs V ppofe that the right line A B be deuided by an extreme and meane proportion IX 4 in the point C. And let the greater fegment thereof be AC. And deuide AC into i. % two equall partes in the point D. Then I [ay that the (quare of the line BD, ts | = quintuple to the {quare of the line D C. Defcribe (by the 46. of the firit) upon the line A Bafquare A E. And defcribe and make perfect the figure (that 1s dinide the lyne AT like unto the diuifion of the line AB by the r0.of the fixth, in the pointes R,H, by which puntes drawe A D ¢ B (by the 3 1.of the firft) untothe line AB parallel lines RM and H N.Solikewife draw by the pointes D,C, unto the line B E thefe parallel lines D L ana CS, ¢ draw the diameter BT). And fora{much as the line AC is double to the line D C, therefore the {quare of i AC, isquadrupleto the {quare of DC, by the 20. of the fixth, that ts , the {qvare RS to the {quare F- uy G. And forafinuch as that whichis contayned under the lines AB and BCis equall to the {quare of the line AC, and that whichis vontayned under the lines AB and BC isequallto theparallelogramme C Ec T the fquare of the line AC isthe {quare RS: wherefore the parallelogramme C Eis equall to the {quare RS. But the fqusre RS is quadruple to the '(quare F G : wherefore the parallelo- gramme CE alfois quadruple to the {quare FG. Agayne foramuch as the line AD is equall tothe line D C, therfore the line A K is equall the line K F, wherefore alfo the [quare G F isequall to the {quare HL: wherefore the lineG Kis equal to the line K L, that is, the line MN tothe line NE: wherefore the parallelogramme MF is equall to the parallelo- grammé F E. But the parallelogramme MF 1s equall to the parallelogramme C G, wherfore the parallelogramme C Gis alfo equal to the parallelogramme F E. Put the parallelogramme C N, common io the both: *w herefore the gnomon XO P is equall tothe parallelogramme CE. But the parallelogramme C E ts priued to be quadruple to G F the {quare, wherefore the gnomon X- O P is quadruple to the fquare G F. Wherefore the {quare D N is quintuple to the {quare F- G. And D Nisthe fquareof the line D B, and GF the {quare of the line D C. Wherefore the {quare of. the tine DB u auintuple to the {quare of the line DC.1 f therefore a right line be deuided by an extreme and meane proportion, andto theleffe fegment be added the halfe af the greater fegment: the{quare made of thofe two lines added together ,is quintuple to the fquare made of the halfe line of thé-greater fegmet. Which was required to be demonjt ated. Ye hhall finde this propoficon an other way demonftrated after the fiueth propofition of this booke. Here foloweth M.Dee,his additions. q ATheoreme. 1. If avight line,gexen,be quintuple in power,to the powre of a fegasent of bin felf:the donble of thas Tii.sy, fegment *Note , how C E and the L2O2ORR XOP ,are pro4 ued equal, for at feruet bin the conuer[e de~ monfirated by M.Dee,here next after. —— —_ ——— aa = = — a a == — -_- ~ = ~i — et Foam — —— ———— = a - = = - +. ——— OF Pn... a ope . .. at ~ a wt f on y PM, ne a > _—— >= = — ~ = simian ate om 4 ta i == = = = ; > ee — — = " —2 _ = = ; > — : =— —— ~ =~ 5§ ™ Aes Sas == = — = = —< = — ms: +} SSS eee ——— a —— —~ == : — —< De. = — — — = ——— : y <2 se. ——- = = =: : * =< = ee = a a b s —— a —— ad om , — ot. «fl 3 __ am ~. ~ — : a ~ — ~~ = = wie, Sp =. = a= == = ——— > -=~ - : = = Bag A ~ —_ = ~. a au - 2 - ne . — > - — >3 aes — - —— oe ek a Et er — —_— ~ a ge > = — ——<¥ _—-+ - Sear ee = ~ 2 ; — = “> = = = : : =—- eee - 2 — + — = SS aS = = re. = = > = = = —= = ee eed ee eS = = FS, SS ie = —_ sets = ES on F = = = 7 a == ~ = — er —= =: oe = = Ss ees ; a — > ee — _ ae — at ee “= = = = ~y — ¥ + a = - eae 7 ee SSS oo ern 4 —= ~~ —— > = =. > ae = Ss -_— : a ae we as Me = - 4 . « . ee ee i —— ae 2 a cs = Sse ee > ils Son ee = - a SE . Se Se eS = = : ——— -- i. - > a nae — a } i 4 K! tf - ' f ; : iv : m| 4 thi | { i Ve - / ‘ I - i { ‘ : ¢ it ( oe rt 4.4 " Wh xP) , ¥ * y te , ia 4 j “ i} RT | aan. Ph ia} H - $))\} ya | ma } oe : , eu) Teele a Edi iu et Sa ate 5 +. eae —e ee E ee = 5 I hn = ~ : — ee 7 = - - 2 ~* —- ——— i = ! R <—g ? = ? - —— = ee — 1 - . ae) cre ae ~_ <= S 2 bisropef- tion, sthe connrfe of the fa'mere % As we bane wotedthe place if the peculer profe therefin the demafration oft ber. T hethirtenth Booke Segment,and the other part réemayning of the firft genentine ; make a line , dinided by extreme and meahe proportion:and that donble of the Jegment ss the greater part thereof. Fotafmuch as,this,is the conuerfe of Euclides third propofition: we will vf the fame fuppofitions and conftructions there {pecified: fo farre,as they fhall ferne our purpofe . Beginning therefore at the conclufion, we muftinfer the part of the propofition, before graunted. It was concluded, thatthe f{qWare of theline DB, is quintuple » tothe fquare of the line DC, his owne fegment. Therefore D N(the {quare of DB)is quintuple, to G F, {the fquare of DC). Bur the {quare of A C( the dou- ble of DC) whichis RS, isquadrupleto G B, (by the fecond Corollary of the 29. of the fixth): and therefore R S,with G F,are quintuple to G F:and {0 itis euident , that the {quare D N , is equall to the {quare R S,together with the {quare GF. Wherefore,from thofe two'equalles , taking, the fquare G F, (common to them both):remayneththe fquareR S , equall tothe Gnomon XO P, Butto the Gno- mon X O P,the parallelogramme*C E,is equall: Wherefore the {quare of the line A C , whichis RS, is equall to the parallelograme C E. Which parallelogamme is cétained ynderB E,( equall to AB: )and CB,the parpremayning of the firft line geuea whic was D B.And the line A B,is made of the double of.the fegmentD:C ;and of C B,the other pare of the line D B, firft geuen. Wherefore the double of the fegment D C,with C B, the part remayning (which altogether, is the whole line A B) isto AC, ( the double of the fegmentD C Jas that fame,A C,is to CB: by the fecond part of the 16.0f the fixth. Ther- fore by the 3.definitié-of the fixth booke,the whole line A Byis deuided by an extreme and meane pro= portion,& A C,(the double of the fegnét D-C) being middell proportionall,is the greater part therof. W eae arightline,be quintuple in power, &c. (as in the propofition ) which was to be demon- firared. ‘Or,thus itmay be demonftrated. Forafmuch as the {quare,DN is quintuple to the fquare GF, (I meanme the fquare of DB the line geué, ‘to the {quare of D C the fegmét): And the fame fquare D N,is equall tothe parallelograme vnder A By, C B,with the {quare made of theline D Crby the fixth ef the fecond: ( for vutothe line A C., equally deuided: the line,C B,is,as it were adioyned). Wherefore the parallelogramme ynder A B, CB, toge= ther with the fquare of DC , whichis GF ,isquintupleto 4 D e B the fquare G F, made of the line DC . Taking then , thac {quare G F,from the parallelogramme vnder.A B,C B : that parallelogramme(vnder A B,C B)remayning alone, is but quadruple te the fayd{quare of theline DC. But , (by the 4.0f thefecond., or the fecond Corollary of the 20. of the fixth } R'S, the {quare of the line AC , is quadrupla to the fame fquare G F. Wherfore by the 7.of the fifth,the {quare & a Hh of the line A C , is equall to theparallelogramme ynder A- B,C B,and fo,by the fecond part of the 16, of the fixth: A- B,AC,and CB, are threelines in continuall proportion. 7g And feing A B is greater thé A C,the fame A C,the double ofthe line DC, fhall’be greater thenthe partB C ,remay- ning: Whertoreby the 3.definition of the fixth,A B,(com- pofed or made of the double of DC, and the other part of 1 DB remaining)is deuided by an extremeand middelpro- T be = 5° portion: and alfo his greater fegmentis A C thedouble of the fegment D C. Wherfore;lfa tight line be quintuple in power &c. as in the propofition:which was to bedemonftrated. onl wih ? | & ATheoveme. 2. . Tf a right line denided by an extreme and meane proportion , be genen ,andte the gras WIENS therof be directly adioyned a line equal tothe Whole line genen;that adioyned line,and the fatd greater De Gan make a line diuided by extreme and meane proportion , whofe greater fegment ss the line aareyned. | a _ Suppofe thelinegeuen ; deuided by extreame and meane proportion ,to be AB deuidedinthe point C,and his greater fegment,let be AC: yvnto AC direly adioynea line equallto A B: letthat be AD: Tfay,that A D, together with A C,( thatis D C)is a denided by extreme and middel proportion whofe greater fegmentis A D, the line adioyned . Devide A D, equally in the point E. Now , foraf- muchas A E, is the halfe of A D,(by conftruétion , ) itis alfo , the halfe of A B(equall, to A D,by con- diruction) :Wherfore by the t,of the thireenth,the fquare of the line compofed ob A Cand A E(which ineis EC)is quintuple to the {quare of the line AE. Wherefore the double of AE > and theline A C See compofed Ss - ; 4 of Enchdes Eelementes. compoted, (4s in ote tight line) fs linede- uided by extreme and meane proportion, by ; | oa the conuerfe of this chird(by me demonitra- eee. Se ted ) : and the double of A E, isthe greater 4 ¢ B fegment .But D C is the line compofed of E the deuble of A E, & the line A C: and with all,A D is the double of A E.Wherfore,D C, 2 is aliné.déuided: by extreme and meane pro-> =: 7 ih ot > at ae portion,and A.Dsiyyhisgreater fegment if aright line,therefore ,demded by extteme and meaheipro- portion, be geuen,and to the greater fegment thereof, be direétly adioyned a line equallitethé whole line geuen,that adioyned line , and the fayd greater fegment , do make a line dinided by extreame and meane proportion , whofe greater fegment,is the line adioyned : Which was required to be demon- itrated, 7 : T wo other bri¢fe demonfirations of the fame. Forafmuch as, p is to A €ras A yisto A C(becaufe & Dis equall to 4 8 ,by conftrudtion ) + bur as anistoac,foisactoc e:by {uppofition. Therefore by the 11.0fthe fifth ,as ac, istoc ®’, fois aD toa c.* Wherefore, asa c,andc s,(which isa )istoc B:foisap,andac(whichisnc)toac. Therefore, euerfedly,as a B,isto a c:foisp Cito a p.. Anditis proued, a p,to betoa c: asa c isto cp.Whereforeasagistoac,andac,toces :folsp c,toavd,andap,toac.Butas,ac,andcs are in continuall proportion, by fuppofition: Whertore p_c,4 D,and a ¢, are incontinuall proportion. Whereféré;by thie:;\definition ofthe fixth:booke;p cis dewded by extremé and ntiddelliproportion, and his greatett fegment,is a po. Which was to be demonttrated. Note from the marke *,how this hath - : two demonitrations.One I haue fet in the margent by. @A Corollary. I. | ‘Opon Exchdes third propofition demonftrated,itis wade euident: that ,of a line deuided by ex- treame and meane proportion sf you produce the leffe fegment ,equally to the length of the greater : the line therby adioyned,together with the fayd leffe fegment make anew line denided by extreame and middle proportion: Whofedeffe fegment isthe line adioyned, For, if A B,be deuided by extreme and middell proportion it the point .C,A C, being the greater fegment,and C B be produced,from the poynt B,makingiajine, with CB, equallto AC, sehich letbe C Q:and the line thereby adioytied,let be B Q:1 fay that © Oy is alinealfo depided by..an éxtreame and meane proportion, in the point B sand thatB Q_ (the line adioyned )is the leffe fegment.Forby.the thirde, itis proued,that halfe A C,( which,let be,C D) with C-Bas one line,compoled, hath his powre or {quare, quintuple to the powre of the | fegment C 1): Wherfore, by the fecond A 9 of this booke, the doyhis re is dé- 53 brie Es sani) 3. B vided by extreme and middell prepor- : - tion ° me che greater fegment thereof, 2 QU fhalbe CB. But, by conftiuction, C Quis 4 —_ | the double of CD, for it is equallto A C. Wherefore C Quis deuided by extreme and middle: prepor- tion,in the pointB:and the greater fegment thereof fhalbe,C:B. WhereforeB Qs 4s the Jefle fegment, which is the line adioyned. Therefore,a line being deuided, by extreme anid middejl proportion, if the leffe fegment,be producéd equally to the length of the greater fegment,the lide thereby -adioyned to- gether with the yd leffe fegment,make a new line deuided, by extreme 8& méan¢ proportion; whofe a lefle fegment,is thelineadioyned . Which was to be demonftrated. <“G ACorollary. 2. if fromthe preater segment of a.line diuided by extreme andteiddle proportion: 4 line sequal te thé leffe fegment be cur‘of: the greater Segment thereby sis alfo denided by extreme and meane propor- sion, whofe greater Jegment foal be now that part of tt which 1s cut of. | Forjtaking from-A.G,alineequallto CB let AR remayne.! fay , that AC, is deuided by an ex- treme arid meane proportion in the point R:and that C R,the line.cut of, 1s the greater fegment. Fort is prouedin the former Corollary that C-Q.is deuided by extreme and meane proportion in the point B.But A.C,is equall to CQ sby conftru@ion:and C Ris-equall to © B by conftrudtion : Wherefore the ‘i Ti ij. = pefidue, *Thereforeby my fecond Thee oreme addel G-~ pon the fecad propofitio2,D C ts deusded ly ex Lrearie and MWCHARE PY OD) Li- foreD Az. greater thes A- C: wherefire if aright lineES ¢. As b# the propee [tion Wek wisxtobe die monfirated 4) S)RRN) Poa as > ee —_—— —— <=" => 9 ee =- a = a en ee ed : z * a Sco a ee = = a — = = == = ae = = <> = SS = ee =. — — ae a aed = 5 = ae a a < 7 = ——— ae FoF SSS ea: r wen = : OF: 3 = = Saad = = = “4. — ne ye Ps = ——_— = = = = oo oy er aes ee ae ~ refidwe,A Ris equall ro B Q the refidue.Seing thesfore the whole A Cis equall to the whole C Q:and the greater part of A C , which is C- yd, hoitod | Ris equalro CB the greater part of | 3 sdlaomsb a C Q:and'the leffe fegm€ét alfo equal A - noreory B to the lefle: and withall feing C Q ts - eyspres oy ‘proued to be diuided by extreme & Be dai bis : uate 8 tre in ces pointB : it % nets | oloweth of neceffity that,A C,is divided by extreme and meane proportion in the point RwAnd fi meer rearerfegmentok GQ. C Rihall be the greacer fegment of AC: Which was to bede= monftrated. 0 iti | be SOLA T hethirtenth Booke % ACorollary. 3. Et is emident thereby, aline being diuided byextreme and meane proportion, that the line Wwhere- by the greater fegment excedeth the leffe together with the lefe feoment , do make atine dinided by extreme and meane proportion: whofe leffe fegment., is the fayd line of exceeffe , or difference betwene she fegments, Tohn Dee. «>» ~ icintogongf Two new wayes to denide any right line geuen by an-extreme perpedscular: Set you may porcese hese tn~ prstecther pa- fasorwill ferues fesbat Dl and AD make an angle for a tri augle to base bss fides propor= tronally cut. €Se. Demoufire- san , acd “poifitH; to the point B;(theoneende ofotr B line gruen)levaright line-be drawen sas Hei: 0 ¢ the liaeAB: fotharitbe alfo parallel to:the: <: “and meant proportion: dem onfirared and added by M : Dee. _ Probleme. \Todenide by an extreme and mane proportion, any right line gewer,in lengthand pofition. o Sucpolt's line geuen in ‘saat and pofition,to be A BI a that ABis to be deuided by an extreme, and meane proportion. Deuide A Binto two equall parts asin the pointC. Produce A B direétly from the point B, to the point D: making B D,équalto BC. To theline A D, and at the point Ditet 2 line be drawen *perpendicular: by the 11. of the firft, which let be D F: (of whatlength you will). From DF and‘ af thé peint D; cur “of thefixth parte of D F: by the:9. of the fixth. And let thar fixth part, be the line D G.VpponD F,as 2 diame- A ver, defcribea femicitcle: which letbe D H- Fi From the point'G; rere‘a line perpendicu-: lar to D F which fuppofe to be GH : and let it come to the circumference of DH F, in the point H.Draw rightlines, H D,and H F.Pro- duce/DH, from the point H, fo long, alla line adfoyned with D H, be equall roH F, which let be DI, equall toH F. From the ; . i 2 5 - Ax B. From'the point I, leta line be draweh,'to): line HB. Which parallel line fuppofero bols:> » K: cutting the line AB, atthe point: Kinkfay) D 4 that A B, is deuided by an extreme & meane proportion, in the point K. For the triangle D KI, hauing HB, parallel to 1 K, hath his fidesD K and D I, cut proportionally, by the 2. of the fixth. Wherefore as 1 His to H D: fo isK B,toB D. And therfore compoundingly, (by the 13. of the fiueth )as D I, isto D H: foisD K toDB. Butby conftruétion DI vis equallte H Pewherefore by thé wof the fifth, D.IistoD.H,as HF.istoD H. Wherefore by the x1. ~ of the fifth; D Kisto DB,as H Fis to. DH. Wherefore the fquare of D Kisto.the fquare of DB, asthe fquare of H F, isto the fquare of DH: by the 22.0f the fixth. But thefquare of H F, is to the {quare of DH: as the line G F isto the line G D: by my corrollary vpon the 6, probleme of my additions to the {econd.propofition of the twelfth. Wherefore hy the 11. of the fifth, the fquare of D K is to the {quare _of DB, asthe liné G Fis to the line GD. But by conftrudtion, G Fis quintuple toG D. Wheréfore the f{guare of DK is quintuple to the fquare of D B: and therefore, the double of D B; is deuided by anex- ‘treme and meane proportié, and B K is the greater fegment therof, by the 2. of this thirteenth: Where- ‘fore feing A B isthe double of D B by conitruction: theline AB is deuided byan extreme and meane p roportion: fe es ee I Baers : ee es of Euchides Elemente.” Fol.394. proportion: and his greater fegment,is the line B K. Wherefore, AB 18 denided by.an. extremeand meane proportion, in the point K. We haue therefore deuided by extreme and meane¢ proportion any line geuen in length and pofition. Which was requifite to be done. ae > . > » The fecond way to exeéuite this probleme. Suppofe the line geulen to be 4 B.Deuide 4 Binto two equall partstias fuppofeit.to be donein the pointc. Produce 4 B from the point B: adioyning a line,equall to.2 C, whichlet be 2... Tothe right line 4 D, and at the point D, erc&t a perpendicular fine equalfto BD, let thatbe D Z«Produce ED tré the point D tothe point F: making D F to contayhe fiue fuch equalfpartés, asD E isone. NoW vport E Fasadiameter,dé{cribe a fémicircle which let be Z KX F:.arid-letithes.. | y Rw ee point where the circumference of £ KF, doth cut.the line,4 8,bethe point X. 1 fay that 4B, is denidedin the point X, by an extreme. and meane proportion. For by the 13.ofthe fixth E D)D XK, &'D F,are three lines irrcontinvatl proportion, (O0-K-being the middle proportionall), Wherefore by the corollary of the 20. ofthe fixth, as ED isto DA, fo is,: the fquare of £ D,to the fquare of D KX, but by conftruction, E Dis fub- uintuple to D F, Whereto:e the {quare of £ D, is flidquitituple'to the < fruareof DK . And ‘therefore ‘the iquare of DK, is quintuple'to the {quareof ED... Add BDisequalkto, ED by-conttruGion; therefore,the {quareefD.X:,isquintuple to the {quare,of B D.. Wherefore the,dou- ble of B.D, is deitided by an extreme ahd mean’ proportion: whofe greater fegmentis BK by the fecond ofthis thittenth’ But by con ftriction, 4 Bis thedauble\of sar W hierelore 4 B, is divided by ex- treme and meane proportion,and his Greater fegment jis B+ and chereby.-X;,,the point ofthe. diuifion. We haue’ therefore deujded by extrehie ad méarié proportions any righting geuen,if length and-po~ © fition. Which was to be done. ox 3" i | ta Mifbouorg zis BW non: | . Ech of thefe wayes,may well be executed: Butin the firft, you haue this auantage: that the diame. e se taken at pleafure. VV hichdn the {econd Way,is cuer iu{t thrife fotong, asthe line geuen to be de. uided. . _ ere ere ee 4 aS , | lokn Dee. S ce WB That emance Ks CHER Orne ke re (6 * 4 3 sai rat.) chi Tas © ' Bo ye Pea DM WARS 1300) See oe PS RES > a Whore 0° Phe aT heoremens so ol he 4Propo/ition, 4 C 4 i» ixy * ha heen , yor «a my cehy Oe 2 . . Mioriiwnnns ‘ — Va tite e Beda d twat s 3 ta J 3Ab5a 2 eo If d right line be denided by anextreame and. meane proportion: the {quares ___ made of the whole line and of the leffe fegmet, are treble to the fquare made Ree Aged aid pa ae lear perc gt ey . ° © ; 4 4 : ealbw ; ; > ———— \Pppofethat the right line ABs bedeuis\ NGA Leal by ait eXtr cane Oe DARE PPOPOPTIDS © ROIS SO es ay Hp itin the point C. And let the greater feg- as ment thereof be AC.Then 1 fay,that the fquares made of the lines AB, and BC’, are treble to the [quare of the line AC.Defcribe ( bythe g6.of the first) vpon the line AB,afquare ADEBsAnd yy make perfect the fizure. Now forafmuch asthe . line AB ,isdeuided by an extreame and meane proportion,in the point C:and the greater fegmeét thereof, is theline AC , therefore that which is contayned under the lines_A B and BC is equalt tothe (quare of theline_AC . But that whichis . > contayned under thelinesA Band C Bisthepaw oo KKk.j. yallelo- Demonfira- t40My.. . i.Dée. This es moft ens dent of my [e- cond Theoreme, added to the third propofiti@. For to adde toa whole line, a bine equall to the greater feg~ met: £5 to wdde to the greater segment a line equall to the whole lines all one thing tn the line produ-~ ced. By the whole line,f meane the line diuided by ex treme and wie hl propor - fiott. T he thirtenth Booke rallelogramme A K, and the {quare of the line A- C is the {quare F D . Wherefore the parallelo- gramme A K is equall to the {quare F D. And the parallelogramme AF is equall to the pa- vallelogramme F E , put the [quare C K common to them both:wherfore the whole parallelograme AK is equallto the whole parallelogramme C- E. Wherefore the parallelogrammes C E. and A- K are double to the parailelogramme A K . But the parallelogrammes A K andC E, are the gno- mon LM N, andthe {quare C K .Wherefore the gnomon LM N and thefquareC K are donble to the parallelogramme AK .But it is prowed that the pavallelogramme A K is equal to the [quare DF. Wherefore the cnomon L M.N and the {quare C K are double to the {quare D F. Wherefore the enomon LM N and the{quares C K and D F ,are treble tothe {quare D F.But the gno- mon LM N andthe {qauaresC K and D F,are the whole {qware AE together with the RD E fquare CK , which are the [quares of the lines AB and BC.And D F isthe {quare of the line A C.Wherefore the {quares of the lines A B and BC , are treble to the {quare of the line AC. If therefore aright line be denided by an extreame and meane proportion, the {quares made of the whole line and of the lee fegment , are treble to the {quare made of the greater fegment: which was required to be proued : Looke for an other demonftration of this propofition after the fifth propofition of this booke. a The s:T heoreme. The s.Propofttion. If aright line be denided by an extreame and meane proportion, and ynto it be added a right line ,equall to the greater fegment ,the whole right line is deuided by an extreame and meane proportion , and the greater fegment thereopsis the right line geuen at the beginning. ee <| denided by an extreame and meane proportion in the point A : and the greater es D C Bb put at the beginning namely, AB, =< an cxtreame and meane proportion a > in the point C,therefore that which - ¢ H F is contayned under the lines AB and BC isequallto the {quare of the line AC.But that which is con- sayned under the lines AB and B- Cis ff. Pea Seen Neral + aa : eed of E:uclidés Elementess\ - F0l.395. C isthe parallelooramme CE and thef{quare of the line: ACasthe(quare C H . Wherefore the parallelogvamme C E is tquall to the [quareC H. Bu¥ vito the {quaré CH is equall the fquare D H by the first of the fixth:and unto the parallelogramme C Ejs equall the pa- rallelocramme H E.Wherefore the parallelogramme D His equallto the parallelogramme H E. Adde the parallelogramme H B,common to then both . Whereforethe whole parallelo- gramme D K is equall to the whole [quare AE. And.the parallelogramme D K , is that which is contayned under the lines B Dand DA, for the line -AD-is equall to the line D Lex the {quare AE is the {quare of the line AB.Wherfore that which is contay- ned under the lines A D and D B is equall to the [quare of the line AB. Wherefore as the line D Bis tothe line B Ay fois the line B Atothe line A D,by the 17. of the fixth. But the line D B is greater then the line B.A . Wherefore thé line B Ais Greater then the line A D. Wherefore the line B D is dewided by an extreame and meane proportion in the point A,and his greater fegment is the line AB . If therefore aright line be denided by an extreame and meane proportion,and unto it be added a right line,equallto the greater fegment: the whole right line is denided by an extreame and meane proportion,and the greater fegment therof, is the right line genen at the becinnine:which wasvequired to be demonftrated. This propofition is agayne afterward demonitrated. A Corollary added by Campane. Hereby it is manifeft that if from the greater fegmsent of aline deusded by amextreame ce meane proportion be taken away thedleffe fegusent’: the faya greater fegment {hall be denided by an extreame and meane proportion ard the greater fegment thereof fhalt be the line. taken away. Aslet' the line as 5 be denided:by an .extreame and meané roportionin the pointc , Andletthe greater fegment, be the “ | fine a csFroma ©, take ciz :making the refidue a p..1 fy that a-4 D c ae ¢ is alfo.deuided by an. extreame and meane. proportion in the + point p,and that his greater portion is Dc. 'For, by the definiti- : ee no(ofa line fo deuided) az,istoac,asacistocs.Butasac— is to c B,fo is A c to p by the 7.of thesifth{fer p \c, by coriftrudtion is equall to c » )wherefore,by the 11.0f the fifth,as a 8 isto a c,foisa ctoc p: and therefore by the 19.0f the fifth, asa B istoac, fois the refidue c 8,to the refiduea p.But c B isto a D,asD c isto AD (by the7.of the fifth) forp c is by ednitridtion eqiiall toc». Wherefore; oc iste c,asp cisitowm and:fo,by the definition ofa line de uided by an extredmeand meane proportion,it appeareth,«, cin the paint p , to be deuided , by an ex- . : treame and meane proportion: which was to be proned. T wo Corollaries( added by M:'Dee following chiefely vpon the veritie, \,,.aad demonitration of his _Additions,vato the 3 porpofition annexed,anad eS partely upon this fifth, by Enclide demonstrated, 9 \ | ACorollary. I. As any line being deuided by an extreame and middle proportion , doth gene vs three right lines, ih contiinall proportion :So either by adioyning direttlytorbe greater fegment a line equal to the first whole lie vor( Secondly sby producing the leffe fegment., equally.conbelength of the greater fegment: or\(thirdh,)by cuttsng from the greater fegment a.partequal to the leffe fegment:or( fourthly )by ad- saying divedilyto the greater fegment.a line equall to the fame fegment :1t 1s manifest that in enery of thefe fower Wazes,Wwe bane two lines deuided by an extreame and meane proportion: (it ts to Weete,one genen,ana the other made Jana With alls euery WAY,We bane ‘fower lines in continuall propertion,) Of thefe two lines, (by extreame and meane proportion:denided,)their.demonttrations jareafter Exclides 3,propofition added:and here in this fifth by Ewcwde proued.But of the fower linesin continu ~ all propogtion ,feing, the demonftration is moft eafy for any man to frame.f will here,butnote'the lines KKk,}). ynte This ts before demonfirated.. most enidentt and briefly by « mM, Dee;after . 10M: « tbe 3, propofie = = : a al ; — + = — — —=> —— —s ~e 7% = eae Ae _ - —s -" ts = P= 2 a a ae = 1S = = SSS ty “3. Se — = — - = E = = = er - : 4 = == —— —s ae = = = —— ———— = = 5a ee =" 25 = =~ = Sa + == = —— — = = = 3 = 2 = : - > SSS <= : —> +a =~ - eS ah Sy a Bt a. - a iM a4 T ony Bot i i t f 4 ia 6S Sa — = "hop - Se = te - " > ~ - —.< > > =a oa ee . = == = —_e — —— SS — — aes = = ~ = = Re SS ne OF So eg ———— = as < { i i 1 3 = Ve. AS eee ge Ua > ee aie PCy jy: a Hp pe ? ; ea * = = ae sir [eerie ts en es ee ee = _ ate * “ = a A tS ae a _ Posi - Se ~ 5 = ee ee 6 GS = 0-8 x a Sy i eo ome SSS pare ‘ Boag es eee Noteg-. 1. Proportional bones. Re 3- 4. Note 1. fwomstd- 2. dle proper= 3. tsonalt. = he Note 4. Weyes of propre|Son,in sheproportiom ofatine diuded byextreme and wsddle proper- Stow, What refelutiors and compofition ss,bath before bene taught in the beginning of the firft booke. T he thirtenth Booke vnto you: asin euety of the fover places;the conftructions haue them lettred and fpecified. As in my firtt way added.after the third prpofition,D C,A D,A C,and C B: are fower lines in continual propor~ D : A Cc B leads a A C | B po nse fy A. C B © — En aennnEeneeenenetl “a A Cc B tion.Andiin the fecond way,A3,C Q.(equall to A C)C B,andB Q:are fower lines in continual! pro- portion,And in the third way,A B,A C,C R,(equall'toC B ) and A R, are the fower lines in continual proportion.And in the fourth vay( by Euclide declared)D B,A B,A C(equallto A D and CB,are tow-. er lines in continual proportion.So,that in the firft way you haue A D,and A C,middle proportionals betwene D C,and C B.In the fecond way,youhaueC Q, and C B,betwene A B,andB Q..In the third way,you haue A C,C R,betweie AB and A R:and in the fourth way,you haue A B,and A C,betwene D B,and CB, A Corollary. 2. ‘fi is alfo manifest that yor may by any of the fower wayes bere fpecified proceede infinitely in the proportion of aline'denided by ixctreame & middle proportion: And inthe firft and fourth Wwayes, en- creafing continually the quantities of the lines made: but in the fecond and third = » dimins {bing continually the quantities of the faya whole lines made( and thereby thes fegmentes ) i nd yet ,ne- - yertheleffe reteyning in euery Ine made ( by any of the wayes ) and in his fegments,ab , and the fame properties which the firft line,and his fegmentes hane. After Which rate of ‘Progreffion , asthe termes in continual proportion do encreafeyand are moe tn number: So, likewife, do the middle proportionalls, (accordingly become moe:Buiener fewer in number,by two,then she termes of the Progreffion are, = gWhat Refolution is. 3c ‘Rifolunionsic the affumption or taking of thething whichis to be proued,as graunted, and by hinges which nece(firily follow it,t0 pafse unto [ome truth graunted. «vou Kbat Compofition is, Compofition,is an a ifumption or taking of a thing graunted, and by thinges which of neceffity a it,to paffe vato the finding out of the thing fought orto be proued. Refolution of the firft T beoreme. Sippofe that a certaine right line A B,be diuided by an extreame cy meane proportion in the port C,c> let the greater ‘[eemet therof. be AC,vntowhich adde a line equal to the halfe of the line A B,and let thatlne be A D.Then 1 [ay that the {quare of the line CD és qui tupletothe fquare of AD . For ‘foralmuch as the fquare of the lineC Dis quintuple to the [quare of A-Dbut the [quare of the line uncctne, See C D is(by the 4. of the fecond ) equall to p | A ‘ = that which is compofed of toe {quares of esse - of Euclides Elementes..\ Fol396. whith is contayned under the lines C A,and.A D twife. Wherforethat which is comepofed of the {quares of the lines C AeA Dytacether with that which is otained under the lines C- A,¢y A D twifeis quintupletothe {quare of the line A D.Wherfwe;that which is compofed of the {quare of the line C Atogether with that,which is contayned under the lines C A, and uA D twifeis quadruple to the {quare of theline_A D.But vit, that, which is contayned under the lincs C A, and AD, twife,isequall that which is conteyned under thelines.C A, and AB, for the line A Bis double tothe line AD. Anduntothe {quare of the line A- C, is equall that which is cotayned under the lines A Bcy B C,forthe line A B, is by [uppofi- tion diuided by an extreme and meanéproportion,in the point C.Wherefore, that, which is contayned under the lines A B, and AC, together with that which is contayned under the lines A B,and B C,is quadruple to the {quare of the line. A D.Bst that , which is compofed of that which is contayned under the lines A B,and AC,togetherwith that which is coutay- ned under the lines A B,andB Cis the {quare of the line AB(by the 2. ofthe {econd.)Wher- forethe {quare of the line A B is quadruple to the {quare of the ine AD. And fo is itin deede:for the line A Bis double to the line A D:as was at the fir$ [uppofed. Compojition of the firft T heorene. Now foralmuch as the [quare of the line A Bis quadruple to the {quare of the line AD but the {quare of the line A B,is,that which is contayned under the lines A B,and A C,toge- ther with that which is contayned under the lines BA, and BC. Wherefore that which ws contayned under the lines B A,and AC, together with that which is contayned under the lines B 4,and B Cis quadruple to the [quare of the line AD. tut that whichis contayned under the lines B A,and A C;75 equall to that which is contayneaunder the lines D A, and A C twife( by the 1.of the fixth),and that which is contayned uncer the lines A B,and B Cis equall to the {quare of the line AC, by the definition of aline dinided by extreme and mcane proportion.Wherefore. the {quare of the line AC,tocether vith that, which is contay- ned under the lines D A,and AC twife,ts quaduple tothe [quar of the line D A.Wherfore shat which is compofed of the [quares of the lines D A,and A C,ogether with that which is contayned under the lines.D A,and AC, twife,is quintuple to the{quare of the line D.A.But that which ts compofed of the [quares of the lines D A, and AC, together with that which is contay ned under the lines D A,and AC twife,is equall to thifquare of the line C D ( by the 4.of the fecond) Wherefore the {quare of the line C Dts quintuple to the {quare of the line A D: which was required -kobe demonstrated. ‘Refolution of the 2. I’ heoreme, Suppofethat acertayne right line,C D,be quintuple toa fegnet of the fame line namely, to D.A-:and let the double of the line D.A,be AB.Ehé 1 faythat toeline A B ts divided by an extreme and meane proportion. imthe paint C:and the greater feqmet therofis AC, whichis thereft of the right tine put atthe bevinning F or forafmuch as the line A.B is dimded by an extreame and meane proportion in the poynt Cand the greaterfegment thereof és the line A C, therefore that which is contained. under the lines A.B and hC,is equall tothe {quare of the line AC . But that which is contaye ned under the lines B A, and AC ,isex® A G>'ys B qual to that which is contayned un-. | ; der the lines D A,and AC twife: for the line B A,is doubletotheline AD. Wherefore that which és contayned under the lines A B,and B C together with thht which is cotayned under she lines B A,and A C,whichris the {quare of the line A B(by thez.of the fecond) is equall to KKk.ty. tine T hethirtenth Booke that whichis contayned undery lines D A,cy AC, twife together with the fquare of the line AC.But the fquare of the line ABs quadruple to the [quare of the line D AWherfore that whichis tontayned onder thelines D.A,and AC, twife,together with the [quare of the line AC,is quadruple tothe (quare of the tine A D Wherefore the {quares of the lines DA, and. A C,together with that whichis contayned under the lines D A,and AC, twifewhich is the {qware of the line D C,are quintuple to the fquareaf the line D A. And foare they in. deede by [uppofition. | | | Compofition of the 2.1 heoreme. ow forafmuch as the quare of the line CD, is quintuple to the {quare of the ine DA, sole 5 of the line a D,is ¢: whichis. copofed of the fquares of y lines DA, AC, togetherwith that which iscotained under the lines DA,¢ AG, twife:Wherfore y [quares of the line D A,@ AC tagether with that whichis cotayned under the lines DA, AC, twifejare quintuple to the [quare of the line DA Wherfore,by diuifio,that which is cotained under the linesD A,and A.C, twife together with the {quare of the line C A,is quadruple to the fquare of the line AWD. And the {quare of the line AB, is quadruple to the {quare of the line AD » Wherefore thatwhich is contayned. under thelinesDA, and A.C twife, whichisthat, which is contayned under the linesB A, and AC once, together with the {quare of the line A C,is equall to the {quare ofthe line A B.But the [quare of BA, 1s that which is contayned under thelinesBA ,and AC, together with that which is contayned under the lines B A,and B C.Wherfore that which is contayned under the lines BA,¢> A- C together with that which is cotayned underithe lines AB,C- BC, isequall to that which is comtayned vader the lines B A,and A C together with the [quare of the line AC. Now then taking awaythat whichis common to them both namely, that which ts cotayned under the lines B A,and A C,therefidue,namely that which is. contayned under the lines AB, BC és equal to the (quare of the line A C.. Wherefore as the line BA, is to theline AC, fo istheline A C.tathe line CB-But the line BAszs greater then the line A C wherefore the line A C.alfo is greater then the line C B.Wherefore the line A B ts, dinided by ‘an ex- streame,and meane proportion in the poynt C ,.and the greater fegment thereof 4s the line ) W ) | } { re mur : i) Wy - } : Ae | af rf a) ety ‘ + i at i : : way’) 5 ‘tab is - hit 7 ta if ’ ‘oe a cE ; : 1 bi hi a A. C, which masrequired to be demonfirated. ein “eS = SS leat = A > Refolution of the TF heoreme. Suppofe that.a certaynevight line AB jbe-dinidded by ‘nn extreame , and meane propor. | tion in the point C:and let the greater feement thereof be the line A C, and let the halfe of the litte AC; be the line CD. Then Nay that the Jquare of the B D is quintuple to the {quaveof the line CD.F or forafmuth as the (quare of the line BD,is quintuple tothe {quare ‘of theline C D But the {quare of the ine D B; is that whichis contayned under the lines ‘A Bana BC ,,tovether with es of thelineD.C (by the 6.0f the fecond). Wherefore e LR DME S : wee ” 7 > - ———— : : = Soak —_ a. ae ae = a “ 30 sR a a 1.5 - SA Seis oct th aetesal “ got a = S-type 3 . Fre = =~ 25 a . imate emma “that whichiscontayned under the linesA B,andBC; . tocether with the{quare of thelineD.C , is quintuple: A ertinety to the fquare of the line D C. Wherefore,that whi ies Dene wig contayned under the lines A'Bsand BC , is quadruple. oo | to the [quare of the lineD C . But unto that whichis wwe ese - contayned wader the lines A Byard BG, %s equall the fquare of the line A C for the line A B;is dinided by an extrtame and meane proportion tn the point C. Wherefore the fquare .of the line A Cis quadrupletathe{quare.of the line) Crand fois it in deede., forthe line A Gisdouble to the lime DC. Conte of Euchdes Elementes.. Fol.397- Compofition of the 3.T heoreme. Forafmuch as the line A C is double to the lineD C , therefore the fquare of the line A Cis quadruple to the {quare of the line D C (bythe 20.0f the fixth) But unto the fquare of the line A C,is equall that whichis contayned under the lines A ByandBC, by [uppofi- tion:wherefore that which is contayned under the lines AB, andBC , is quadruple to the {quare of the line CD . Wherefore,that which is contayned under A B, and BC,to- ther with the {quare the line D C, which is the {quare of the line D B ( by the 6. of the fe- cond)is quintuple to the {quare of the line C:which was required to be demonitrated. Refolution of the 4.I heoreme. Suppofe that a certayne right line AB , be dinided by an extreme and meane proportion in the point C._And let the greater fegment thereof be. AC. Then 1 fay that the '[quares of the lines A B,and BC,are treble to the {quare of the line A C. For forafmuch as the [quares of the lines AB, and BC, are treble to the {quare of theline AC, but the {quares of the lines A B,and B C,are that which is contayned under AB , and BC,twife together with the Square of the line A C(by the 7 .of the fecond). Wherefore that which is contayned under the lines A B,and B C,twife,together with the {quare of the line A C,is treble to the '{quare of the line Meal She SSNs a ee AC.Wherefore, that which iscontayned under ~* . - the lines A Bey B C,twife,is double to the {quare of the line AC . Wherefore that which is contayned under the lines AB, and BC , once,is equallto the {quare of the line AC. And [oit isin deede. For theline AB is diuided by an extreme,and meane proportion in the point C. ‘Compofition of the 4.T heoreme. Forafmuch therefore as the line AB , is dinided by an extreme and meane proportion in the poynt C,and the greater fegment thereof is the line A C,therfore that which is contay- ned vuder the lines A B,and BC is equall to the {quare of the line A C.Wherfore that which 4s cotayned under the lines A B,and BC twifeis buble to the {quare of A C.Wherfore that which is contayned under the lines A Band BC , twife,together With the {quare of the line AC, is treble to the {quare of the line A C.But that which is contayned under the lines A- B,and BC,twife,together with the [quare of the line .AC,is the fquares of the lines .A B,and BC (by the 7.0f the fecond). Wherefore ot bare of the lines A B,and B C, are treble to the Square of the line.A C:which was required to be demonstrated. Refolution of the s. I heoreme. Suppofe that acertaine right line AB , be diuided by an extreme and meane proportion in the point C. And let the greater Jegment therof be the line AC. And unto the line AB, adde a line equall to the line AC , and let the fame be. AD. Then I fay that the line D- B,is dinided by an extreme and meane proportion in the point A. And the greater fegment ther of is the line A B.F or forafmuch as the line DB is diuided by an extreme Cy meane pro- portion in the point A,and the greater fegment thereof is the line AB , therfore as the line D B,is to the lineB A fois the line B- A,to the line _ AD: buttheline AD, A c B +s equall to the line AC : wherefore as the 3. = —— a oe ad = tT = eee2.--G = ¢ E ~ = —S<3 Sa > . Sa 2 <9 <== 3 ~ = seine = —> ro Se int. SS - Ss - =s 2 = a ne z at tec z : =— ——— ae =. = = — gh gs SSS SS = = 2 ee ee ee =I : = ee i ee = - p= ee ea ee a = = = = : 5 ~ e * bs iH. ~ e VE ARS ee ere SS > a ss - — — a ee 2 a 2 a . = - —_ _ " = ies nd PR M2 =e a ee “2 = m= —— =i < Ce a= Vi — 3 a = = = - See > le a = Teh ene ~ | ae = == a = = = = — i = ‘ Sn cniee Ss ‘ 5 , a { : ' H i) ee v L} wy { ? i 4 “ : > ha f ' Libs ' n } Pe at ¥ q ¥ q me, q ti ‘ ' d iF ie : Pi : \ af f é y al) | ih " vs NESS 7 if Pa | ; a ; 2 af be . ry k i * i he | mi, * itt | wh a) | “Wg ) ' ee i mn! 4 i. ¥. r A | ir) 4 : . . in Se as, 3+ = — See ee ee a ee * Proclusin the Greeke: in the 58, page. 2? 32 2? 7 T beshirtenth Booke she line D B,is tothe line B A,fois the line BA to the line AC Wherfore by conuerfio as the line B D isto theline D A,{ois the line AB to the line B C(by the corollary of the 19. of the fifth) wher fore by diuifion;by the.17 :of the fifth;asthe line’ B'A,ts to theline AD, fo 1s the line AG tothe lineC B.But the line. ADs equall tothe line A.C Wherfore asthe line B A, istothe line AC,fois thelineA tothe line C B.And foit isindeede, forthe line ABs, by fuppofition,diuded by av extreme and meaneproportion tn the point C. ) Compofition.of thes. Ibeoreme. Now forafmuch as the line AB , 15 diuided by an extreme and ‘meane proportion in the point C:therefore as the line B A ts to the line A C,fo is the line AC tothe line C B: but the line A C is equail to the line AD Wherefore as the line B Aviso the une AD , fois the line A Ctotheline C B.Wherfore by compofition (by the 18 of the fifth) as the line B Dts to the line D CA, fois the lint A Bt0 theline BC.Whereforeby conuer fion( by the corollary of the 19. of the fiueth) as the lineD Br to the line BCA, fois the live B Ato theline AC bet the line ACistquall to the line A DwWherefore at the line D Bis to the line BA, fois the line BA wtheline AC. Wherfore the line D Byis dentdéd by anextreme and meant proportion in the point A’? and his greater fegment is tie line AB : which was required to be demon- firated. | 4 7 * An Aduife by Tobn Dee added, @Eing,it is doubteles,that this parcel of Re/olstion and Compofition,is not of Exclidtes doyngrit éin not intly be imputed to Exehde,that he hath,thetbyseyter fuperfiuitie or any part difproportionedunt his whole Cotepo/iion Elemental!» And-though, for one-thing , one good demontftration well fufrifedh: for ftablifhing ofthe yeritie: yet,of.one thing diuerfly demonftrated : to the diligent examiner ot the diterfe meanes, by Which, that varietié arifeth, doth grow good occafions of itttentmg demon" trations ,where matter is more {traunge, harde, and ‘barren .Alfo, though refohitiomwere cotinall: Euclide before vfed: yet thankes are to be geuen to the Greke Scholie writter, who did leaue both the definition, and alfo, fo fhort.and eafy-examplesof a Method, fo auncient , and fo profitable. Theantiguity of it, isaboue 2006; yeares: it is to were, Center fince’ Plato his time, and the profite, therof fo great,that thus I finde in the Greeke recorded. *Mefodor Si éuws wapad idovlat: xaddisy . c ? 3 © “a : EN sintec ue 3 ‘3 : fz) ‘ y diarag avandoees’, to dpydoucrcye dyin, yey Boor o Cinlodsduoysiv xy 6ITAaT ey (ws past) ; : ie , pe < : mt A Fs } rs Re 2 : Asodapecyet elped WHEY. aD. ng KgY EXeVOg TOhAWY x glee, YEaMETescy dupélys isoeit] ous yeveodat. Proclus hauing fpoken of fome by. nature, excellent in innenting demonitrations ,pithy and breif fayeth : Yet are there Methods geuen [for that purpofe].Andin dede, thatthe beft, which, by Refolution,reduceth’ the thine inquired of,to ari vndoubtediprinciple.W hich Method;Plato, taught Leodamas (asisre-. ported)And he is régiftred, tlfereby ;to hanebenetheinuenter of many things inGeometry,." Sa And,verely 3.in Problemes , itis the chief ayde for winning and ordring a demonftration ¢ firlt by Suppofition,of the thing inquired of, 'to be done: by due and orderly Refolution to bringit to aftay; at an yndoabted veritic. ii which point of Att , great abmadance ofexamples, are to be {ecn., in.that. ' : . . . . * , e bs ™ ° x am | . excellentand mighty Machemaucien sArchitmedes +. un his expofitor,Eutocius, in Menecnimus likes wife:and in Diocles booke,de Pyrijs:and in many other. And now, for asm uchas, our Excisdein the Jatt fix Propofitions of this thirténth booke propoundeth ; and concludeth thofe Problemes , which\ were the ende,Scope, and principal purpofé,to which all the premiffes of the 12. bookes,and the reit of this thirtenth, are direéted and-ordered,$ It fhall be artificially done,and toa great commodity, by Refolution , backward , from thefe 6. Probleives , to returne'to the firlt definition of the firft booke:I meane,to the definition ofa point. Which,is nothing hard to do.And I do counfaile all fuch,as defire toattéine , to the profound knowledge of Geometric , Arithmeticke, or any braunche/of the {ciences Mathematicall , fo by Refolution ,.(difcrearly.and aduifedly)to refolue,vniofe, vnioynt and diffeauer euery part of any worke Mathematical, that, therby, afwell, the due placing of euety verity 4 and his proofe:as alfo,whatis either fuperfluousjor wanting ,may euidently appeare.For fotoinuent,é there with to ordér théir Writings, was the cuftome of them,who in the old time,were moft excellent, And L.(formy-part)in,writing any Mathematical. conclufion, which reguireth great difcourfe, atlength haue found, (by experience) the commoditie of it,fiuch: that to dovothier wayes.were to mea confn- fion,and an vimethodicall heaping of matter together: befides the difficulty of inuenting the matter to be difpofed and ordred.I haue occafion,thus to geue you friendelyaduife,for your behofe:becaufe fome, of late,haueinueyed againit Ewclide,or Thee inthis place, otherwife than I would with they had. | | r The = £. of Euclides Elementes. Fol.398. The s.T heoreme. ___ The 6.Propofition Ifa rational right line be dtuided by an extreme and meane proportion: eyther of the fegments, is an irrationall line of that kinde,which is called arefiduall line. AV ppofe that AB; beyng a rationall line be deuided { y anextreme and meane proportion inthe point G,and let the greater fegment thereof be:AC.Then I fay that eyther of thelines A C ,and CB , is anirrationall lineof that kinde, SEE) which is called a refiduall.Extend the line A B,tothe point D: and let the line AD,beeguallto halfe of theline AB. Now foras much as the right line A B;is-p . a 3 diuided by an extreme cy meane PIED OT = nn te lee} tion in the point C,and unto the greater | fegmét AC is added a line.A D,equall to the halfe of the right line A B:therfore(by the tof the thirtenth the [quare of the lineC D, is quintuple to the {quare of the line A D.Where- fore the {quare of the line C D hath to the fquare of the line AD that proportion that nuber . hath to nuber. Wherfore the {quare of the line C D is comme/urable to the {quare of the line A D.But the {quare of the line DA is rationall, for the line D A is rationall,forasmiuch as it is the balfe of the rationall line A B. Wherefore the [quare of the live C D,is rationall. Wherefore alfo the line C Dis rationall And foralmuch as the '{quare (A the line C.D hath not to the {quare of theline AD , that proportion that a {yuare number hath tou Square number ,therfore( by the 9 of the tenth) the line C D ,is incommen|urable in length , tothe. line A D.Wherefore the linesC D , and D Aare rationall commenfurable in power only. Wherfore the line A.C is arefiduall line,by the 73.0f the tenth. Againe, foralmuch as the line.AB,is denided by an extreme and meane proportion,and the greater Segment thereof is A C,therfore that which is cotayned under the lines AB,¢ BC,is equallto the {quare of the. line A C.Wherefore the [quare of the line A C,applyed to the rationall line AB, maketh the bredth B C. But. the {quare of a refiduall line, applyed toa rationall line maketh the bredth afirst refiduall line(by the 97. of thetenth) . Wherefore the line CB, isa firftrefiduall line. And it is proued that the line.A C, is alfo a refiduall line. If therefore a rationall right line be diuided by an extreme and meane proportion either of the fegments,is.an irrationall line of that kinde,whichis called a refiduall line,whichwas required to be demonftrated, «A Corollary added by Campane. F ete itis manifeft , that ifthe greater fegment be a rationall line: the leffe fegment fhalbea refi- wall line. F For if the greater fegment A C,of the right line A C B be dinided into two equall partes in the point D,the fquare of the line D B, fhalbe quintuple to thé = of the line DC, (by the 3. of this booke.) And forafmuch as the line CD , (beyng the halfe of the rationall line fuppofed A C) is rationall by the 6. diffini- tion of the tench’: : And vnto the fquare of thelineD C, A D C B the fquare of theline DB is commenfurable ; ( for itis ered quintuple.ynto it ) wherfore the {quare of the line D B, is rationall.W herfore alfo the line DB is rationall. And : ; forafmuch as the fquares of thelinesDB & DC sare not in proportion, as a fquareniberis toa fquare number ; therefore the lines DB and DC are incommenturable in length ( by the 9. of the tenth.) Wherefore they are commenfurable.in power only : Wherefore by the 73. of thetenth, the line BC, which is the leffe fegment,is a refiduall line. Ther. heoreme. The 7-Propofition. Tfan equilater Pétagon have three of his angles whether they file in ore | .t, is Conftrn tions Demonfiras t20%e aE | i) |" 44] in PRD Ss ~ — Sa —~- ——— ~——— ene > " ) TP ~~ ee —— a a ——— =m z —— = ——— ~~ - ——— ne 7 aoe —.—ers —— — = i z —_ = —— == ~ = ~ —~ = ews = . 5 —_ es a res ago — = = — Se os a = a = eee | ee : aes ee 5 = —= ee eee =* — Se : = toss Sc Oe 4 aoe pes: ts =< SwEre _— == i = > {= Se. ne ee = ——— = Tre Saas Se ee z mu Rha. CDs comalttethe angle F ED. (ly the 8. of: pHa Sh ee CSS TTP foft:Andit isproned that theungleb C A3ts bg ab S* <3 Wy ee tothennale AE B. Wherefore thewbole angle BC Disequallto the whole angle AE D.But the ancle BCD yi fuppofeA to beequall to the angles A, and B. Wherefore the angle A E- D, is equal} to the angles A amd Ban outward angle of the triangle ABH» Ang, Ko. the angle E~A-Gis double to the angle B. AG. ( bythe. laft of the fixth).F or the circumference EDC is Adit» ». ble tothe circumference C/B. Wherefore the angle H- A E isequallto the angle A H E .Wueréfore alfothe tight lime EE. is( by the. of the firft.equall tothe ..\ Eee Ces ee right lineE A, that istotheline AB,.And forafmucbas, the right line BA isequall tg the right line AE therefore the angle AB Eis equalltotheangle A EB ..Butit és proucd that. theangle ABE is equal tatheangle B.A. wherefore alfothe angle BE A: is equall to the angleB.A A..And tn the tao triangles A BE andsd BE the angle BE is common to them both wherefore theangleremayning, nauscly BA Bssequall totheangle remajuings namelysto HB hy the.cordlarpoftheg2of the firsh)..Wherfore the triangle ABE} I5.e quiangletothetriangle ABH Wherefore proportionally asthe line E B.. is tathe,line B Ay: foisthelsaesaB tothe. lineB Hd bpthegsof thesirth ), But the line B.A is equall tashe line E-HWherefoveas the lint BB istatheline bHfaisthevine EH to the line H B.. But the line B Exs greater shenthelineB Arwlierefore the line hA yalfois greater. then theline H- BWhereforethetineB Ejsdiuided pan extreme and méane proportion in the paint H (by the 3.dif fircitionof the fixth and bes oreater fegment Eas equallte the fide of the Penta. contin like fart alfo may we prone that the line A Cisdiuided by anextreme and meanepre- portion inthe point Hand that his greater fegment GH, isequall to the fide, of the Penta- von. (For théwhale lined Cis equalltotkewhole line B E,and at hath bene proned that the | parts tekenumapB Hand A Hear eeguall suber fore therefidue CH is.equall to the refidue E Huby thespsakthe fifth) Wtherefoxeinian equilater and equiangle Pentagon two right lines do {nbseaid-imo of the apgles following in order: thofe lins.doo dinide. the one the o- ther by an extreme and meane proportion : and the greater fegwents of thofe lines are echt equal tothe fide of the Pentagon:which was required to be demonftrated. q Ihe 9.T heoreme. ~ The 9.Propofition. Tf the frdeof dn 'equilater hexagon and the fide ofan equilater decagon or teangled figure; which both are injeribed imone xs the felfe Jame.cercle, be ‘added Together : the whole'vight line made of them is a line dimded byan extreame and meane proportion , and the ERP Jegment of the — is . Y. the Confirnfione Demonstra- tion. » 252 —_ — AS z 2%. - = . = ie! _—_—— —vPe = SS =e ee ee ee SS SSS SS SS ee eee ——— = = SSS a =e ——_ = — ~~ = = ~ : - ie ——— ae = = aS en z = oo sie =: —— ——— == = * +. 2= . Coustrattion. Demon/t ra- t80%e This Corollary tg the 3-propo~ fiticn of t be 14.bo0ke afe ter Campante T he thirtenth Booke the fide of the hexagon; an Gs Vppofe that there ben circle ABC. And let the fide of a decagon or tenangled ‘figure in{cribed in the circle'A BC,beBC,and let the fide of an hexagon or fixe angled figure in{cribed in the fame circle,be C D.And let the lines B C and CD be fo ioyned together directly that they both make one right line, namely, BD. Then I fay: thatthe line B Dis diuided by an extreame and meanesproportion in the point C:and that the greater fegmet ther of ts the line C D. Take( by the r.of thethird) the centre of the circle..And let it be the point E:and draw thefe right lines E B,EC,and ED . Andextend the line BE tothe point A.Now fora{much as BC isthe fide of an®\- equilater decagon;therefore the circumference or femitir- cle AC Bis quintuple to the circumference CB .Where- fore the circumference AC is quadruple to the circtmfe- rence C B. But as the circumference AC isto thecircum- ferenceC B, fois the angle AEC to theangle C EB,by the laft of the fixth . Wherefore the angle AEC ts quadruple totheangleC EB. And forafmuch astheangle E BC is equallto the angle E C B( by the 5.of the firft,)for the line E Bis equall to the line E C,by the diffinition of acircle, therefore the angle A ECisdonble £0 the dnORE-C B sby the 32.0f thefirft: And forafimuch asthe right line E Cis equall tothe right tine OD \by the corollary of the t5.of the fourth (for esther of them is equall to the fide of the hexagon in{cribed in the circle ARC) therefore the angle C E D is equall ta the angle C D Ev wherefore the angle EO Besdonble to theangle E D C,by the 32 of the fir ft. But it is prowed thas the angle A E Cisdoubletd-the angle EC B, wherefore the angle AE Gis qua: druple to the angle E DCs And it is proued that the angle AEC is quadruple tothe angle BE CWherefore the angle E D'Cis equall to the angle BE Cs And the angle E B Dis com: nion to the two trianeles BE Cand BE D : whereforethe augleremayning BE Dis equall to the'wneleremiayning EC B,by the corollary of the 32. of the firft . Wherefore the triangle EBD is-equiangle to the triangle E BC.Wherfore,by the 4.of the fixt,proportionally,as the line B.D istothe line BE, foistheline BE to theline BC . But the lineE Bis equallto the line D2Wherefore asthe line BD is tothe D C ; fois theline DC tothelineC B . But the line BD is ereater then the line D C's wherefore alfo the line D C is greater then the line C- B. Wherefore theright line BD 1s dinided by an extreamtand meane proportion in the point C:and his greater fegmentis DC. If therefore the fide of an equilater hexagon , and the fide of an equilater decagon or tenangled figure, which both areinforibed in one and the felfe fame circle\be added toeether:the whole right line made of them, is alinedinided by an extreame and meane proportion,and the greater feement of the fame is the fide of the bexa- gon:which was required tobe promed. © aa A Corollary added by Fluffs.. Hereby it is manifeft, chat the fide ofan exagon infcribed in a circle being cut by amextreame and meane proportion, the greater feginent thereof is the fide. of the decagon.in{fcribed ‘in the fame circle. For ififrom the right line D'C be cut ofa right line equall to theline CB, we may thus reafon , as the whole D Bis to the whole D C; (6 is the'parrtaken away DC to the part taken away C Bis .wherefore by the i.oftheififth, the refidue is coithe refidue as the wholeis to the whole .. Wherefore the line D- C is cute like vinto the line D B: and therefore is cut by an extréame and meane proportion. ea Campane Campane putteth the conuerfe of this propofition after t this maner. | If aline be dinided by an extreme and meane proportion,of Wwhatcircle the greater fegment is the fede of anequilater Hexagon, of thefamefhall the leffe Segment bethefide.of aneguilater Decagon, And of what cireletheleffe fegment isthe fide of an equilater Decagor,of the fame is the, greater jeg ment the fide of anequilater Hexagon... | « For the former figure remayning,{uppofe that the line B D be dinided by an extreme and meane proportioniithe point C :and let the greater fezment therof be D C. Thé I fay tharof what circle the line D. Cis the fide of an equilater Hexagon; of the fame circle is.the line CB the fide of an,equi- later decagon:s and ofwhat.circle, theline B C is the fide of an equilater Decagon,of the fameis: the line D C the fide of an equilarer Hexagon... For if the line D.C be thé fide of an Hexagon infcribed in the'circle,then by the corollary of the 15. of the fourth , the line D C is e- qual to the line BE.And forafmuch as the proportié of the lineB D to the line 1D Cis as the proportion of the line D- C to the line C B, by fuppofition:therfore (by the7.of the fifth)the propottion of the line B Dto theline BE ,isas the proportion of thé line B Etotheline BC . Wherefore (by the 6. of the'fixth ) the two trianglesD E B,E BC are equiangle ( for the angle B is common‘ to eche-triangle). Wherefore the angle D is equallto the angle CEB : for they are fubtended of fides of like proportion. And foraf- muchasthe angle A EC is quadruple tothe angle D ( by the 32. of the firlt twife taken , anid by the s.ofthefame) thereforethe fameangle A EC is quadruple co the angle CEB. Wherefore ( by the laft of the fixth) the circumfe- rence A'C is quadruple to to the circumferéce CB. Wher- fore the line B Cis the fide of adecagon infcribed in the circle ACB. But nowif the line BC be the fide of a decagon in- | {cribed in the circle A B C,the line C D thalbe the fide of an Hexagon infcribed in the fame circle.For Jet D C be the fide ofan Hexagoii inftribed in the circle H. Now by the firft part of this propofition the line B C fhalbe the fide of a decagowinfcribed in the fame circle.Suppofe thatin the two circles A C+ BandH be infcribed equilater decagons,all whofe fides fhalbe. equall.to. theline CB. And -forafmmuch as euery equilater figure in{cribed ina circle is alfo equiangle stherefore bothe the decagons are equi angle.Andforafmuch asaltheanglesoftheonetakento- . gether dre eqiid}! 6 al the anglesof the othér taken toge=. ° ther,asitis.ealy tebe proned» by that which is added afe ter the 32.0fthe frit, therefore one.of thefe decagons is equiangle to the other: and therfore the one is like to the other by thé diffinition of like fuperficieces » And for that if cherebetwa liké redline figuresinfetibédinitwo cit ‘ cles, the proportion of tlie fides.of like proportion of thofe Rgures,{halbe as the proportion of the-Diameters; . of holeeieles, ast is eafy ‘to'prowe by the corollary of | * the 20.0f the fixth jand firlt of this booke sbucthe fides of: ) the like decagons. infcribed inthe two dircles A B.C-and... H are equall: therefore theyr, Diameters alfo are equall. Wherefore alfo’ theyr femidiameters aré equall . But the fernidiameters and thedide\ofthe Hexagon are ‘equall, by. © the corollary of the 1s .of the fourth » Wherefore the line. D Cis the fide ofan hexagon:inthe circle ABC, asalfe.._ it is the fide of an hexagon infctibed in the circle F,which =~ is equall to the circle A B C: which was'required to be proved; ae gf The 10. T heareme. = The ‘T0, Pr ope tion ‘ Ifin a circle be deferibed an equilater Pentagon, the fide of the Pentagon coutdineth in power both the fide of an hexagon and the fide of a decagon, | DD ay. * being’ Demon ftra- tion of the first parte Demonftrati- on of the fem cond part, an t | ; = [we ee a — ai > = So te : =, A St ee = — mgt Meeks way — - oe ne > Ry 3a —~— ee . = — 2 * — >a -_ = ee ee - = a pt e . Sn Wi <= eas = : Tn a SS — ~ —— = or === = 3 E Oe eee +-s - -* ee —% ~ ; ow = a —. - 7 —— —_ = - — _ — ——< ———— a i, pene =i eS soa gata . = = = “2 Se eS eS —SS — + — -=F we - = == —— “== oe ee nes Pe ane a a ee > Sa ee Se ES SS SSS SS Se ee eS ee oe eS 2 ~ - nee . a ~ on e +e . —"- - on - = > — “ ce — Ld - wes =—— —— - - = —— = ~ ‘= = _ ——— = = = -< — — = = _~— ———_ > _ x ~~ + z = =~ es — ————— a - = ee eae — ¢, onfir KCtIONe Dem :nfira- S20% » . ; we’ *. SOLA T he thirtemtbBooker 0 being alkdefiribed tone and the felpefume civele. od le ©4,) QV ppofe that ABC D E beacircle. And inthecircle ABC D E defcribe ( by Lo \ithe 11 .0f thefourth) apentagon fieuresA BCD E:T hen 1 fay; that the fide of iS ‘the pentagon figure 1 BC DE containeth in power borh the fide of an hexagon pKa! frure, und of a décacon ficure, being defcribed in the circle ABC DE. Take ( by the 1. of the third ) thecentre of the circle, and let the famebe F . And drawing a right line from the point A tothe point Fxextend it tothe pointG. And drawea right line from the puint F to.the point B und from the point F drawe.( by the 12.0f the first): anto..the line ABU perpendicnlar lineP H 2 and extend it tothe point. K. And drawe the right lines A K and K B. And againe, fromthe point F draw ( by thé fatne) unto the line A K,a perpendi- _ cular line FN and extend F N to the pomt M which dine let cut the line AB inthe point L . Andetraw a right line from the 240 Gs2lets, drudie point K to the point L . Now foraf- vr Pee" ip onze t much as the circumference ABCG B00. sy us equ4ll to the circumference AE DG, of which the tircumference-A BCis °*° equal to the circumference AED, — therfore the rest of the circumference, ~ namely, C G, is equallto the reft of the circumference namely,to DG . But ©. the circumfertwceeD is [ubtended of «2h: the fide of a pentagon : Whetcfore the -= %\ circumference C G is fubtended of the fide of a decagon figure . And foraf- much as the line BH 13 equall tothe -"\* baci SEY b wt SE Ses te: j dhs ey: ‘ ey { the line F Flt conanion tothemB0Ws: 5 00) 4+ ai biel . - On 4 ¢.26.0f the shird) . t PSS SSIES) * 60 CII OC OR Bigs oa f.3 22 2 double tothe circumference BK . But the circumference C D 1s double to the. circumference 32,0f thefirst) the angleremayuing A Soy ey aoe ana BLE. Where. he triangle A-B Fise tangle tothe triangle B F L..Wherefore (bythe g. of the fixt) proportionally as thevight, te isto the right line BF; fois the fame right line F B to 27126 ve oe ee eee | Pe New the ~ 4 eu — audi of Euchdes Elementes. Fol.4 03. the right line B L: wherefore that which is contained under thelines AB and BL is equall to the {quare of the line B F »by the 17 .of the fixt. Againe foral{much as the line A L is equall to the line L K U for by the laft of the fixt,the angle K F L is equall tothe angle A F L,which equall angles are contained under the lines F K,F Land F A,F L,¢y the line F K is equall to the line F A, and the line F L is common to them Loth . Wherefore, by the 4. of the firft, the line AL is equall tothe line L’K } and the line L N is common to them both, & maketh right angles at the point N, and (by the 3. of the third ) the bale A N_is equall tothe bale K N.Wherfore alfo the angle LK N is equall to the angle L A N.But the angle L AN is equall tothe angle K BL (by the’s.of the first) . Whirefore the angle L KN is equallto the angle K BL . And the angle KA Lis common toboth the triangles A KB and AK L. Wherefore the anele remaining A K Bis equall to the angle remayning A L K (5y the Corel- lary of the 3 2.0f the firft). Wherefore the triangle K BA is equianele tothe triangle KL A. Wherefore(by the z.of the fixt) proportionally as the right line B Ais to the right line A K; fois the fame right line K A to theright line A L. Wherefore that which is contained under the lines B A and A L is equall to the {quare of the line A K (by the 17.0f the fixt). And it is roued that that which is contained under the lines A Band B L, isequallto the {quare of the line B F. Wherefore that which is contained under A B cy B L together with that which is contained under B.A and AL (which by the 2.0f the [econd,is the [quare of the line B A) is equall to the {quares of the lines A F and AK’. But thelineB Ais the fide of the pentagon figure, and A F the lide of the hexagon figure (by the Corollary of the 15 .of the fourth), and AK thefide of the decacon fieure. Wherefore the fide of a pentagon figure, containeth in power both the fide of an hexagon fizure, and of a decagon figure, being defcrived allin one and the felfe [anze circle : which was required to be. demonfirated. ‘@ A Corollary added by Flu/fas. | cA perpendicular linefrom any angle drawen to the bafe of a pentagon, paffeth by the centre. “For we drawe a right line from the poynt A'te-the poynt C., and an otherfrom the poynt A to the point Dy¥ thofe right lines fhail be equall, by the 4,of the firlh sand. therefore in the trianglew\ C D the angles at the points C and Dare,by the s.of the firft,equall But theangles made at the point where the line A G cutteth the line C D, are by fuppofition right angles : wherefore, by the 26. of the firft,the line C D is by theline AG diuided into two pe repel 3.and itis alfo diuided perpendicularly: wherefore by the-¢orollary of the firft of the thirdin theline AG is the centre of the circle:and there- fore the line.A G pafleth by the centre. qUbe 1y.Theoreme. —..\. The i1:Propofition. If in a circle hauing a rational line.to his diameter—beinfcribed an equilae ter pentagon:the fide of the pentagonis an trrationall line , and ts of that kinde which is called alefSe line. °°" « line ConStruSs¥ion. ~ - : : al { : a # t H fil ' : is . ¥ We ', th r iv xy ‘ A if i | Se Se _ “a aw —-~ ery ee een Ve Ce ’ - Si tant . = = z —— . = - ny = i aoe — a = =— o x =a or aa “ wee Oe = = reat SE sn eee < " Se one = a ed 2 —— —— = oy es of ee ae - <= a Demonftrae B30R, T be thirtenth Booke line F Kis rational. And the line or {emidiameter B F is rational. Wherefore the whole line B K is rationall. And foralmuch as the circumference ACG is equallto the circumference A D G,0f which the circumference AB Cis equall to the circumference AE D , wherefore she refidue C G is equall tothe refidue GD. Now if we drawe a right line from the point A to the point D,tt ts manifeft that theangles ALC aud AL D are right angles. F or foraf- much as the circumference C Gis equall to the circumference G D , therefore ( by the laft of the fixth) the angle C AG is equalltotheangle D A G.And theline AC is equall to the line AD, for that the circumferences which they fubtend are equall,and the line AL is common to them both , therefore there areswo lines dC and A L equall totwo lines. A D and AL, andthe angleC AL is equall to the angle D A L.Wherefore (by the 4. of the firit) ihe bafe C Lis equall to the bafe L D , and the reit of theangles tothe reft of the angles , and the lineC Ds double to the line CL. And by the fame reafon may it be proued,that the an- gles at the point M are right angles,and that the line A C is double to the line C. M.Now for- afmuch astheangle ALC is equallto theangle AM F for that they are both right angles, and the angle L AC is common to both the triangles A LC and AM F: whereforethe an- gle remayning,namely,A C L,0s equalto the angle remayning A F M,by the corollary of the 32.0f tine firft.Wherefore the trzangle AC Lis equiangle tothe triangle AM F.Wherefore proportionally, by the 4.0f the fixthasthe line L Cus tothe line C A, fois the line M F tothe line F A. And in the {ame proportion alfo ure the doubles of the antecedents L Cand MF (by the 15.0f the fifth). Wherefore as the double of the line L C isto the line C A, fois the double of the line M F tothe line F A. But as the double of the line M-F isto the line F A, {0 is the line M+F to the halfe of the line F A,by the 15.0f the fifth,wherefore as the double of the line L Cis tothe line € Afois the line M F to the halfe of the line F A,by the 11.0f the fifth. And in the fame proportion by the 15.0f the fifth,arethe halues of the confequents namely,of C A and of the halueof the line AF Wherefore asthe double.of theline LC is to the halfe of the kine AC, fois the line M F to the fourth part of the line F A. But the double of the line LC 4s the line D C,and the halfe of the line C. Ais the line C M,as hath before bene proued, and she fourth part of the line F A is the line F K ( for the line F K is the fourth part of the line F H by conftruttion) Wherfore asthe line D.C is to the-line.C.M.fois the line MF tothe line F K Wherfore by. compofition( by the 18 :0f the fifth as both the lines DC andC M are tothe line CM, fois the whole lin@M K tothe tine F K. aa abbas Wherefore allo (by the 22. of the fixt).as the. - | {qaares of the lines DC and CM aretothe{quare of thelineC M, fois the {quare of theline MK to the {quare of the line F K.. And forafmuch as (by the 8 .of the thirtenth) a tine which is [ubten- ded under two fides of a pentagon figure , as is the line AC, being dinided by an extreame & meane proportion »the greater feement isequall tothe fide of the pentagon figure , that is, unto the line D C:and (by the 1. of the thirtenth ) the greater Segment having added unto it the halfe of the whole, is in power quintuple to.the fauare made of the halfe of thewhole : and the halfe of the . | _woole line AC isthe line CM . Wherefore the {quare that is made of the lines D.C and GM, thatis,of the greater [egment and of the halfe of the whole as of one line,is quintuple to the {quare.of the line C.M,that is , of the halfe of thewhole.But as the [quare made of the lines DC and CM , as of one line,is to the jue: thelineC.M , Je ts it proued , that the [quare of the line M K is to the {quare of the line F K. Wherefore the {auare of theline M- K isquintuple to the {quare of the line F K . But the{quare of the line K F is wie »as : we | ; =e | of Euchdes Elementes. Fol.402. hath before bene proued wherefore alfo the fquare of the line M K is rational, by the 9 diffi nition of the tenth, for the [quare of the line M K hath tothe {quareof. the line K F that proportion that number , hath to number , namely , that 5. bath to1.ana therefore the fayd {quares are commenfurable,by the 6 . of the tenth. Wherefore alfo the line M K is rationall. And forajmuch as the line B F is quadruple to the lineF K (for the femidiameter BF is equal tothe femidiameter F H), therfore the line B K is.quintuple to the lime F K Wherefore the {quare of he line B K is 2. times fo much as the {quare of the line K F,by the corollary of the 20.0f the fixt . But the {quare of the line M K ,is quintuple to the [quare of the F Kas ts proued.Wherfore the {quare of the line BK is quintuple to the fquare of the line K M .Wher- forey {qusre of the line B K hath not to y [quare of the line.K M,that proportio that a [quare umber nth to.a(quare number,by the corollary of the 2$.of the eight. Wherefore (by the 9. of the tenth ) the line B K ts incommenfurablein length to the line K M , and either of the kines is raionall . Wherefore the lines B K and K M are rationall commen{urable 1n power onely.Butif fro a rationall line be taken away a rationall line being commenfurable im power onely to the whole, that which remayneth is irrationall,andis (by the 73. of the tenth) cal- led a refiduall line. Wherefore the line M Bisa refiduall line. And the line conueniently t0y- ned vntoit, is the line M K. Now 1 fay that the line B Mis a fourth refiduall line . V nto the exce(fe of the {quare of the line BK aboue the [quare of the line K M, let the {quare of the line N beequall( which exceffe how to finde out,1s taught in the af[umpt put after the 13 pro- pofition of the tenth ) . Wherefore the line BK is in power more then the line K M by the fquare ofthe line N. And forafmuch as the line K F is commenfurable in length to the line F B, for tt is the fourth part thereof , therefore (by the 16.0f the tenth ) the whole line K B is comimenjurable in length to theline F B But the line F Bis commen{urable in length tothe line B H namely, the [emidiameter tothe diameter : wherefore the line B K is commen{ura- ble in length to the line B H,by the 12. of the tenth . And fora{much as the [quare of the tine BK is.geintuple to the fquare of. thelineK M,, therefore the {quare of the line B K hath to the [quare of the line K M that proportion that fime hath to one .Wherefore by conuerfion of proportion (by the corollary of the 19.0f the fifth) the {quare of B K hath tothe {quare of the line N,that proportion that fine hath to fower:c therfore it hath not that proportio that a fquare oumber hath toa [quare number sby the corollary of the 25.0f the eight . Wherfore, the line 8 K isincommen|urableinlength to the line N (by the\9 .of the tenth) Wherfore the line B Kis in power. more then the line KM, by the [quare of aline incommen{urablein length torhe line BK . Now then forafmuch as the whole line B K is in power more then the line conueniently ioyned,namely,then K.M, by the (quare of a line incomenfurable in length to the like BK sand the whole line BK is commenfurable in length to the rationall line genen B H:theveforethe line M Bis a fourth refiduall line,by the diffinition of a fourth refiduall line.Buta rectangle parallelogramme contayned under arationall line anda fourth refidval line,is ivvationall, and the line which contayneth in. power the fame parallelogramme zs alfo ivrationill,and is called aleffe line (by the 94.0f the tenth ) . But the line_A B contayneth im power the parallelogramme contayned onder the lines H Band BM (for if we drawea right line from the point A to the point the triangle AB H fhall be like to the triangle ABM, by the 8.of the fixth.F or from the right angle B AH ts drawen.to the bafe B H aperpendicu- lar line. And therefore as the line B Histo the line B A; foisthe line AB to the line BM. this folbmpeth al{o of the corollary of the fayd 8.of the fixth. Wherefore the line A B which is the fide of the pentagon figure, is an wrationall line of that kindewhich is ealled a leffe line. If therejorein a circle hauing a rationall line to his diameter be infcribed an equilater pen- tagon,toe fide of the pentagons an ivrationall line,and is of that kinde which is called ale(fe line: which was required to be demonstrated. T he 12.Propofition. q Ihe 12. F heoreme. If in a circle be de{cribed an equilater triangle: the fquare made of the fide of | MMm.1. the 4 oat — RE RRR * nine i 3 y = ae ee = ’ %.. % Sea ~ fe? ~% eee, ee ie , rh 4 eee en aa™ wT ' P T he thirtenth Booke the triangle is trebleto the quare made of the line, whichis drawenfrom the centre of the circle to the circumference. LENG V ppofe that AB C bea circle , and init defcribe an equilater triangle ABC. aN <9 T hen fay that the [quare made of the fide of thetrianele ABC is treble to the ONE |quare made of the line drawen from the center of the circleA BC to the circum- Confiruttion. ference.T ake( by the 1 .of the third)the centre of the circle,and let the fame be D. Aud draw Demon ra- $407. aright line from the point A tothe poynt D,and extend it to the point E.And draw a right line from the point B to the poynt E.Now fora{much as the triangle AB Cis equilater, therefore eche of thefe three circumferences AB, | A C,¢y BEC is the third part of the whole circum- ference of the circle A B C:wherefore the circumfe- rence BE is the fixth part of the circuference of the circle Ufor the circumferece of the [emicircle A BE 4s equallto the circumference of the femicircle A- C E, from which taking away equal circumferences A Band AC, the circumference remayning BE Jhalbe equal to.the cirenmference remayning EC): wherefore the right line B E ts the fide of an equi- later hexagon figure decribed in the circle.Where- foreit is equall tothe line drawen from the centre of the circle tothe sircumference , that is vato the Line D E(by the corollary of the 15.0f the jixth) . Aud forafmuch as the line A E is double to the line D E, therefore ae quare of the line A Eis quadruple to the [quare of the line DE (by the 4.of the [econd):that is,to the [quare of the line B E.But the {quare of the line AE as equall to the f quares of the lines A Band B E ( by the 47.0f the firft) for the angle AB E és(by the 31.0f the third)a right angle Wherfore the [quares of the line AB Gy BE are qua- druple tothe [quare of the line B E . Wherefore taking away the {quare of the line B E,the {quare of the line A B fhalbe trebleto the [quare of BE : but the line B Eis equall tothe line D E.Wherefore the {quare of the line A B is treble ‘to the [quare of the line D E.Wherefore the {quaremade of the lide of the triangle,is treble tothe {quare made of the line drawen fro the centre of the circle to the circumference :which was required to be proued. | x A Corollary added by Campane. °° Hereby it is manifeft, thatthe line B C which is the fide of the equilater trianglé,diujdeth the femi- diameter D E into two equall parts.’ For letrhe pe of the diuifion be E. And fuppofe a line to be drawen,from the poynt D to the B,and an otherfrom the poynt D to the poynt C . Now it is manifeft (by the 4.of the firlt)thar the line B Fis equall ro the line F C,and therefore ( by the 3 . of the third all the anelds at the poynt F are right angles. Wherefore(by the 47. of the firft) the fquare of the line B D is equallto the {quates of the lines B Rand FD, and by-the fame the fquare of the line B E is equallto the iquares of the lines B Fja nd EF Essbut the line B Dis equalltotheline BE (ashath before bene proued). Wherefore by the common fentence the twoo {quares of the two lines BF and FD are equalito the two {quares ofthe lines BF, and F E . Wherefore taking away the fquare of the line B- F which is cémen to them both :the'refidue,namely,the fquare.of the line D F fhalbe equall to the re- fidue namely, to the {quare of the line F E. Whertorealfo the line F Dis equal to the line F E. Wher- fore hereby.itis manifeit that aperpendicular line drawen from the centre of acircle to the fide ofan equilater triangle infcribed in it,is equall to halfe of the line drawen from the centre of the fame circle, to the circumference thereof. : : x A Corollary added by Fluffas. The fide of an equilater triangle is in power fefquitertia to the perpendicular line which is drawé from one of the angles to the oppofite fide.For of what parts the line A B contayneth in power 12.0f & >» 2 ‘S of Euchdes Flementesax < Fol.4.03. fuch parts thé line BF whichis the halfe of A B contayned in power 3. Wherefore the refid ue,namely, thé perpendicularhine A F contayneth in power of fuch parts 9.(for'the {quares ofthe lines AF;and BE are‘by they7.of the firitequall to thefquare of the line A By Now t2.to 9.15 fefquitertia: wherfore the powenofirhé line A.B isto the powerof thé line A F insfefquitertia proportion. _-Moreouer the fide.of the triangle is the meane proportional! betwene the diameter and the per- pendicular line: For(by the Corollaty of the s.ofthe fixth )theline AF is to'che line AB asthe line A B isto the in@ AP. i OW awa ous ¢ VBA YO. wi wee _ Farther the\perpendiculanline.drawen fram the angle diuideth the bafe into two equal parts ‘and paflertt by the,ceuter, For if there fhouldbe drawen any other right line fré. the point Ato the poynt F, thé chat which ts drawen by the point D,two fight lines fhould inclade'a fuperhcies, which is'impoffi- ble. Wherefore the contrary followeth mamély, that cheJine,which being drawen from thé angle pai- {eth by. the centensisa perpendicular line to\thé bafe(by.the 3,of the third). | The 1.Probleme. LT he13.Propofition, « Fomakea* Pyramisyandto comprehend it'imaphere genen:and toproue that the diameter of the [phere is in power fe/quialtera to.the fide of the yd V ppole that the diameter of the (phere genen be AB and dinide A B in the poynt C, é! Jothat the line A C be double to the line B C(by the 9. of the fixth) . And upon iris the line A B,making the center the point N »defcribe a femicircle AD B.And(by the 11.0f the firft-) fromthe point C rayfe up vyatothe line AB a perpendicular lineC D. And drawe aright line from the-point D to the point A. And de[cribe a circle E F G hauing This orollary sa Pheri » POP > ftom of theza beske after Campane, This C orollary 4s the 3 .Coroliee ry afier the 17.4 propof:tron of the 14. beoke af= fer Cc Am pane * By the name of a Pyramis both here €§ in this booke fol- lowing Gndere fland a Tetrae hedron. Firft part of the confiruction, his femidiameter equall to the line CD.And de{cribe in the circleE F G,an equilater triate.. . gle EF G( by the 2.of the fourth). And by the 1.0f the third) take the cétre of the circle,and let the fame be the point H. And draw\thefe.°*\"* right linesPH,AF cp H-GsAnd¢bythe 12. °° of the eleuenth )fro the point H rayfe up unto : the playne fuperficies of the circleE F Gaper-°. pedicular line HK cy let the line HK be equal to the right line AC. And draw thefewight *~ * lines KE,KF,¢7 K G.Now forafmuch asthe A MG B line HK is erected perpedicularly to the plaine Ae | Juperficies of the circle E FG, therfore(by the 2. definition of the elenenth ) it maketh right -— stn angles with all the right lines that touch it, , and which are in the felfe fame fuperficies of the circle E F G. But it toucheth enery one of thefe right linesyH E,H F and HG: Where- fore the right line H K is evetted perpendicu- larly to euery one of thefelinesH E,H F, and HG. And forafmuchas the line. A Cis equal tothe lineH K ,-and the lineC D tothe line ~~ H E,and they comprehende right angles there. fore Perdew ‘A is equalliathe bale RK E(by an the 4.of-t Ps nd by the [ame reafon ech “rs of the lines K F, and KG isequall to the line ~* D A.Wherefore thethreelines K E,K F and set KG are equall the one tothe other. And ‘foralmuch asthe line AC is double to the linec B, (by conflruction therefore the line AB is treble to the line BC: but as the line.A Bis to the 3 MM gy. line D fd » Firft pare of the dsmonfiratien, , = - — — — ~— = _ a FSP a ae Si Fey WEE = = SS ——— ~ _ = — es = =z eee eee ee = a ee = == it men nd — —_ — = 2. = = a ee aa ae = T he thirtenth Booke This ffump? os line B. Cfo is the {quare of the line A D tothe [quare of the line D ‘m8 (Whiche may thus be siugaiteat., proedlt is manifeft(by the Corollary of the 8.of fixth) that the line C D isthe meane pro- the endofshes, portional betwene the lines AC and C B.Wherfore (by the corollary of the20. of thefame) pres > the fqnare of the line A Cis to the {quare of: the line CD , as theline AC is tothelinec B: € provofi= 5, ysis | sion prewed. ,, wherefore by compofition (by the 18 .of the fiueth ) the {quares of the lines AC and CD are slew Ds tothe fquare of thelineC D, ‘as the line A B is to the'line BC. But the fquares of the 3 Tines AC and D are (by the 47 of. the firft equall to the {quare of the line A D: wherefore ’ (by the guof the fiueth the fquare of the line A D isto the [quare of the line D.C , as the line ;, AB ts tothe line B C.1 Wherefore the (quare of the line AD is treble to the {quare of the line D Cx And forafmuch as the line K Eis equall tothe line A D ( as it hath ben proued) and the line H E is put equallto theline G D: thereforethe {quare of the line K Eis triple to the {quare of the line H E.But unto the [quare of the fame line H E is( by the 12. of the thir- senth) the fquare of the line F E treble ‘wherefore the line E.F 1s equall to the line KE, Now the lines K.E,K'F and K G are equall the one to the other,as it hath before ben proued., and foals are the lines E F, F GanaG E, for that they are the fides of an equilater triangle. Wherefore enery one of thefelines E F, F G,andG E,is equall to euery one of the lines K E, K-F and K.G.Whercfore thefe fower triangles E F.G,K E F , K F Gand KG Eareequila- ter Wherefore there is made a Pyramis confifting of fower equall and equilater triangles, whofe bale is the triangle E F G and toppe the eae dia 5 poynt K. } ae Now it is required to. comprehende the [ame Pyramis in the phere Geue , and to proue. that the diameter of the [phere is in power fef- Second pert of quialtera to the fide of the pyramis. Adde vn- the demiftratis. tothe right line K a rightline directly, na- mely H Lsand let the line H K beequall to the *teokeatthe the lineBG.* Now forthat as theline AC endof thisde- is tothe/line CDi, fois theline CD tothe es pecens "line CB(by the corollary of the 8-of the fixth). firuéion and OU the line AC is equal to the line KH the. demonftration lineC DtothelineH E, cy thelineC Bto the of this fecond ine HL.T herfore as the line KH is to the line | part after Fluf- | : : Rae 5 E, fo is the line H E to the line H L.Where- ._ ore that which is contayned under thelines . HK and H Lis equalltothe {quare of theline EH.Andether of the angles KHE,@ EH L is avightangle, wherefore a femicircle defcri- bed vpon the line KL fhall paffe by the poynt * Readethetwo E.* For if we draiwv a right line from the point i ae gg E to the poynt L , the angle LEK fhalbe a ie Pia gt jer Vight angle,for that the triagleE LK is equt- atthe end ofthe angletoeither of thetrianglssELHandE-. demonftraton HK (by the 8.of the fixth.) Now thenif the _ Be ae ee air a — the diameter K L abiding fixed,the femicircle be turned round about, vntilit returne Us. ter Gnderflan- tothe elfe [ame place from whenfe it began to be moued, it hall alfo palfe by the pointes F ding of shisrea- and G.F or drawing a right line from the ponit F ta the point L,and an other from the poynt aie L tothe point G,which alfomaketh at the points F and G right angles , the pyramis fhall be » ww © BD Second part of she Confirudio. o™ WSs In e = — J . tease Te ae a ogit aaigamnle OM, > =a S non — - - = ~_ on == of Euchdes Elementes. Fol.404.. {In the femicircle A D B of the former figure drawe theliné DN. And diuide the linéK L) ihto ewo equall partsin.the point M.And draw aline from Mo G And fotafmuch as by \conitructionthe <4* other con- line K H is equall to:the line A.C,and the line HL to the line C B: therefore the whole line A B is equal firuttion and to the whole line K L: Wherefore'alfo the halfe of the line KL, namely, the line L M;is equal tothe fe- demonfiratson midiameterB N: wherefore takingaway from thofe equal lines,equall parts B CandLH, the refidues of the fecond N C,andM H, fhalbe équall,. Whereforein the two triangles MH G and.N C D,thetWo fidesabout part after Fluf- the equall right angles D C N and GH M,nam ely the fides,D.C,C N,and GH »HM,, are equall, wher- fas. fore the'bafes M G and N Dare equall (by the 4.f the firft.) And by the fame reafon may it be proued that right lines drawen'from thé poynt M to the points E and Fare equal to the line N D. Buttheright line N D is equall to the line A N,which is drawen from the centre to the circumference + wherefore the line M G is equall to the line M K,& alfo to the lines M E,M F and M L. Wherfore making the cétre the poyntM,and the fpace M K or MG defcribe afemicircleK, GL :.and the diameter K L abiding fixed let the fayd femicircle KG, L'bemoued rounde about vneillit returne to the fame place from whence it began to be moued:and there fhalbe defcribed a {phere about the centre M ( by the 12.diffi- nition of the eleuenth) touching euery one.of theangles.of the Pyramis which are at the points K,E,F, G: for thofe angles are equally diftat from the centre of the {phere,namely, by the femidiameter of the fayd fphere,as hath before bene proued. Wherefore in the {phere geuen whofe diameter is the line K- Lyorthe line AB,isinfcribed a Tetrahedron EF GK-} Now I (ay,that the diameter of the [phere is in power fe{quialtera to the fide of the Py- pind nars of ramis I or forafmuch as the line A Cis double tothe line CB( by coftructio)therforethe line rhe demonfira- A Bis treble tothe line B C.Wherfore by conuerfion by the corollary of the 19 .of the fiueth) *** the line A B is{efquialtera tothe line AC . But as the line B Ais to the line AC, fois the {quare of the line B A,to the (quare of the line A D.F or if we draw a right line fro the point B,to the point D,as the lineB D isto the line AD, fois the fame. D tothe line A C,by realo of the likenes of the triangles D AB,¢ D AC(by the 8.of the fixth):¢ by reafon alfothat as the firftisto the third [ois the {quare of the fof to the {quare of the fecod(by the corollary of the 20.0f the fixth) Wherfore the {quare of the line B.A;is fefquialter to the {quare of the line A D.But the line B Ais equal to the diameter of the {phere geut namely ,to the line KL, as hath bene proued,c> the line A D is equal to the fide of the pyramis Sid in the {phere. Wherfore y diameter of the {phere isin power fe{quialter to the fide of the pyramis:}¥herfore there ismade'a Heyyy comprchended in a [phere geuen ; antl the diameter of the fpberess [elquialtera to she fide of the pyramis; which was required to be done and proucd... «An other demonftration to proue that as theline A Bis to theline BC, fo is the {quare of the line A D to the {quare of the line DC, Let the de{cription of the femicircle A D B be as in the firft de{cription.And upon the line AC defcribe (by the 46.of the first).a{quare E Cyand make perfecte the parallelograme F B. Now fora{muchas the triangle D A B is equiangle to the triangle D AC (by the 32.0f the fixt therfore as the line B A istatheline persica A D, foisthe line D Ato theline A C, by the 4..of the fixt. Wherefore that whichis contai- ned under the lines BA and AC, is equall to the [quare of the line A D, by the 17. of the fixt.And for that as the line A B 1s to the line BC, fois the parallelogramme E B to the pa- rallelogramme F B, by the 1. of the fixt: and \A the parallelogramme E B is that which is con- tained under the lines B A and AC ( for the line E Aisequalltothe line AC): and the parallelogramme B F is that which is contat- ned under thelines AC and BC. Wherefore asthe line AB isto theline BC, fois that which is contained under the lines B A and .\p ~ i " {hen 4 ) ¥ , Wi it ie P| er I ’ ; wis { . : 54 a tt rh | eae at hi }) th ' ii! , | ” ah ne : ee i wee \4 i? ‘) ii? 4 ne iy pe py al 4 \e > , Ae 5 Bt Te peice ii a. le i t 4 oh pe) LF \ x ah qa gt ' E te Re Ry Mk a Ae : i 4 a +» Square of the line C.D for the perpendicular line D Cis the OLAS The thirtenthR ooke + u wh, tathatwhich is contained under the lines::AG.andCB . But that-which is contai- » wedondey the lines BA and AC;3s equall to the {quare ofthe line AD, bythe Corollary of the saphena thas which itanthined vader the lnied 416 and CB, is equall the ¢ seane proportionall betwene the fe pentes of the bafe, namely, ACund CB s by the former Corollary of the 8 of the fixt , for that theancle A D Bis aright angle Wherefore as the line AB is to\theline BC, fois the Square of thehue.A:D to the fquare of the line DC, by the 11.0f the fifts. which was requi- ved to beproucde 0 v2 oh: eb a ‘on on 2. Jt wo Aflumptes added by. Campane. “a- Suppofe that vpon theline A B be ere€ted perpendicularly thedine D C; which line DC lethe the . *eageproportiosall berwene the partes ofthe line AB, namely, AC &CB = fotharas thelineA Cis to thedine G D}dolet.the line CD beto theline CB. And vpon the line.A B defcribeafemicircle, Thé i fay; that the circumference ofthat femicircle hall paffe by the point D,which is theend of the perpé- dicular line. But fot, chen itthalleirher cut the y HSS aySioa dine, CD, or it thall pafleaboue it, and includeit oei\ gail we nottouchingit , Firitlet it cutiitin the point B. > And drawethefe tightlines EBand B.A. Wher- fore by the 31.0f the third, the whole angle\A\E* Bisaright angle... Wherefore by, the firtt part of the Corollary of theg.of thefixt; the-line A Cis to'theline EC as the tine E Cis tothe line'C B. Bui oy the'siof the fift;theproportion of the line A Spasteslnck ©; is-greater, then, the praporti- : ‘ai se on of thedamedine’A-C to the line CD (for the _ ee th ss hal eae line CE s leffe then the line © D).-Now for thar. A “i Svs Cc 8B theliné CE isto uit line C B, as theline AC is ne Sty forhaling B,and the line CW iste the line C B,'as theline A C.is.to the line C D,, therefore by the 13.05 th® ft, thre proportion @f the line EC to'the line C B, 1s greater then the proportion of the line C D to the line CB i Wherefote by the to.of the fift, the fine E Cis greater rhen the line D C,namely, the part greater then the whole : which is impoifible . Wherefore the circumference fhall not cut the line C DiNow lay, chat it thdll not pallé aboue the line C.D, and.not.touch it inthe point D 4 Fer if ie be poffible, let itpafle abone it,and extend the line C D to the circaference, and let it cut it in the point F. And draw thé'finés FB and £'A >and ie {hall followe’as before chat the line CD is greater then the line CF: be oe isimpoffible . Wherefore that is manifeft which was required to be proued. > nie ki .% F ¥ . o & . is % + -* 6 See vy #e 9 4% a { [ie & i« a ’ a : ors St ts 4 iF there bea right angle unto which a bigs ss Jubieniledsant if wreaet the fame:be deferibed afte emicirele : the corcumference thereof faall paffe by the\poine of theright angle. © es, For fippofe that there-be a right angle A B C, vitowhieh fubtend thebafe A Cand vpon theline AC defcribe'a femicircle . Then I fay, that the circuniferenee thereof fhall paffe by.the point B. For if not, tien 1¢ fhall paffe either abotte. the point B,or vnder the-point B « Firftlet it pafle vnder the poine B, and lot the circumference be A E C-. And (iby the i2.of the firft) from the point B “drawe vite the line A.C a perpendicular line B D,which fet cut the’ \\ su SG att De circumference of the femicircle in the pointE.And_, cake drawé'thefe right lines E A and EC: Nowitis ma- ge nifeit, by the 31.0fthe third, thatthe angle AEC . : Easing is aright angle . But (by the 21.0f the firft) thean--8.) *\ ay Aan gle AEC isgreater then the angle ABC: which PI POa~ is impofhible, by the to.common fentence . Where- fore the circumferéce of the femicircle paffeth not // vnder the angle B , Now I fay, thar it paffeth not a=)» “7 //" boue the angle B . For ifit be poffible, let it pafle a —— —— ae ST -- ~——s r 3 a = — 5 = = s ee - = 2 ED nae ad - ts eee 2 =5 = eS ee Pn I a 7 - ies sae 3 = < , Sad a [Sus St eee os FS eins a> 9 = a S a T he thirtenth Booke ~ Ake the diameter of the former {phere geuen, which let betheline AB: and WE 2 << diuide it (by the 10. of the firft into two equall partes in the point C. And de- SS | ate) feribe vpon the line AB a femicircle A D B.And(by the 11 of the firft ) fro Dy REA the point C rayfe up unto the line A Ba perpendicular lineC D .And draw a “ we right line from the point D tothe point B. And defcribea [quare EF GH haning enery one of his fides equall to the line B D. on : PD * For the 4. an- gles at the post rollary of the I5.0f the firft: and thofe 4.8 gte, are.equall the one to the o- ther by rhe8.of thefirft:and therefore ech 4s rh, : saht pea the [quare of the line EH . Wherefore alfothe line Second part of the demiftratia. fora{much astheline AC is equall to theline BC,therefore theline A Bis double to the line B C,by the diffinition of a circle. Butasthe line A B isto the line BC, fois the fquare of the ee ee of Euchdes Elementes. Fol.406. hint AB tothe{quare of thelise BD.by the corollaries of the 8.and 20.0f the fit. Where- fore the {quareof the line A.Bis dowbleio the {quare of the lineB D ..And it is proued that the {quare of the line L M is duuble to the (quareoftheline L E.Whereforethe{quare of the line B D is equall to the {quari of the line LE» For thelineE H whichisequall tothe line L Fis put to be equall to the Ine D B. Wherefore the.{quare of the line.A Bisequall to the Square of the line L M.Whergjore the line A Bis equall.totheline LM, And the line AB is the diameter of the [phere geut,wherefore the line L\Mis equallto the diameter.of the Sphere geuen Wherefore the octocdros is contayned in the [phere geuen: * and it is alfoproued that the diameter of the {phere zs ixpower doubletothe fide of the ottohedron.Wherefore there is made an octohedron,and it is wmprehended in the {phere genen wherein-was comprehended the Pyramis:and it is proued that the diameter of the [phere isin power double to thefide of the ottohedrn: which was required to be doone,and to be proued. Certayne Corollaries added by Flu//as. sp Firft Corollary. T he fide of aPyramis isin ower fefquitertia tothe fide of an oitehedron infivibed in the fame Sphere. 3 For forafmch as the diamete:isin power double to the fide of the o€tohedron , thereforé of what partes the diameter contayneth inpower 6.0¢ the fame, the fide of the of&tohedron cotayneth in power 3. but of what partes the diamete: contayneth 6.ofthefame, thefide of the pytamis contayneth 4. by the 13.0fthisbooke . Whereforeof what partes the fidewof the pyramis contayneth 4. of the fame the fide of the octohedron contaynetl 3. | The go smon bales of thefe 2yramids are fet vpon ewer) fquare'contayned of the fidesiof the oc- * For the [quare of the line AB, which 1s proued equal to the [qeare of the bine L Mss dota ble ta théfauare of the live B D; which ts alfo ea guall to the [q#are of the line LE, This Corohary #8 the 16. propee tohedron, ypon whith {quare are-etthe 8: triangles of the otohedron’: which pyramidsareby thes, /#tom of the 14, diffinition ofthe eleuenth equall'aid like. And the forefayd fquare common to thofe Pyramids, is the thalfe of thefquare ofthe diameteiof the fphere,for itis the {quare of the fide of the oohedron. eseatys 2 swe Third Corollary. | The three a snes of the ofohedron; do cutte the onethe orber perpendicularly $70 two equall parts,in. the center of the {phere Which contayneth the fayd oftehedren. LEN? o : As itis manifest by the threediameters E.G, FH;andLM which cutte the one the other in the center K equally and perpendiculaly. | g The 3.Pribleme. L’he-15. Propofition, Toimake a Jolide caled a cube,and to comprehend it in the {phere genen, namely,that Sphere wherein: he former two folides were comprehended: and fo pr oue that the diameter of the {phere isin power treble tothe fide of the cube 4 Ake the diameter of ihe {phere cenen, namely, AB and diuide it in the point C . So that let the line A Che double to the line b C by the 9. of the fixt.And vpon the fi NNw.4, line beoke after ¢. ampane 5 ——— _ = — os eh Se a ee = - soa A Ts = se eee nen = ste _— SSS oe i : — = a = =r “ite. YE el PPAR | re ib 8 Firft part of the demonftreattor. Second part of she Confirudio.- Second part of the demonftra- ESOR « % By ther. Af- fumpt of the 13. ofits booke. Third part of the demonftra- $508. LOSS Ke the bat G F foal] be'ertéted perpendicularly to the line K F fy the 2. ‘cunena wenth. ded by the fame reafan agayne a femicircle deferibed upon the line G-K fhall pafvealfo T he thirtenth Booke line AB deleribe a (emicircle ADB: And by the rr.of the firsh) from the poyanC ray fer unto the in AB Se salle lineC D.And a ech we And dejeribe i EF GH; haning enery one of bis fides equalt v | Vb to theliwe D B, And from the pointes E, F,G, Hvayfe-vp by the 12. of the eleuenth ) unte the playne fuperficies of the fyuare EF GH perpendicular lines E K,F LGM, und HN: and let euery one of thelines EKyFL,GM, and HN beput equallto one of the lines E F, F G,G 5 or HE, which arethe fides of the fquare; and draw thefe right linesK L,LM, M N,andN K.Wherfore thereis madea cube namely F N_which is contayned under fix e- guall {quarcs . Now it is required to compre- hend the fame cube in the [phere genen, and to proue that y diameter of the [phere isin power ble to the fide of the cube. Draw thefe right dines X Gand EG “And forafmuch as the an- gle K EGisa right angle, for that the line K- E isereéted perpendicularly to the playne fu- perficies EG,aud therefore alfo,to.the right - line E.G; bythe 2: diffinitia ofthe elewenth, wherefore a fersicircle defiribed wpon the line K Gfhall*paffe by the poynt E. Agayne for- afmuch astheline FG ss erected frp pai cum larly to either of thefe lines F L ana F E,by the diffinition of afquare, Or bythe 2 diffini- . D > PR kw Ae) F » ; Ma i’ tion of the elenenth, therefore the line F Gis erected erpendiculart to the le - funerfict , C2é PoKibythegof the eleuenthWhereforesf we draw aright line froin the nt Je Fel os iffinition of the ele- by the point F. And likewife Jhall it pafce by the reft of the pointes of the angles of that cube. If now the diameter K G abiding fixed thefemicircle beturned round det hh it ead into the felfe fame place from whence tt began firft to be moued, the cube falbe compreheded . i ee eee =. 2 ieee of Euchdes Elementes. Fol.4.07. x. An other demonftration after Fluffas. Suppofe that the diameter of the Sphere geuen in the former Propofitions, be i let the center be the point C, vpon which defcribe a femicircle A DB . And SE domce: AR cut of a third part B G, by the 9.0f thesfixe. And from the point G raife vp vnto the line AB a perpendi- cular line D G, by the 11. of the firft. And draw thefé right lines D A,D C,and DB. And ynto the righe line DB putan equallrightline'Z I: | & and vpon the line Z1 defcribea {quare : E ZIT .And fré the pointes E,Z,1,T, ereéte vnto the fuperficies E Z1T per=* pendicular lines EK,Z H,I M, T N(by the t2. of the eleuenth):and put euery one of thofe perpendicular lines equall to theline ZI. And.drawe thefe right. lines KH, HM, MN, andN K, ech of which fhall be equall and parallels to the line ZI, and to the rett of the lines of the {quare,by the 33.0f the firft-And moreouer they fhall containe equall angles(by the 10,0f the eleuenth): and therefore the angles are right angles, fot that E ZIT is afquare: wherfore thereft of the bafes fhall be fquares. Wherfore the folide EZITKHMN being cétained vnder 6.equall {quares, isa cube; by the 21. definition ofthe eleventh. Extend by the oppofite fides K Eand MI of the cube,a plaine KE I- M : andagaine by the other oppofite fides NT and HZ, extend an other plaine H ZT N.Now forafmuch as ech of thefe plaines deuide the folideinto two equall partes, namely, into two Prifmesequalland like ( by the 8. defi- nition of the eleuenth); therfore thofe plaines fhall cut :the, cube by thecen- tre, by the Corollary of the 39. of the eleventh. Wherefore the cémon {etti- on of thofe plaines fhall paffe by the centre. Ler that common fection be the line LF. And forafmuch as the fides HIN and K M of the fuperficieces KE- IM and HZTWN do-diuide the one “= other into ewe pe: partes, by yay the Corollary of the 34. ofthe firlt, and fo likewife do the fides 2 TandEY : feftion L Fis drawen by thefe feétions, and diuideth the plaines KEIM fe H 8 an — ‘aean partes, bythe firlt of the fixe : for their bafes are equall, and the altitude is one and the fame; namely. the altitude of the cube . Wherefore theline LF thall diuide into ewo. equall partes: the diameters of his plaines, namely, the right linesK1, EM, Z N,and NT, which are the diameters of the cube.W her fore thofe diameters fhall concurre and cut one the other in oneand the felfe fame poynt, let the fame be O. Whe'fore the tight lines OK, OF; O 1,OM,OH,'OZ,;O T, and O N, fhall be equall the one to the other, for'that they are the\halfes of the diameters of equall and like rectangle parallelogrames. Whereforemaking the centre the point O, and the {pace any of thefe lines OE, or O K.&c. a Sphere defcribed, thall paffe by euery one of the angles of the cube,namely, which are at the pointes E,Z,1,T K\H,M,N,by the 12.definition of the eleuenth,for that all the liries drawen from the point O to the ini gles ofthe.cube are equall . Bue the * e line E Lcontaineth in power the two equall rightlines E Z, and ZL, by the 47.0f the firft,. Wheretore the fquare of the line E Lis double ee the fquare jof the line Z1. And forafmuch as the right line KT fubtendeth the right angle K EI (for that the right line K Eis erected perpendicularly to the plaine fuperficies of the right lines E Z and Z T(by the4.of the eleuéth)s therefore the.fquare ofthe line K 1 is equall to. the fquares of the lines B Land E K,)but the {quare of the line E Lis double to the {quare of the line E K (for itis double to the fquare of the line Z I,as hath bene proued, and the bales of the cube are equall {quares ) . Wherefore the {quare of the line KI is triple to the {quare of the litte K E, that is, to the fquare of the line Z1. But the right line ZI is equall to the right line D B,by ee ynto whofe {quare the {quare ef the diameter A B is triple, by that which was demonttrated in the 13.Propofition of this booke. Wherefore the diameters K 1 & D Bare equall, Wherefore there is defcribed a cube K I,and itis comprehended in the Sphere geuen wherin the other NNa.jj. folides = ee en EE re ~-—— il yt — => rie SSS - —- aos a es = 3 = ca — ——S TSS = j — - — — ~ — 4 == = == eS eee 2 SS Ss ge i NS ae = =z = fs etree PORE = i a . T hethirtenth Booke folides were contained, the diameter of which Sphere is the line AB. And the diameter Kor AB of the fame Sphere, is proued to be in power triple to the fide of the cube,namely, to the line D B, or ZI. Se qj, Corollaryesadded by Flufias. : 3 Firft Corollary. Hereby it is manifeft, that the diameter of a Sphere containeth in power the fides bath of a pyra~ esis and of a cube infcribed init. For the power of the fide of the pyramis is two thirdes of the power of the dianseter ( by the 13.0f this booke ) . And the power of the fide of the cube is , by this Propofition, one third of the power of the fayd diameter. Wherefore the diameter of the Sphere contayneth in power the fides of the pyra- mis and of the cube.. @| Second Corollary. All the diameters of acube cut the one the other into two equall partes in the centre of the fphere which containeth the cube . And moreouer thofe diameters doin the felfe fame point cut into twoe- quall partes the right lines Which ioyne together the centres of the oppofite bales. As itis manifeft to fee by the right line L O F. For the angles L K O,and F I O,are equall, by the 29. of the firft : and it is proued, that they are contained ynder equall- fides : Wherefore ( by the 4.of the’ firft) the bafes L O and F O are equall . In like fort may be proued,that the reft of the right lines which loyne together the centres of the oppofite bafes do cut the one the other into two equall partes in the centre O. q Ihe 4:Probleme. Ihe 16. Propofition. Lo make an Icofahedron and to comprehend it in the Sphere genen where in were contained the former folides, and to proue that the fide of the Icoe fabedron is an irrationall line of that kinde which is called a lefse line. 1 wy) Ake the diameter of the Sphere,namely, theline AB: and denide it inthe oN point C, fo that let the ine AC be quadruple tothe lineC B, by the g. of the 4) fixt.And defcribe vpo the line A B a femtcircle.A D B.And(by the r1.of the eNKS first) from the point C raife ~ untothe line AB a perpendicular line C D. € point D tothe point B. And defribe a circle Firft part of the confiruction. tS ~ __- ¥ er oa eg _— ~ yy oe _— ee —_ ree te ‘ . . : — - _— = _— rae Se oem = whe 2 "ED | ’ i: ; ‘a i # ha \ i) { thy ul ie ae : i if rth ‘ i] : f et hid +i : uJ au i 4 Vig 3 i | he ot eu. a vara Hie f 1 ea / {4 | 1 " fy ' { ay : | : Mn, it : Ht) } Ni 1 + a iy Hy 1 ee t ay ani , if - ; iio : ' 1) ’ | eae \ 1 : } leet hy { i ¢ 4 ‘ a. iit t - \‘g uP Fe : Hf ti me by Hey Poo - “ i 1 ui | H " ul j uy i " mn | i *) \ } a : i .| tt i 4 , Ta! 1) : | i | mo : } f " : Ne : ‘y aah | an) Aue ¢ : ni vik ty 0G SM iMate Pa fi + | hea «4 . hare , ie re | Peg | Poe \) as ' { f ) ae nt ane i : t A} 5 mt He «) +h LI , F |) ae wt ee nt ie % Ve HP d Ts , fee |) iif : }) ie ry ¥} ri i { : ee - | aa } i Did Y yi - - an a” + if ii: §y te i, hi ‘ 7 t i - 4 ’ hh : ah rh ~p om fl a re ae . - —_ en! = — oy mF And draw aright line from t EF GH-K whofe line fromthe centre (which let be the point Z ) tothe circumference, let beequall tothe line D B. Andinthe circle EF GH XK defcribe (by the 11.0f the fourth) an equilater and equiangle Pentagon figure EF GH K’. And deuide the circumferences E F, FG,GH,H Kyand K E, into twoequall partesin the pointes L,M,N,X,0. Draw alfo thefe right lines EM,MN,;NX,X 0,andO L: and moreouer thefe lines O E,E L,LF,F M, : MG,GN,NH,HX,XK, and K 0, and they {hall be the fides of an equilater decagon in- Seribed inthe circle EF GH XK, by the 29.0f the third. Wherefore the figure LL MN XO 4 s amequilater pentagon ,by the 29.0f the third,and the right line E 0 is the fideof a decagon 4 or tem angled figure. Raife up (by the 12.0f the elenenth) from the pointes E,F,G,H,K ,and the centre Z,vntothe plaine {uperficies, of the circle,perpendicular lines EP,F R,GS, HT, | KV, and ZW, and let ech of them be putequall tothe line drawen from the centre of the circle E F GH K, tothe circumference, namely, tothe line Z E . Wherefore right lines : drawen fromW toP, from W toV, fromW toT, fromW to, from W to R, fhall be equall and parallels to right lincs drawen from Z to E, from Z to K, from Z toH, from Z to G, and from Z to F, by the 6.and 7 .of the elenenth, and 33. of the firf? . Wherefore the plaine fuperficteces EF GH K, and P RST V, which are extended by thofe parallel lines, are pa- | . rallef . ; ¢ Z [ = 2 eae : > : a: mets ————— et peng ere or eae ee 3 Sa eee = fainter, Se meat Boe ~4 - ee ee % on a | 4 “ ee | 2 a Se = es ae ie a8 ee, ES — + r “ . “ se : et! an» Ki PO pS See . Pere = ; mt ~ = toate tid 2 - - = - a Es Ree = - - - — “ - woe nt . * . ie — 3 : z ER aes = 4 = 4 it x < - =" _< , ~ a Ss — aS: PS eS a ee SE EE HET ; oe Ces i corn ty sy iect sate pied 2 : of Euclides Elementon< Fol.4.08. rallel (uperficieces, by the 13.0f the eleuenth. Wherefore making the centre the point Wand the fpaceW Por WV, defcribe acircle, and it {hall pafseyby the pointes TS ;R, and fhall be equal to the circle EF GH K «For the femidiameters of echeare equall. And drawe thefe right lines P R,RS,ST,T VV. Pj and they {hall make a pentagon, whofe fides fhall be e- quallto the fides of the Pentagon OL M N X,by the 29.0f the first «For ech of them doth fubtend two fides of the decagonsor / —. | S.04 the fift part of equall circles. From... U3 ty totaly the upper pointes P;R,S,;T Vydrawd eh. thefe lines PO,P L,RL,RM,SM, SN,TN, TX, VX,V 0: which fhall [ubtend right angles cotained under the fides of the decagon EL - F (MGNHX KO, andthe per- pendicular lines PE , RF ,SG, TH, VK. a. ce th a a -— es r . a = ‘Now forafmuch as the perpendicular lines P E, RF,SG,T-H,and VK, are pute. 7#h ar tindag quallto the line Z E drawen fromthe centre, therefore they are equall to the fide of an equi- later hexagon infcribed in the fame circle (by the Corollary of the 15. of the fourth). Where- fore the right lines PO, P-L;V-0,andV X( whith {ubtend the right angles contained vn- der thofe perpendicular lines and the fides of the decagon) containe them in power,by the 47. of the firft. Bat the fide of a pentagon (namely, the fide LO or'PV ) containeth in power the fides of anhexagonand of adecazonin{cribedin one and the felfe fame circle, by the ro. of a , NNY.ty. thts \ ” ee ee , P . —_ = J eee SS ee ee = % rene on ae ae aw - ~ eS ews a = - ~ = a) ee 7 7 ee to = ee ae alee ae a = : mad ee ~ : - - =~ _ — = — —— x ———— -- a ’ dé ~~ 4 i Re he p= ES ss i ore ; Sa he a . E se = —— Pm i — ae ‘4S 3898 « T he thirtenth Booke this. booke “Wherefore the [ubtending lines P 0,.P L3V 0;V X,TX,T NS N,S MRM, RL, containe in power the {elfe fame {quare thatthe fidesof the pentagon O'L.M NX con- taine, or that the fides of the pentagon P RST V containe and therefore thofe fubtending lines are equal to the fides of theforefaid pentagons..Wheréfore the triangles contained of thofefubtending lines and of the fides of the pentagons, and which are ten in number ,name- ly,PLO,OV P,V OX,VXT,TXN,T NS, S N(MSCUR, ROUL, awd RLP, are equilater.Againe produce the right line ZW on eitherfideto the points 9 cy Y ‘and un- to the fide of the decagon, namely ,to the line O E,put the hnesZ and WQ equal: And for- afmuch astheright line 9 TY is ere- | i ) éted perpendicularly to the plaine [u- perficies OL MN X, therefore itis alfo erected perpendicularly tothe o- ther plaine {uperficies P RST V,by the Corallay of the.14.0f the cleutth. And drawe thefe right lines 9 P, QV,2T,25,& QR: andthefe lines alfoY L,YM,YN,7TX, andT Q. Second part of the coftraction. +. “ <-¢ |... ay Twa é ia “* ou + BAL mg Bras ota VBeraee : , ~ See Pe, Me WS Wa IS: Shi Ss SST ST oa Now forafniuch as the lines: P32V5,9T, 2S, and QR do eche fubtend right an- Second part of gles contayned undershe fides.ofan equtlater hexagon Cr of anequilater decagon in{cribed she demoftratt0» 59 the circle P RS ZK or inthecircle EF GH K(whichtwo circles are equall) therfore the fayd lines are coheeg ~p- ? 34 : Lf =. ual to the fide of the pentagon in{cribed in the fore/ayd circle by the 4 : a ’ - ae — . a . : eee an tiie ce, a a ae _ —_ “ . asta oe en of Euchides Elementes., ~ Fol.4.09. of this booke,and aveequallthe one to the ather,by the g.oftheparst, (forall the angles at the poynt W which they [ubtend ave right angles) hey ew the fie triangles OP ¥ , OP R, DRS; 2S TF And QT KV swhicharecontayned wader the fayd lines QV, QP,2 RO §; QT andvoder the fides of the penatagonk P RS Tyareequilater, and equal tothe tex for- | mer triangles: And by the fame reafow the fruc triangles-oppofite unto them; namely; the trt- il angles TIME TMN,I NX,TXO ana T OL are equilaterand equalto the faid ten twiangles. For thelines¥ L,Y MY N3Y XyandV O,dofubtend right angles cotayned vider the fides of an equilater hexagon and of an equilater decago infcribed inthe circle EE.GH K,which us equall to thecircleP RST. Vs. Wherefore there is. defcribed.a folide tontayned under 20.equilater triangles. Whereforedy the lait diff section of the elenenth there es defcribed ain Pf cofabeds On. Now it is required to comprehend iin the (phere geuemynd to prose that the fide.of the Icofahedron 1 an irrational line of that kinde whichiscalled aleffe line: Forafs much as the line Z Wis the fide of an hexagon,cy the line W Q isthe fide of a deeagon, therfore theline Z 9 is dinided by anextreme and meane proportion in the point W, and his greater fegmet is ZW by the 9.0f the thirtéth ). Wherfore ds the line QZ wto the line ZW,fo is the line Z- W tothe line W Q.But the Z W is equall to the line ZL by conftruction, and the lineW 9, tothe line ZX. by conftruction alfo:Wherefore as the line QZ is tothe line Z L,fo isthe line ZLtotheline Z.1,and the angles QZ Land L ZY arerightangles(by the 2. diffinition of the eleuenth):1f therfore we draw 4 right line from the poynt L to the poynt.Q the angle 1LQ |halbe aright angle,by reafo ofthe likenes of the trianglesYL Qand ZL (bythe 3:of the fixth). Wherfare a femitircle defcribed v'po the line 27 hal paffe alfo by the point L (by the affumpts added by Campaneafter the 13.0f this booke). Awd by the ame reafo al. fo,for that asthe line 2 Z isthe line ZW,fo is the line ZW tothe lneW 9,” but the line forthe Line Q- Z Fis equalito the line Y Wyand theline. ZW to the ine PR W>wherefore'as the line YW Ws equal ro gs tothe line WP ,fow the line PWiethe'line W 9D And therefore avayne if we draw a right * Ps = T239 line fromthe poynt P to the point Y the angleY PQ fhalbe aright angle.Wherfore a femi- see . — circle deferibes ypon the line 2 T fhil balf allo by the point P,by the former alfumpts: cy if both. the diameter 21 abiding fixed the jeunicivele be turnediround about, unt:lit cometothe Ths Pert 44 felfe fame place from whencett becanfirft vo be mowed it {hail paffe both by Ho cop P and gerne ajith alfo by the reff of the pointes of the angles af the Icofahedronm,.and the Icofahedron fhalbe rated by Fluf- comprehended in a fphere.t fay alfothat was contaynedin the phere geuen. fas “Distide(by the r0.0f their ft )the line Z W tm to two equall parts in the point a.And fors almuch asthe right line Z 24 dinided by an extreme and peane proportion.in the point W,and-his leffe bom is 2Wethenefore the feement QW hauing added vuto tt. thehalfe of the greater ‘fegment, namely, the line W ais (bythe 3 of this booke in power quintuple to the {[quare ery A the halfe of the greater feement - wherefore the {quare of the line 2 aw quintuple to the fquang of the linea W But. vnto the fauare of the Q a,the [quare of the line © ¥ is quadruple (by the corollary of the 20. of the fixth )for the line 9 ¥ is double to the: line 2-a:and by the fame veafon unto the (quare of theW A the [quare of the lineZWis quadruple - Wherefore the {quare of theline QY # quintupletothe fquare of the line ZW (by thers .of the fineth). Aud forafmuch as the line A Cis quadruple to the line € B , there. fore the line A B is quintuple to the line CB. But astheline AB wtotheline BC, fo ws the {quare of the line A B to the {quare of the line B D(by the 8 of the fixth, and corollary of the 20.0f the fame).Wherfore the [quare of the line A B us quintuple to the fquare of the line B= D.And it is is proued that the {quareof the line QY ws quintuple to the {quare of the line zW,And theline BD is equall to the line ZW,for eather of them 1 by pofition equall tothe linewhich is drawen from thecentre of the circle E F GH tothe circumference.Where- foretheline AB is equallto theY 2. But the line ABzs the diameter of the [phere geuen: Whercforethe line? 2 whichis prowed to be the diameter of the {phere comayning 1 i co- fahedren *. right linestrom the poynt,a,tothe poyntes P.and'G. after this maner. A T herhirtenth Booke fabedron,isequall to the diameter of the {phere gene Wherefore the Icofahedrowis contays ned inthe phere geut.Now I fay thatthe fideof the tcofabedron isan ivrationall line of that kinde which is called a leffe line. F or forafmuch as the diameter of the [phere is rational, and isimpower quintupleto the fquare of the linedrawen fro the centre of thetircleo LM NX: wherefore x 3 the line wirich & drawen from the-centre of the circleO L MN X is rational: wherefore the diameter alfa being comenfurable tothe fame line ( by the 6.of the tenth) ts ra- tionall.Butif in a.circlebauing a rationall line tovhisdiamacter be de{cribed an equilater. pentagon, the jideof the pentagon iby the 11.0f this booke)anirratianall line,of that kinde which iscalled aleffe line. But the fide of the pentagon OL M NX is alfo the fide of the Ico- Sehedron defcribed,as bath before ben proved. Wherfore the fide of the 1 cofabedro tan irra- sionall line of that kinde which is called a leffe line . Wherefore thercis defcribed awtcofahes. dron.and itis: contayned inthe phere geuen, and itis proued that the fideof the! Itofa- hedron wan irrationall lineof that kind which is called aleffeline. Which was required to be dont,and to be proved. \.\ 0 ha ts stots, ) atReS sacl A Corollary: Hereby it s manifeft thatthe diameter of the fphere,is in power quintuple to the line which is dra'wen from the centre of the circle to the circuithferenite on which the Icofabedron 1s decribed. And that the diameter of the phere +18 coms pofed of the [ide of an bex 420n aud of t wo fides of adecagon, defcribed in one BA beey CYEY ANGE soos wiv \o.41 shi son wasted pd habane regan sia 83 Tae aN e) a % - 32 Ni 5 34) shh | ' Vic rays wis 5x4 2h . ; maa 5" % 4 ‘3 1B PAS Fluffas,prou eth the Ieofahedron deferibed,to.be'cétayned in a {phere} by drawing ' mes | VS bebe : Lb S ay OMA: ” »” EAS 3 * 4 u ‘3 : pA se | 8 VN 7 4%e4 _ Forafinuch as the lines Z W,W P are put equal ro-the line drawen from the centre to the circum - ' ference,and the line drawen from the centre’ to thé’ circumference is double to'the lirie a Weby con= firuGtion ; thercforethe line W Pils 2lfo double to the fame lined Wi Wherefore thefquareofthe line > W Pisiquadtupletothe{quare ofth¢line'a W (by the corollary of the29.0f the fixth). And thofe lines P W and W acontayninga right angle P. W a (as hath befote bene proued)are fubtended of the'right- line a P- Wherefore (by the 47 ‘of the firft ) the line’a P concayneth in power thelines PW’; and Wa: *““< "Wherefore the rightline a P isin’ papheie eos to the line W, a. Wherefore the rightlines a P, and C a.Q being quintuple to. one and the famedine W a,are(by the's.of the fiuéth )equall . In like forte alfo may We prouc that vnto thofe lines a P anda Q ,are equall the reft of the lines drawen from the poynt ato the reit ofthe angles R,S,T,V..For they fubtend right angles contayned of the line W a,and of the lines drawen from the centre to the! circumference » Arid forafminch'as ynto thé line Wa is equall the ding V.a,which is likewife erected perpendicularly yntothe other plaine fuperficies O\L.M N X:there- fore lines drawen from the point ato the angles O, L,M,N,X,and fubtending right ancles at the point Z contayried vnder lines drawen fr6 the centre to the citcumference,and vnder the line'a Z,are equal not onely'the'one to the other, buralfo to the lines;drawen fré the fayde poyata,to the former angles at the poyars, P,R,S,T,V,. Forthelines drawen fr6 the centre to the circumference of ech circle are equall, & the line a W is equall to the linea Z,But the. line a P is proued equal to the linea Q._, which is the halfeofthe whole Q:Y. Thereforé the refidue aY is equall to the forefaydlinésaP,aQ &c. Whetéfore making the centrethe poyntiajand the {pace one of thofe lines a Q,aP,&c.extende the {u- perficies of a {phere,& it fhal touch the 12:angles of th¢ Icofahedron,which are at thé pointes O;L,M. N,X,P5K,S,T,V,Q,¥: which {phere is defcribed,if vpé the diameter Q Y, be drawena femicircle,and the we fcinicircle be moued about, till it returne ynto the famé place from whenfe it began firft to be. moued. Ses PS SA TAS sd in gee ‘ et eu aot | \aatotwp ut BK seeti sb Ae Senne) Se eee “ ertan\soe sapPA Corollaryadded by Figfase The oppolite fides of an Icofahedron are ‘parallels.For the diameters of the {phere'do fall vpon.the oppofite angles ofthe Icofahedron ; as it'was manifeft by the right line Q_Y . If therefore there be ima- gined to be drawen the two diameters P N,and O M they fhall concurre in the point F: wherefore the night lines which ioyne them together,P'Vjand'L N,are in one and the felfe fame playrie fuperficies by cakes the of Euchdes Elementes: Fol.410. che 2.of ete elewenth ..And forafmuch as the alternate angles. at, the endes of the diameters are équall (by the 8.of the firft): for the triangles contayned vnder equall femidiameters and the fide of the Ico- fahedron axe equiangle:th:refore ( by the 28.of the firft)the lines P V and LN are paralles. q I he s.Probleme. T he 17.Propofition. T'o make a Dodecahedron , and to comprebend it in the [phere genen, wherin were comprehended the forefayd folides:and to prone that the fide of the dodecahedron is an irrationall line of that kind whichis called are fiduall line. = a) ¥ Ake two playne fuperficieces or bafes of the forefayde cube, which let be-the two aN R > fquares ABCD and CBEF, cutting the one the other in the line.B G. pers M4 (Lb pendicularly according to the nature of acube.. And (by the 10.0f she firft-). di- uide enery one-of the lines AB,BC,C D, D A, EF,EB, and F C into two.equallpartes in the poyntes G, H, K, L, M, N,X And drawe thefe right lines GK and H L,cutting the one the other in the point P and likewife draw the right lines M:Hand N X cutting the one the other inthe poynt O. And dinide enery one of thefe right linesNO,OX, HP,andLP, by an/extreme and meane proportion in the | points R,S,T ,ct,and let their greater fegments be RO,OS,T P,andP 4. And ( byther20f » the eleuenth) fro the poynts R, 0,58, rayfevp to the outward part of the playne [uperfictes E- BCF of the forefayd cube, perpendicular lines RV 0° & SZ: andlet eche of thofe perpedi- cular lines be equall to one of thefe lines RO, OS orT P , which perpendiculars [halbe paral lels ( by the 6. of the elenenth) , ana Likewife from the pointes T ,P ,, rayfe-vp unto the out- ward part of the playne [uperfices ABC D of the fayd cube,thefe perpendicular lines I W, P. st;and & I,eche of which perpendicular lines utequall allo tothe line OS, orO.RorT P : andthe fayd perpendiculars foalbe parallels (by the forefayd 6. of theeleuenth) » And draw thefévight lines Y H ,HW,BIVWC,C Zand CB. Now. fay that the pentagon. figure V BWCZ isequilater and inoneand the felf fame plaine fuperficies, and moreoner ds equi- angle: Draw thefe-right lines T B,RB; ana SB. And forafmuch asthe right line NO dinided by an extreme and meane proportion # the pont Rand his greater fegment isthe line : ROstherefore the {quare of thelines N O and NR are treble tothe {quare of the lineRO (by the 4.of thisbooke). But the line ON equalltothe line NB , and the line O R to the line RV Wherefore the {quares of the lines B Nand RN are treble to the (quare of theline RY. Bat unto thefquares of the lines B Nand NR ts equall the (quare of the line BR (by the 47. of the firft) . Wherefore the [quare of the line BR is treble to the [quareof the line RV\ Whereforethefquares of the lines B Rand RV are quadrupleto thefquare of the line RV Bus vntothe {quares ofthe lines BRand RV isequall the [quare of the line | 000.1. BY First part of the construe... tt0%s ‘The pentagon VBW CX, proued equs- latere ad ip ess oss eee a en = SS ee \ nen a 4 a ao = = = | We en “= ~ BSS ag Es % ThelimeSR ss equaléS pa- rallel to the liwe ZV (by the 33. of the firft ) for Shey soyne toge- ther equall and parallel lines, aud are inthe fame furerficres with them,by the 7 .of ibe ele- weuth, The pentagon PEBWCR, proued to bein one and rhe felfe fame playne [u- perficses. “Sn Sd HIROT 7 y The thirtenth Booke BY (by the 7 of the firft) forthe angle BRV ts-avight angle (by the'z. diffinition of the: elenenth) Wherefore the Sauart of the line BV ws quadruple to the [quare of the line V R.’ Wherefore the line B V ts double to thethe line R V (by the Corollary of the 20.0f the fixth). And the line ZV is.al{o double to.the live RV (for™ that the line’S R w donble to the line o- R, that is,to the line RV which ws equall to the line OS) Wherfore the line BV is equall to the lineV Z...And fora[nzuch.as the twolinesB Ne. pi : and N Rare equal to the two lines B Hand I1- T namely, the wholes and the leffe (egmets, and they comprehend richt ancles , namely, of the {quares,B O,and B P, therefore (by the 4.of the firft) the bafes B Rand BT are cqualt. And for- afmuch\as thelines B R;and BT areequall,and.. the two lines RV, and T W arealfoby confiru-, t on eguall ,and the angles BRV and BT Woes sf are by fuppofitiomright angles:thereforeacaine ©.) (by the 2:0f the firft) the bales BV and-B W are equall: but the line BV , isproued equall to the lineV BW herforethe ling BUCis alfocequall.to the lineV Z. In like fort alfo may we proue that either of thefe lines WC,CZ is equal to the [ame lineV Z.Wherefore the pentagon figure BV Z- CW ts eguilater. Now Lfay that it isin one and the [elf [ame playne [uperficies . Fora{muchas the line Z Vis F a parallell to the line S R(as was before proued) 4 S\06 AY ee - ; but unto the [ame line S Ris theline CB.a oho oa cee parallell( by the 28.0f the firft). Wherfore (by. ens ina -G o> the 9.of the eleuenth ) thelineV Z is a parallell eee | ba 4337 Dis to the line C B.Wherefore,by the feueth of thee~ leuenth,the right lines which ioyne the together © 3 8. | are in the felfe [ame playne wherein are the parallel lines. Wherefore the T rapefium BY Z- C isin one playne.And the triangle B WC 1s iw one playne(by the 2.0f the eleuenith). Now te prose that the Trapefium BV ZC & the triangléBW Cure in one and the felf fame plaine, sve must proue that the right lines Y H, and H Weare wade directly one right line: which thing isthus prouede F ora{much as the line Pts deuided by anextreme and meane pros portion in the point T,and his greater fecmentisthe line eT, therefore as the line H Pista . theline P T,fois the line PT tothe lineT H.Buttheline HP is equallto the lineHO ,and the line P T to either of thefe lines T Wand 01 .Whereforeas the line H.0.istothe line Ow Y [01s thelineWT to theline T H.But the lines HOund TW being fides of like proportion are parallels(by the 6.0f the elewenth): | Foreithehof them is erected perpendicularly tothe plaine {uperficies B D)sand the lines T. H and OT are parallels, which are alfo fides of like proportion, by the {ame 6.of the eleuenth, ( For eitherof them is alfo-erected perpendicu- larly to the playne fuperficies B F.) But whenthereqretmotriangles , haning two fides pros portienallsotmo fides,fo fet.vpin oneanglesthattheir fides of like propertioare alfo parallels Aastha tangle YO. anhH TW are) whofe tivo fides; OH. HT; being ix the two ba- fes ofthe cube mnking.an angle at the point H,the fides remayning of thofe triangles thal by the 32v0ftheljixth) bein bnevight line Wher fore thé lines TH GH W make both ont right line ikut euery right lineis(by the ssof the elenenth)in one cy the {elf [ame plaine fuperficies, Wherefore if ye drama right line fromB toX , there fhalbe made a triangle BW YT, which foalbetn one and she felfe fame plaiye( by the2-0f the eleuenth) .._And therefore the:whole L209 pentagon of Euckiles Etementés,\© pentagon figureV BWC 2 is in one and the felfefameplayme {upérficies. Now alfo I fay that it is equiangle.F or fora{muchastheright line NO.is dinided bj\an extreame and meane proportion in the point Rydnd bis greater fegmentis OR, therefore as Fol.4.tt, The pentagon VBWCk, és both the lines N O and O Radded together 1sto,the lige OSN3 fo (by the swof this. beoke)ss: proued equee the line ON to the line. O R.But the line O R is equalltuthelineoS Viherefore as the line engles S Nis to the line RO fois thelineN O to thelineoSherforethe lineS Nisdiuided by anextremeand meane proportion inthe poinbO,andbis ercntir feament inthe timeNOs Wherefore the [quares of the lines N-S and S O\aretrebletothefquare of theline IN: O: (by the 4. of this booke) . But the line N O.1s equal tothe NB; and thesine S O-tothe. lise § Za wherfore the {quares of the lines NS and Z Sare treble tothe {quare of thedineM Biwhers fore the {quares of the lines ZS,S.N and N Byarequadripletathefquare of the tine N By But unto the {quares of the dines SN¢> NB (bythe x7 .0f the first )is equal the fquareofthe line S B: wherefore the [quares of the lines B Sand S Z;thatessthe fanare of the line B Zyby the 47.0f the firft,(for the angle ZS Bis aright aneleby pofitio ) is. quadruple tothe {quare of the line N B.Wherfore the line BZ 1s doubletothelineB. N(by the Corollary of the 20.0f the fixth) But the line B Cis alfo double to theline BN Wherefore theline BZ 1s cquall to the line B C.Now fora{much as thefe two linesBV and¥ Zare equall to thefe two lines B« Wand W C,and the bafe B Z is equattto the bafe BC, therefore (by the 8 .of the fir(t the an- ele BV Zis equall to the angle B W C. And imtske fort ( bythe 8 ofthe first.) may we prowe that the angleV ZC is eqnall to the angle BWC { prouing firft that thelines GC Band CV are equalgmhich are prouedequal by this,that the line N S 1s equal to the line X R,and ther~ fore the line CR is equal to the line B S,by thegz.of the firft:wherfore alfoby thefamey line CV is equal to the line BZ ,that is,to the line B C(for the lines BG Gy BZ are prouedequat.) Wherefore the three anglesBWC,BV Z,and¥ ZC areequall thé oneto the other. But if 1m an equilater pentagon figuie there be thre angles equall the oneto the other , the pentagon ts (by the 7 of the thirteth)equiangle.wherfore the pentagon BV Z CW is eyuiangle. And it is alfo proved that itis equilater. Wherforethe pentagon B ¥ Z CW is both equslater G e~ ‘guianele. And itis madevpor one of the fides of the cube, namely,vpon BC.” If therefore upon enery one of the twelue fides of the cube bevfed the like conftruction,there hal then be made a dodecahedron contayned undertwelue pentagons equilater and equiangle. Now itis required tocomprehend itinthe {phere genen,and to proue that the fide of the dodecahedron is un irvationall line of thatkinde which is called a refiduall line Extend the line 0,andletthe line extended be ¥' Qs now thenthe line YQ fhall light vppon thediameter of the cube;and {hall dinide the one the other intotwo equall parts . For thisis manifest to fe bythe39.of the eleuenth . ¢ Forif by the 1wolines NX and M H be drawer two playnes perpendicularly to the bafes; and cutting the cube , the common fection of thofe playnes fhalbe the lineY o-produced : for their common fectionis from the poyntO erected perpendicularly to the plaine E BC F bythe r9.of the eleuenth).Let them cut the one the o- ther inthe point Qewherefore Dis the centre of the phere which comprehendeth the cubes and YQ isthe halfediameter of the {phere by shat which wasdemiftrated in the 1$.0f this booke:. wherefore the right lines drawen from the centre Q to all the angles of thecube fhalbe equall. And draw aright line from the point V tothe point Q. Now foralmuch astheright line NS is divided by an extreme and medne proportion ix the point 0 and his greater feg- menvistheline N O as hath before ben proucd, therefore the(quares of the linesNS and S 0 are trebletothe [quare of the line N O,by the 4.0f this booke. But theline N Sisequal to thelineY Q (for the line NO is equal to the line OQ as hath before ben proued,¢y the line 10 tothe line OS) being both leffe fegmentes:but the line OS is equall tothe line V , for the line R O is equall thereunto: wherefore the {quares of the lines Q ¥ and V are treble to the [quare of the line NO.But unto the {quares of thelines QY @ TV the {quare of the line YO is bqsiall (by the 47 of the first): wherefore the {quare of the line¥ Qs trebleto the ; 000.4. {quare * Looke for @ farther con. Siruction after Fluffas at the ende of the de- monfiration. That the do. decahedre ig contayned in the Sphere LEHENs Tan abatt Z sit SSS ee == my r <= = 2s a5 EEE Sn ee a = = cna Ee — - —p~w a nl be > ew —— ] pak a bale = ahaa -- eS SS ee ———— = ar an caine saa ati See EO ice = if JS eee ae SS x = — SS Se en —— > SS SSS SSS = te > - - _ . ~ - _ “+ - Feo . - — = — 2 eter ise 2 a —— — ———— —— ---— - - ———— =4 ee =. —ee a aaah Be ee ee = Ss ee ose —---- met SS Se eae pe a , == = : Jekgoed =r — — ~~ - : oe te ae ee, Ae anomie may gee = oes es ~~ = - . rs Se gt as SS esr: es a = =? wa 8 on a 2 Se ee - a ete mee 4--, Tah eRe), eK, :. = 2a +o Brome a Fetal " 2. That the fide of the dodeca- bedron ts 4 vefeduall lines T he thirtenth Booke {quare of the line NO . But thefemidiameter,» of the {phere copreheding the faid cube ssin pa wer treble to the half of the fide of the cube. For we hane before(in the 15.0f this booke) taught how to make a cube ,and to comprehendeitina — fpbere,and hane proued that the diameter of the fphereis in power treble to\ the fide of the cube. Now in what proportio y whole is tothe whole, in the [ame is the balfe tothe halfe(by the 15.of the fifth) .But the line NO is the half of the fide of the cube. Wherefore thelineV- 9 is equall to the [emidiameter of the phere copreheding the cube.But the point 2 is the centre of the {phere coprehending the cube. Wherefore the point V, which is one of the angles of the dodecahedron, toucheth the {uperficies of the [phere geuen . In like fort allo may we proue , that enery one of the rest of the angles of the dodecahedrom roucheth the fuperficies of the phere. Wherefore the dodecabedronis comprehended in the (phere eucn. . Now lay, that the fide of the dodecahedron is an irrational line of that kinde which is called a’ refiduall line . For forafmuch as. the line N O. is diuided by an extreme and meane proportion in the point R, and his greater fegment is the line O R, and the line 0 X is alfo ds- wided by an extreme and meane proportion in the point S, and his greater fegment is the line o S.. Wherefore the whole line N X 1s diuided by an extreme and meane proportion, and his greater fegment is the line RS . ( F or for that asthe line O N isto the line O R, fois the line O Rto the line NR, andinthe {ame proportion alfo are their doubles ( for the partes of eque- multipliceshaue one and thefelfe(ame proportion with the whole, by the 15.of the fifth ) . Wherefore asthe line N X is to the line RS, foisthe line RS to both the lines NR andSX added together. But the line NX 1s greater then the line RS, by both the lines N R and S$ X added together. Wherefore the line NX ts diuided by an extreme and meane proportion,ana his greater fegment isthe line RS . But the line RS is equall tothe lineV Z, as hath before bene proued . Wherefore the line. N_X 1s diuided by an extreme and meane proportion, and his greater fegment isthe lineVZ . And forafmuch as the diameter of the Sphere is ratioe nall,and is in power treble to the fide of the cube, by the 13.0f this booke, therefore the line NX, being the fide of the cube, is rationall . But if arationall line be dinided by an extreme and meane proportion, either of the fegmentes is ( by the 6.0f this booke) an irrational line of that kinde which is called arefidwall line. Wherefore the line V Z being the fide of the do- decabedron,is an irrational line of that kinde which is called a refiduall ine Wherfore there ts made a dodecahedron,and it is coprehended in the Sphere genen, wherein the other folides were contained, andit is prowed that the fide of the dodecahedron is a refiduall line : which was required to be done, and alfoto be prowed. | od q Corollary. i Hereby it is manife/t, that the fide of a cube being dinided by an extreme and ‘ % _ - a iene > ———— sadlinesio, aaa vet camagseins SPs of Eenchdles Elementess Fol.4.12. and meane proportion ,the greater feoment thereof is the fide of the dodee cabedron . Asitwas manifett by theline V 2 which was proued to-be the greater fegment ofthe right line N X,namely, of the fide ofthe:cube: A further conftruction of the dodecahedron after Fluffas. Forafmuch as it hath bene proued thatthe pentagon BV ZC W is equilater and equiangle and toucheth one of the fides of thecube.Let vs fhow alfo by whatmeanes vpon eche of the 12 -fides of the cube may in like fort be applyed pentagons ioyning one to the other,and compofing the 12.bafes of the dodecahedron. Draw in the former figure thefe rightTines A I,I D,I L,# K.Now forafmuch as the line > L was in the point ¢& diuided like vnto the lines P H,O N.or O X,and vpon the pointes T,P,@,were erected perpendicular lines equall ynto the line O Y,and the reft: namely, vnto the greater fegmét:and the lines T W and &I were proued parallels , therefore the lines WI and T &are parallels, by the 7, of the eleuenth,and 33.0f the firft. Wherefore alfo, by the 9. of the elenenth;the lines W land DC are parallels. Wherefore by the 7,0f the eleuenth CW I Disa playne fuperficies.And the triangle AL Dis a playne {uperficies,by the .of the eleuenth.Now it is manifeft that the right lines ID, & I A are equall to the right line W C.For the night lines A L & U & (which are equall to the right lines B H,8¢ H T)do make the fubtéded lines A # and B T equall by the 4.0f the firft.And agayne forafmuch as the lines B T and T W contayne aright angle B T W, as alfo doo the right lines A @ and &1 contayne the right an- ele A &1( for the rightlines W T ; and1& are ereéted perpendicularly vnto one and the felfe fame playne AB C D by fuppofition) . And the fquares of the lines B T and T W are equall to the (quares of the lines A & , and &I(for it is proued that the line B T is equall to the line A @, and the line T’W to the line &1).And vnto the fquares of the lines B T and T W is equall the {quare of the line BW,by the 47 .of the firft: likewife by the fame vnto the {quares of the lines A Gand & Lis equall the fquare of the jine A I. Wherefore thefquare of the line B W is equall to the {quare of the line A Fplinresore alfo the line B T is equall to the line AI. And by the fame reafon-are the lines 1Dand WC equall to the fame lines. Now forafrauch as the lines A Land 1D, and the lines A L and L D are equall , and the bafe I L is common to them both,the angles A L Land DL I fhalbe equall by the 8.of the firft: and therefore they are right angles by the 10.diffinition of the firft.And by the fame reafon are the angles W HB, and W- H C right angles.And forafmuch as the twolines H T and T W are equall to the two lines L @ and @- I, and they contayne equall angles, thatis, right angles by fuppofition , therefore the angles W H-- T,aud I L &,are equall by the4.of the firft.. Wherefore the playne fuperficies A I D is in like fort incli- ned to the playne fuperficies AB C D,as the playne fuperficies B W C is inclined to the fame playne A- BC D,by the 4.diffinition of the eleuenth . In like fort may we proue that the playne W C D1 isin like fort inclined to the playne A B C D,as the playne B V Z Cis to the playne EB C F. For that in the trian- gles YO H and # PK which confift of equall fides ( eche to his correfpondent fide) the angles YH O, and ## K P,which are the angles of the inclination, are equall.And now if the right line # K be extended co the pointa, and the pentagon C W I Dabe made perfect, we may,by the fame reafon, proue that thar playne is equiangle and equilater , that we proued the pentagon B V Z C W to be equaliter and e- qguiangle.And likewife if the other playnesB-W-1 A and'A LD be made perfé@ , they may be proued to be equall and like pentagons and in like fort firuate,and Be are fes ypon thefe common right lines B- W,W C,WI,AI,andID,And obferuing this methode , there fhall vpon euery one of the 12.fides of the cube be fet euery one of the 12 .pentagons which compofe the dodecahedron. ay Certayne Corollaryes added by Flufias. 3. Firft Corollary. T he fide of a cube,is equall tothe right line which fubtendeth the angle of the pentagon of a dode- cahedron contayned im one and the felfe fame {phere with the cube. For the angles B WC and AID, are fubtended of the lines BC and AD. Which are fides of the Cube. @ Second Corollary. In a dodecahedron there are fixe fides enery two of Which are parallels and oppofite, whofe fetti- ons into two equall partes , are coupled by three right lines , which inthe center of the {phere which contayneth the dodecahedron ,deusde into two equall partes and perpendicularly both them felues and alfo the fides. OOo0.ij. For Draw in the former figure thefe lines, & 4,4 L, ti D, ae ae == oe = ee esse a — -—< ea ne = c- = <4 Ree n 1s . San rere = ie a ne: — ai Peet Boe ee ers a ee i 4 4 ) ab 0 5 yatey th ie ' | i ‘ 1 om iu \ ey j ; | ’ , frie | RRS | i ha i ian iu ) Oe ‘i Dae A PB dal} ie ie Bel i ' y ’ " i uy Ar | Ae iP ae dat i fl PP ey a iy yi My i vot iy. ‘tie i iy fia 4 my te Uy 7 | tm , 4 oA 2 as : a » a th Ef : ul a! P ih yi + oy ' “ bt i a 4 ee 1 Vad " i tae ee ah ) | - tht | it ae FST PPM ET IIE «OH z = Ss > a 3 we: ROSE ASE Mi = = ~ ee = a a ee es ee — ss. ~~ Ay = -« i} f t = lesen here ig — sted _— Fes = —— Pee 5 Nee at The fide of 4 pyramis, The fide of a cnbe, a The thirtenthBooke ows ~\Rdr vpn the fixabafes of the cube are fet; fixe fides of the. dodecahedron, asit hath ‘bene pro- ued (by the lines Z V,-W 1 &c. ) which are cutteinto two equall partes by tight lines, which ioyne réaether the centersof the bafesef the cube,as the line Y O produced; and the other like, Which lines coupling together the centers of the bafes are threé:itmnumber, cutting the one the other perpendicu- larly(for they are parallels to the fides of the cube ) and they cutte the one the other into two equall partes in the center ef the {phere which contayneth the cube ( by thatwhich was demontftrated in the 15.0f this booke )-And vatothefeequall lines toyning together the centers of the bafes of the cube,are without the bafes added equall partes O Y, P #, and the other like , which by fuppofition are equall to halfe of the fide of the dodecahedron). Wherefore the whole lines ;;which ioyne together the feGtoins ofthe oppofite fides.of the:dodegahedron , are equally arid they cut thofédides into two equall partes and perpendicularly. ) | TRL -. 5 Third Corollary, ou Aight ineioyningtogerberthe poynts of the feibions of the oppofite fides of the dodecahedron in- to.tied egucll partes beiig dinided by anextreame and meane proportion : the greater fegment there of paige fide of the éube ; andthe leffe fegiment the fide of the dodecahedron contayned in the felfe ame (phere. } .Forit was proved that theight line ¥.Q_ is diuided-by an extreame and meane proportion in the poynt.O, and thathisgrearer fegmerit O- Qus-halfe the fide of the cube ,and his leffe fegment O Y is halfeof the fide V Z{ whiehus.thefide of the. dodecahedron).Wherefore it .followeth (by the 15.0f the fifth). that their doublesare in,the fame proportion. Wherefore the double of the line Y Q which ioy- neth the poynt oppofite ynto the line Y,is the whole: and the greater fegment isthe double of the line O QO wwhichis-thefide ofthe cube: 8a the lefle fegment isthe double of the line YO, which is equall to thefide of the dodecahedron, namely,to'the fide V Z. a q dhe 6. Probleme. Lhe. 13. Propofition. "To finde ont the fides of the forefayd fine bodies, and to compare them -A sry te pephep.c beailates 9 We Seat | amen the the diameter of the Sphere geuen, and let thefamebe AB, and dinide it yes inthe point C, fo that let the line AC be equallto CB, by the 10: of the firft: CER ndin the point D fo that let A D be double to DB,by the 9.of th d oh din.the-point D,f ) soy the 9.0f the fixt.An Phy ENS upon the line'A B defcribeafemicircle.A EB. And fromthe pointes C and ELIS Fh vail ) ae: dee Sere Dsraifeup (by the r1.0f thefirft) wnto the line A B perpendicular lines C E ana DF. And draw thet yight lines AF, F.B;andBE . Now fora{much as the line A D is double tothe line D B , therefore theline A B ts treble to the line D B . Wherefore the line B A is [efquialter to the line A.D ( foritis as 3.,t0,2.).But astheline B Ais tothe line AD, fois the {quare of the line B A to the [quare of theline A F (by the 6.of thefixt,or by the Co- rollary of the fame, and by the Corollary of the20.0f the {ame ) : for the triangle _A F B ts equiangle to the triangle .A F D .Wherefore the {quare of the line BA is fe(quialter tothe fauareof-the line AF.,.But the diameter of afphere isin power fe{quialter to the fideof the pyrameis by the 13 .0f this booke and theline-A-Bis the diameter of the {phere . Wherefore the line A F is equall to the fide of the pyramis. = Yaguineforafmuch asthelineA Bis treble tothe line B D:but as thé line ABisto the line BD, foisthe {quare of the line AB tothe {quare of the line F B , by the Corollaries of the 8. and 20.0f the fixt. Wherefore the {quare of the line A Bis treble to the {quare of the line F B. But the diameter of a {phere is in power treble to the fide of the cube ( by the 15.of this booke) aud the diameter of the|phere is the line A B. Wherefore the line BF is the fide of the cube. . And forafmuch as the line A Cis equall to the line CB, therefore the line A B is double to the line CB:But as the line A B.is tothe lineC B, foisthe {quare of the line AB tothe [quare of the line B E (by the former Corollaries) . Therefore the {quare of the line A B is double ie 104 ito the of Enctides Elementes\ Fol.413. the fgiee of the'line BE . Butthe diameter of the phereisin power double to the fide of the octohedron; and A.B is the diameter of the (phere geuen: wherefore the line B E is the fide of the ottohedrou... Raife up (by the 11.0 the firft ) fromthe point. A unto the right line AB a perpendicular line AG. And put the line A G equall to the line AB .<:And drawe aright line from the point G to the point Cs And let the line GC cut the tircumference in the point H. And (by ther2.of thepirft ) fromthe point A drawe vito the line AB a perpendicular kine HK. New forafmuchas.the line. GA 1s double tothe line AC (for GA, is equall to AB ).. ButasG.Aisto AC, fous HK to KC ( by the 28.0f the first; and Corollary, of the 2.0f the fixt ): whereforethe line K is double to the ine \K C.Whereforethe{quare of the line H-K is quadruple tothe {quare of the line K C; bythe Corollary of the. 20. 0f the fixt. Wherefore the {quares of the linesH K and K C, which are all one with the [quare of the line H C, by the 47.0f the first, is quintuple to the fqauare of theline K C ..But the line HC ise guall tothe line CB, by the definition of a circle. Wherfore the{quare of the line BG is quin- tuple to the fquare of the line ge C.K. And fora{much as the line A Bis double to theline BC »of which the line AD is double to. the lineD B.Wherfore the refi- due,namely,B Dis doubleto the refidue namely, to D.C (bythe 19.0f the fift ) . VV herefore the \ line B C is treble to thelineC D. Wherefore the {quare of the line BC is nonecuple to the (quare of | the line C D, by the Corollary of the z0.of the fixt.But the {quare of BC is onely quintuple tothe fquare of CK . Wherefore the quare of CK is greater thé the fquare of C D, by the 10.0f the Sift. Wherefore alfathe line C K is greater then the line D C.¥n- | to the lineC K put (by the 2.0f the firft ) an equaillineC L «And from thepoint Lraifevp unto the line A Ba perpendicular line L M. And drawe aright line from the point M tothe point B . Now forafmuch as the {quare of the line C Bis quintuple tothe fquare of the line C Kjand the line AB is double tothe line BC; and the line K Lets double tothe line CK s therefore the [quare of the line A B is quintuple tothe {quare of theline K Ly bythe 15.0f the fift. But the diameter of a [phere isin power quintuple to the line which.45 drawen from the centre of the circle to the circumference on which the Icofahedron is de{cribed,by the Co- vallany. of the x6 of the fixt .Avadthe line AB isthe diameter of the fphere.> mebereforethe line:K Lis the femidiameter ofthe circle on which the Icofahedron is defcribed. Wherefore theline K Lis the fide of an hexagon. figure defcribed in the fame circle; by the Corollary of thexzs.of the fourth. And foralmuohas the diameter of the {phere is made of the fide an hex acon figure, and of two fides of a decagon being ech of them de{cribed in.ont and the felfe ame circle (by the Corollary of the 16 of this booke).: andthe line ABds the diameter of the ‘pbere, andthe line K Ls the fideof the bexagon,and the line AK is equall tothe line L Be wherefore cither of the lines AK and LB is the fide of adecagon defcrebed in the circle om which the Icofabedronis de{cribed( that is,in the circle whofe fermidiameter is the ine KL) And forafimuch d3theline L Bis the fide of a decagon,and the line M L of an hexagon (for M LisequallteK L, for thats isequall to KH by the 14.0f the third, for they — ‘ . iftant { { Wa A K S DL & The fide of ste otiohedron. « ‘ Thethirtenth Booke : diftant from the centre, and ech of thelines H K and K Lis double toK ©). Wherefore the line MB isthe fide of a pentagon,by the10.0f this booke » Butthe fide of the pentagon is alfo The fide of an sh, fide of the Icofahedron, by that which was demonfirated inthe 10 of this boake. Where- —* fore the line M B is the fide of anTcofabedron. \ The fides of And foraf{much as the line F Bistheyide of a cube, let itbe dinided by an extreme and f meane proportion i the point N, and let the greater fegment therof be NB. Wherefore the dodecahedron Teed nw the fide ofa Dodecabidron, by the soles ofthe 17. ae babii 8s vk Comparif And forafimuch as it hath bene proucdyby the 1 z.0f this booke, that the diameter of the the fuchider [phere is in power {e{quialter to AF thejide of the pyramis,and isin power doubleto BE the of the forefayd fide of the octohedron,by the 14.0f the fame, and isin power treble to F Bthe fide of the cube, bodies, by the 15.0f the fame .Wherefore tt followeth, that of what partes the diameter of the {phcvé containeth fixe, of fuch partes the fide of the pyramis tontaineth fower : and the fide of thé ectohedron three :.and the fideof the cube to -Wherefore the fide of the pyramis 13 in power. to the fide of the octohedron in fe(quitertia proportion : and isin power to the fide of the cube. in double proportion . And the fide of the octonedron is in power to the fide of the cubein fefs quialtera proportion . Wherefore the forefaid fides of the three figures, that is, of the pyrae mis,of the octohedron,and of the cubc,are the one to the other inrationall proportions: Wher- fore they are rationall. But the other two fides namely ,the fides of the Icofahedr on and of the Dodecahedron, are neither the one to the other ,nor alfo to the forefaid fides,in rationall proe portions : for they areirrationall lines, namely,a lefSe line, and arefiduall line. awit Thefide ofthe Butthat MB the fide ofthe ¢ at Icofahedron 1¢ofahedronis greater then N B proucd greater the fide of the Bodecahedron then the fide may thus be proued. Foralmuch -f the dodeca~ 45 the triangle F DB ts equian- casa gle tothe triangle FAB, bythe ~ 8.of the fixt, therefore proportt- onally, as the-line BD is tothe line B F, fois the line B F tothe ’ line BA. And fora{much as there are three right lines pro- portionall, therefore as the firft ésto the third fois the {quare - made of the first to the {quare nade of the fecond by the Corala~ lary of the 20. of the fixt. Where ~ fore as the line-D Bistotheline™ BA, foisthe [quare of the line DB tothe fquare of the line RTA SyNe BF .: Wherefore*( by conuerfion , by the Corollary of the.4. of the fineth) as the line AB is tothe line BD fo%s the ‘[quare of the line F B tothe fquare: of. the line BD. ‘But theline A B is treble tothe line BD, as hath before bene proned, Wherefore the {quare of theline F Bis treble to the (quare of the line B D . But the [quare of the line AD is qua- ie to the {quare of the line DB, by the Corollary of the 20. of the fixt; forthe line AD és double to theline D B.. Wherefore'the fquare of the line AD is greater then the fquare of the line F B; by the to of Sa it alfotheline AD is greater then the line F B.. Wherefore the line AL ts much greater then the line F B. Andtheline AL being dinided by an extreme Cy meane proportion, his ereater fegmentistheline K L,by the o.of this bokes ( for the line EK isthe fide of an hexagon,and the line KA is the fide of a decagoninfcri- bed in oneand theJame circle, as hath before bene proued ) : and the line FB being dinided ruil ? by an Ader. K. G ie Oe B. ~ ee ke ae of Euchides Elementes. Fol.4146 by an extreme and meane proportion, his greater fegmentis NB. Wherefore the line K Lis ——e greater then the line N B.(* For two lines dinided by an extreme and meane proportion,are ,, pee ais enery way like proportional ). But the line K L is equallto the line LM ..Wherefore the line proved in the LM is.greater then the line NB . But the line M1 B is greater then the line L 0 Far par ee eee i = Sm ag OT PR ea Tbe thirtenth Booke agayne it fhould be contayned under fower right angles. Wherefore much lefse can any folede angle bevontayned under more [quaresthen fower Vnder three equilater and equianele pen- tagons i contayned the folide angle of a dodecahedron. But under fower it is impof|ible. For forafmnch asthe angle of a pentagon is a right ancle- and the ‘fift part more of a right angle,the fower angles fhalbe greater then fower right angles: which is impoffible. And ther- fore much leffecan a folide anglebe’compofed of more pentagons then fower . Neither can a folide anglebe contayned under any other equilater and equiangle figures of many angles, forthat that alfo fhould be abfurd.F or the more the fides increafe , the greater are the angles which they contayne,and therfore the farther of are the (uperficiall angles contayned of thofe fides from compofing of a [elide angle . Wherefore befides the fore[ayd fine figures there can not be made any folide figure contayned under equall fides and equall angles : which was re- quired to be proucd, ) ; An Affumpt. But now that the anele of aw equilater and equianele pentavon is a right angle and 4 That rhe angle fifth part more of a right angle,may thus be proued. Suppofethat ABC D E be an equilater 0 pe equilater and equianele pentagon. And ( by the 14.0f the fourth )defcribe about it a circle ABC DE. Senate sek And take ( by the 1.0f the third) the center thereof, tight angle and andlet the (ame be F.. And draw thefe right lines A Peedi ts F ASF BSF C, F D,F E.Wherefore thofe lines do bs eae | ~ hint BD aif Giutie | the anoles of the pentagon intotwe equall , v4 “i before proved partesin the poyites A,B,C, D, £; by the f- of the | fot inthe corollary fir “And forafmuch asthe fiue angles that areat of the g2.ef the abe poynt F are equal to fower right angles , by the i frf. : : a? ; Pr), J corollary of the 15. of the first ; and they are equalt \ the one to the other by the 8:of the fir/t: therfore one \ \ of thofe ancles,as for examplefake,the angle A F B \\ \ isa fifth part lefse then a right angle.Wherfore the Aas ancles temayning namely, F AB,cy AB Fare one vight angle and a fifth part ower . But the angle F- A Bis equallto the angle F BC. Wherefore the whole angle ABC being one of the an- gles of the pentagon is aright angle and afifth part more then x right angle: which was re- quired to be proned. Seo 3 PASE TIRE | eam - ew « A Corollary added by Fluffas. Now, let vs teach bow thofe fine folides baue eche like inclinations of theyr bafes. “a Firftlet vs take a Pyramis, and diuide one of the fides thereof into two equail parts: and from the | two angles oppofite vnto tharfide,draw perpédiculars, which, fhall fall vpon the fe€tion,, by the co- The fides of the ‘ollary. of the 12.0f the thirtenth,and@e the fayd poynt of diuifion(as may eafily be proued). Wherfore anole of the in-a they {hal containe'the ange of the inclination of the plainés,by the 4.diffinition of the eleventh which clination of the atle isfubtended of the oppofite fideof the pyranzis:.: Now forafmuch as the reftof the angles of the fuperfeieces of i0eknation of the playnes of the Pyramis,are contayned vader two perpédicular lines of the triangles, the Ferrahedré andarefubtended of the fide of the Pyramis , it foloweth , by the 8. of the firit, thatthofe angles are areprowedrati- °Guall; Wherefoie(by the 5 diffinition of the eleuéth)the fuperficiéces are in like fort inclined the one Sit to the- other, ->- : : ; | acts) 5 Xs The fides of the One of the fides ofa Gube-being divided into two,equall parts,if from the fayd fection be drawen uncle ofthe in. atwe.ot the bafes thereof, two perpendicular lines,they fhalbe parallels and equall to the fides of the - clination of the {quare which cotayne'a rightangle, And forafinuch asall the angles of the bafes of the\Cube are ri he fuperfciecesof angles; therefore thofe perpendiculars falling ypon the fettion of the fide common tothe two bafesys she cxbe proved Thalbcontayne a night angle (bythe 10, of the eleuenth) t which {elfe angle is the angle: of inclination.» ag B (bythe 4.ditiinition of the eleuenth and is fubtended of the'diameter of the bafe of the'Cube. a,” i the — of Enclides Elementes. Fol.415. the fame reafon may we proue that the reft of the angles of the inclination of the bafes ofthe cube are right angles. Wherefore the inclinations of the fuperficieces of the cube the one to the other,are equal (by the’s.diffinition of the eleuenth), In an Odtohedron take the diameter which coupleth the two oppofite angles. And from thofe oppofite angles draw to one and the felfe fame fide of the OGtohedron, in two. bafes thereof,two per- pendicular lines, which fhall diuide that fide into two equall parts and perpendicularly (by the Corol- lary of the 12.0f the thirtenth). Wherefore thofe perpendiculars thall contayne the angle of the incli- nation of the bafes(by the 4.difinition of the eleuéth):and the fame angle is fubtended of the diame- ter ofthe Odtohedr6. Whertore the reit of the angles afrer the fame maner defcribed in the reft of the bafes,being comprehended and fubtended of equall fides, thall (by the 3.of the firft) be equall the one to the other.And therefore the inclinations of the playnes in the O@ohedron, fhal(by the 5.diffinition of the eleuenth)be equall. In an Icofahedron let there be drawen from the angles of two of the bafes , to one fide common to both the fayd bafes perpendiculars , which fhall contaynethe angle of the inclination of the bafes (by the4.diffinition of the eleuenth) : which angle is fubtended of the right line which fubtendeth the angle of the pentagon which contayneth fiue fides of the Icofahedron, by the 16. of this booke : for it coupleth the twoo oppofite angles of the triangles which are ioyned together . Wherefore the relbof the angles of the inclination of the bafes being after the {ame maner found out, they fhalbe contayned vnder equail fides,and {ubtended of equall bafes,and therefore (by the 8.of the firft) thofe angles fhal- be equall. Wherfore alfo al the inclinations of the bafes of the Icofahedron the one to the other fhaibe equall,by the 5 .difinition of the eleuenth. | In a Dodecahedron,from the two oppofite angles of two next pentagons draw to theyr common fide perpendicular lines,pafling by the centres of the fayd pentagons,which fhal,where they fal,diuide the fide into two equall parts by the 3 ofthe third.(For the bafes of a Dodecahedron are contayned in a circle) And the angle contayned ynder thofe perpendicular lines is the inclination of thofe bales (by the 4.diffnition of the eleuenth) And the forefayd oppofite angles are coupled by a right line equal to the right line which coupleth the oppofite fections into two equall parts of the fides of the dodecahe- dr6(by the 33.of the firft).For they.couple together the halfe fids of the dodecahedr6, which halfes are parallels and equall,by the 3.cerollary of the 17. of this booke: which coupling lines alfo are equall by the fame corollary. Wherefore the angles being contayned of equal perpendicular lines,and fubrended of equalhcoupling lines, thall(by the 8.of the firit)be equal.And they are the angles of the inclinations. Wherefore the bafes of the dodecahedron are in like fort inclined the one to the other ( by the s.diffi- nition of the eleuenth). bo 4 Flujfas after this teacheth howto know therationality orirrationality of the fides of the triangles, which contayne the angles of the inclinations of the fu perficieces of the forefayd bodies, "In a Pyramis the angle of the inclinatié is contayned vnder two perpédicular lines of the triangles, and is fubtended of the'fide of the Pyramis Now the fide of the pyramis is in power fefquitertia to the perpendicular line,by the corollary of the 12.0f this booke: and therfore the triangle cétained of thofe perpé dicular lines.and the fide of pyramis,hath his fides rational & commenfurable in power the one zo'the:other. Forafimuch as the twoo fides of a Cube (orrightlines equall to them ) fubtended ynder the dia- meter of one of the bafes;doo make the angle of the inclination : and the diameter of the cubeisin power fefquialter to the diameter of the bafe , which diameter of the bafeisin power double to the fide( by the 47 .of the firtt): therefore thofe lines are rationall and commenfurable in power, In an O¢tohedron, whofe two perpendiculars of the bafes contayne the angle of the inclination of the Odtohedron,whichangle alfo is fubtended of the diameter of the O&ohedron , the diameter isin owerdounble to the fide of the Octohedron,& the fide is in power fequitertia to the perpédiclar line, * the 12.0f this booke:wherfore the diameter thereof isin power duple fuperbipartiens tertias to the perpendicular line. Wherfore alfo the diameter.and the perpédicular line are rational] and commen{u- table (by the 6.of the tenth.) As touching an Jcofahedron,it was proned in the 16.0fthis booke’, that the fide thereof isa leffe line,when the diameter of the {phere isrationall. And forafmuch as.the angle of the inclination of the bales thereof , is contayned of the perpendicular lines of the triangles, and fubtended of the right line which fubtendeth the angle of the Pentagon which contayneth fiue fides of the Icofahedron : and yn- to the perpendicular lines the fide is commenfurable (namely , isin power fefquitertia vato them, by the Corollary of the 12.0f this booke) : therefore the perpendicular lines which contayne the angles are irrationall lines,namely,leffe lines( by the1z05.of the tenth booke.) And forafmuch as the diameter contayneth in power both the fide of the Icofahedron,and the line which fubtendeth the forefayd an- gle , iffrom the power of the diameter which is rationall , be taken away the power of the fide of the Icofahedron which is irrational , it is manifett that the refidue which is the power of the fubtending line fhalbe irrationall.For ifit fhoulde be rationall , the number which meafureth the whole power of the diameter,and the part taken away of the {ubtending line, fhould alfo, by the 4.common fentence of PPp.ij. the That the plate HES of az odte- hedron arein Lske fort snelix ned, Thatthe plate 7ES Of an leofa- hedrom are in leke fort incls- ned, That the plas- nes of a Dedeca hedron are tn Like fort 1nclt- ned. The fides ofthe angle of the iit< clination of the [uperficieces of the Tetrahedré are proned ratio enall, The (ides of the angle of the ine= clination of the [uperficseces of the cube proued rationall, The fides of the angle €§c.of the ottohedron proned ration wall, The fides of the angle €5¢.of the Icofahedrow proued srrati- onal, The fides of the. angle €5c. of the dedecahedro proded irratro- wetll, Hew to know whether the an- gle of she incli- watio bea right angle,an acute angle, or an ob- lique angle. T he thirtenth Booke’ the feuenth meafure the refidue,namely,the power of the fide: which is irrational! for that it-is a leffe line,which were abfurd. Wherefore itis manifeft that the rightlines which compofe the angle of the inclination of the bafes of the Icofahedron are Irrationall lines.For the fubtending line hath to the line contayninge,a greater proportion,then the whole hath to the greater fegment. The angle of the inclination of the bafes ef a dodecahedron, is contayned ynder two perpendicu- lars of the bales of the dodecahedron,and is fubrended of that right line, whofe greater fegmentis the fide of a Cube in{cribed in the dodecahedron, which right line is equall to the line which coupleth the feétions,into two equal parts, of the oppofite fides of the dodecahedron. And this coupling line we fay is an irrationall line,fer that the diameter of the {phere contayneth in power both the coupling line, and the fide of the dodecahedron: but the fide of the dodecahedron is an irrationall line, namely,a re- fiduall line(by the 17.0f this booke). Wherefore the refidue namely , the coupling line is an irrationall line,as it is eafy to proue by the 4.comon fentence of the feuéth. And that the perpédicular lines which contayne the angle of the inclination are irrationall,is thus proued. 3 Suppofe that there bea Pentagon ABC D E,and draw in it the perpendicular line, AG, and let the line fubtending the angle of the pentagon be A-C.Now forafimuch as the right line A C is the fide of the cube,and C D the fide the Dodecahedron infcribed in oneand the felfe fame fphere , by the 2. Corollary of the 17.0f this booke: but the line A C is commenturable to the diameter of the {phere,by the 1y.0f this booke, and is therefore rationall,by the 6: diffinitié of the tenth: & the right line C D which isthe fide of the dodecahedron tsirrationall (by the 17.0f this booke). W herfore the lineC G which is the halfe of the line C Dis irrationall by the 103.0f the 10.boke.And the right line A C contayneth in power the two right lines “ AG and G G(by the 47.0f the firft).Iftherfore fromthe §& YE power of the right line A C being rational! be také away the power of the line C G beingirrationall , the power: remayning , namely , the power of the line A G, fhall of neceflitie be irrationall. Forifthe power of the line AG taken away fhould be rational! , and the whole power of the line A C1s rationall, the refidue, namely, the power ofthe line C G fhould be alfo rationall, and fhould be meafured by the felfe fame numbers, by the 4.common fentence of the feuenth. But itis proued that the line C- G is irrationall , for it is the halfe of the whole refiduall line C D(by the 17.0f the thirteenth): which is impoffible. Wherefore the perpendicularline A G is ir- rationall . Wherfore the angle of the inclination of the dodecahedron , which is contayned vnder two perpendicular lines of the pentagon,and is fubtended ofa right line,which coupleth the fe€tions into two equall parts of the oppofite fides of the dodecahedron (by the 2.corollary of the 18. of this booke) which line we haue proued to be irrationall(for that it is equall to the two lines A C,and C D by the 4. corollary of the 17.0f this booke)is contayned vnder irrationall right lines. By the proportion of the f{ubtending line(of the forefayd angles of inclination) to the lines which containe the angle,is found out the obliquitie of the angle, For if the fubtending line be in power dou- ble to the line which contayneth the angle,then is the angle aright angle (by the 48.0f the firft. ) But if it bein powerleflethen the double it 1s an acute angle ( by the 13. of the fecond) . Butifir bein ower more then the double,or haue a greater proportion then the whole hath to the greater fegmét, the angle fhalbe an obtufe angle(by the 12.0f the ened and 4. of the thirtenth ) . By which may be proued that the fquare of the whole is greaterthen the double of the {quare of the greater fegment. This is to be noted that that which Flas hath here taught:touching the inclinations of the bafes of the fiue regular bodies, Hypficles teacheth after the s.propofition of the 15.booke. Where he confef- feth that he receiued it of one Ifidorus , and feking to make the mater more cleare ,he endeuored : himfelfe to declare, that the angles of the inclination of the folides are geuen, and that they are either acute or obtufe,according to the nature of the folide:alchough Exelide in all his ry. bookes hath not yet fhewed, whata thing geuen is. Wherefore Fluffas framing his demoftration vpon an other ground procedeth after an other maner,which femeth more playne, and more aptly here- to be placed then there.Albeit the reader in that place fhal not be fruftrate ofhis alfo. (-*«) ~“ Cc eo ay Theende ofthe thaseaeeooke of Euclides Elementes . ee ee er = >. : 3 -Fol.aré. @ the fourtenth booke of | Euclides Elementes. SS SS SSS —_ a — N this booke , which is commonly accompted the 1 4. | Jbooke of Exchdeis moreatlarge intreated of our prin- wba feipall purpofe:namely, of the comparifon and propor- The argument “A\ ¥-iition of the fiue regular bodies (cuftomably called the o/'*he fourrenth EY 's7115.figures or formes of Pythagoras )the one to the other, booke. dtandalfo of their fides together , eche to other: which - +|{thinges are of moftfecret vfe,and ineftimable pleafure, 14 \landcommoditietofuchas diligently fearch forthem, -A| {and attayne vato them. Which thingesalfo vndoub- , tedly for the woorthines and hardnes thereof ( for thinges of moft price are moft hardeft ) were firft lear- ched,and found out of Philofophers, not of the inferi- aie : ESS Hor or meane fort , but of the depeft and moft grounded Philofophers , and beft exercifed in Geometry . And albeit this booke with the booke following,namely,the 1 5:booke,hath bene hetherto of all men for the moft part, and isalfo at this day numbred and accompted amosgit Euchdes bookes,and fuppofed to be two ofhis,namely,the 14.and 15.in order:as allexemplars (not onely new and lately fet abroade, but alo old monumentes written by hand ) doo manifeltly witnes : yet it is thought by the beftlearned in thefe dayes , that thefe two bockes arenone of Ex- clides,but of (ome other author,no lefle worthy, nor of leffe eftimation and authoritie, notwithftanding, then Exclide, Apollonius a man of deepe knowledge a great Philofopher and in Geometrie maruelous (whofe woderful bookes writté of the fections of cones, which exercife & occupy thewittes of the wifelt and beft learned,are yet remayning ) is thought,and that not without iuft caufe, to be the author of them, oras fome thinke Hypficles him felfe.For what can be more playnely,then that which he him felfe witnef- feth in the preface of this booke, Bafilides of T sre (fayth Hyp icles) and my father together, feanning,and peyfing a Writing or bookeof Apollonius , whith was of the comparifon of a dodecahe- dronto an Icofahedron infertbedin one andthe felfe fame [phere', and what proportion thefe figures bad the one to the other, foundthat Apollonius bad fayled inthis matter. But afterward (fayth he) I found an other copy or booke of Apollontus , wherein the demonftravion of that matter was full and perfect,and fhewed ut untothem, whereat they much reioyfed . By which wordesit femeth to be manifeft that Apollonius was the Grit author of this booke , which was afterward fet forth by Ayp/icles.For fo his owne wordes after in the fame preface feme to import. oe tenon = — . cementite a ee a . ainarerreannanael — 7 pe —— > — s ~ - Na SS meee = a == ee = - SS — 5 ae a —— - ems a = - -- > allie

: ~ ~ Nie = - — <= Lene eas Se a ee SS ite a Ste eee > ae * ’ aan pe = eT ea a aa oe = P “ee ss PRN += a etl be CES a EE A SE eee Za rin RA —— Firft propof- tton afier Fluffase Demonftra- $30, T he fourtenth Booke right demonSiration of that which they fought for: which when they faw,they much rei0y~ fed.As forthatwhich Apollonius wrote,may befeneofiall men,for that itis in cuery mans hand. And that which was of vs more diligently afterward written agayne ,1 thought good to fend and dedicate unto you,as to onewhome Lthaught worthy commendation , both for that deepe knowledge which I know yeu haue in all kindes of learning,and chiefely in Geome- trie,fo that you are able vedily toindce of thofe thinges which are [poken , and alfo for the greate loue and good will which you beare towardes my father and me. Wherfore vouchfafe gently to accept this,which I fend vutoyou.But now isit time to end our preface, and to be- vin the matter. 45 : q Ube 1.T heoreme. Lhe 1. Propofition. AA perpendicular line drawen from the centre of a circle to the fide ofa Pentagon defcribed in the fame circle: isthe halfe of thefe two lines names ly, of the fide of an hexagon figure, and of the fide of a decagon figure bee mig both defcribed in the felfe fame circle. = | TWSe | V ppofe that the circle be A BC. And let the fide of an equilater EVA SS | Pentagon defcribed in the circle ABC,be BC. And ( by the 1.0f a QAR) the third) take the centre of the circle,and let the fame be D.And S EOD e\| (Ly the 12.0f the first) from the point D draw unto the line BC / 4 bs ' unto the line E F put an equalllineG E. And draw aright line from the point G tothe point C . Now fora{much asthe circumference of the whole circle is quintuple to the circu. ference B F.C ( which is fubtended of the fide of the penta gon )-and thecircumference ACF is the halfe of the cir- cumference of the whole circle, and the circumference C F ( whichis fubtended of the fide of the decagon ) is the halfe of the circumference BC F: therefore the circumference ACF is quintuple to the circumference C F ( bythe 15.0f the fift) . Wherefore the circumference AC is qradruple: to the circumference F C . But as the circumference AC uw to the circumference F C,fois the angle A D C to the angle = F D G,hy the laft of the fixt .Wherefore the angle ADC | . quadruple tothe angle F DC. But the angle A DC is doubleto the angle E F C,by the 20. of the third : Wherefore the angle E F Cis double to the angle G D C.But the angle E F C equall ta the angle E GC, by the 4.of the firft . Wherfore the angle E.G C is double to the an- gleE DC, Wherefore the line D G is equall to the line GC (by the 32. and 6.of the fir). But the line G Cis equall to the line C F ,by the 4.0f the fir ft. Wherforethe line D Gis equall’ to the ling CF . And the line G E is equall to the line EF ( by construction ) . Wherefore the line DE is equallto the lines E F and F C added together. Vuto the lines E F and F Cadde the line D E.. Wherefore the lines D F and F C added together, are double to the line DE. But theline DF isequalltothe fide of the hexagon : and FC tothe fideof the decagon. Wherefore . of Ennclides Elemente’: Fol.417. Whereforethe line D Ess the halfe of the fide of the hexagon, and of the fide of the decagon being both added together and de{cribed sumone aud the felfe fame circle. It is manifest.” by the Propofitions of the tiirtenth booke, that. a perpendicular line drawemfrom the centre of acivcletothe fide of anequilater triangle deferibed in the fame circle, ishalferof the femidiameter of the circle. Whereforeby this Propofirson, a perperidscwlar dru- wen from the centre of a circle to the fide of @ Pentagon, ss equall to the perpendicular drawen from the centre te the fine of the triangle, and to halfe of the fide of the decagon def{cribed inthe fame circle: q Ihe 2. I beoreme. Lhe 2.'Propofition. One and the felfe fame circle .comprehendeth both the Pentagon ofa Daz decahedron, and the triangle of an Icofabedron, defcribed in one and the felfe/ ame S phere. tales His Theoreme is defcribed of Arifteus tv that booke whofe title is, The | , comparifon of the fite figures, and 1s de{cribed of Apollonius a his [e- pene edition of the copari[on of a Dodecahedron to an Icofahedron, which is, o.* that as the fuperficies of 4 Dedecabedrti3 tothe fuperficies of an Icofabedron, fo 1sthe Dode- ~ > wade of the fide of the pentagon, and of that right lime which 1s fub- teuded Grider two fides of the pentagon, ave quintuple to the [quare o f the femidiameter of the circle. Suppofe that AS C he 4 circle. Andlet the fide of a pentagon in the circle ABC,bé AC. And take¢by the tof the third ) the centre of the circle and letthe famebe D . And (by the 12:0f the firft) from \ the point.D draw unto the line A.C a perpendicular ine D F\) CAnd extend the line DF on either fideto'the pointes Bund E> And draw a right line from the point © Ato the point B. Now 1 fay, that the [quares of the lines B Aand AC are quintuple to the (quare of the line D E. Drawe aright line from the\point Ato the point E. Wherefore theline/AE isthe fide ofa decacon ficure. ) otk And forafmuch as the line’ BE is double to the line D E : thereforethe[quare of the line BE 1s quaarupleto the [quare of DE ¢ by the 20.0f the fixt ). But unto the {quare of the line BE; aréequall the {qiares of the lines B Aand A E (by the 47.0f the fi ff, for the angle BY Eis aright angle, by the 31 .0f tbe third). Wherefore the {quares of the lines B A and AE, are quadruple tothe [quare of the line D-E.VWherfore the [quares of the lines ‘A BSA E, and DE, are quintupleto the {quare of the line D’E-. But the [quares of the lines DE and AE, areequall tothe {quare of the line AC ( by the’ 10. of the thirtenth ). Wherefore the {quares of the lines B A and:_AC, are quintuple to the {quare of the line D E. T his being thus proved, wow isto be demonflrated that one and the felfe [ame circle co- prehendcth both the pentazon of a dodecahedron,cy the triangle of an Icofahedron defcribed inonecy the felf famercincle. T ake the diameter of the phereye let the {amébe AB: And in the fame fpherede{cribeadedecahedron,c> al{o.an keofahedron\. dud let. one of thepitagons of the dodecahedron be C DER GC let.one.of the txiangles of thelcofabedron be K LH . Now Lfaythat the [emidiameters of the circles which are de{cribed about them are eq — ; SS that Bb * This is mans fet by the 12, propofsrié of the thirtenh book is Campane well gathereth 573 a Corollary of t he fame. Thea th een s he ty f OP OLLIE after Fluffas. * This ss after- ward preued in the 4, propoft- Li0B. This Affumpt vs the 3 prepofi- tronufier Fix fas. Cent rudion of the Affummpt. Demon ratior of the Agumpts Confirudson of the propofition. < pee = Sa ee —— — —_—— = = a ‘ 2 > = = 2 ~ ee en ee - — — = < gy as 4, = Se Sa Sse : 2 > 3 phe 2 = = St ei pee bf a Hey hy BP Hy Ne Ki Ra a (thy ‘ tii / eee ee ed sae a ly ao areas he--- 68 en 5 —— T he fourtenth Booke that is,that one ana the [elfe fame circle conta | at is, ife fas yneth both the pentagon C DE FC hepa . L o age ari in ie the point D to the ee G Wherfure the ae af acube( by the corollary of the 17.0f the thirteth) Take a cert cht lin And let the {quare of the line A B be quintuple tothe fquare of the Pg a-ha put after the 6.propofitio of the teth.But the diameter t, a {phere is in power quintuple to oe Square of the femidiameter of the circle,on which is decribed the Icofabedro (by the corolla Demonitra- tion of the propofttiore % Here is requst- ved an Affumpt which is after ward prowed in the 4. propoft- to of this boke: namely, that lines densded by anextreme €9 meane propor Fron,arecuery wit) proport tom nall: mbich graunted ther. followeth the reajanapt lye. For the line M- WN és by fuppoft- tion deusded b ry anextreme and wncune proports on, 65 bss grea- ter fegment ts the line MX, and the line D- G deuided by an an extreame and mie dne pro- portion, “bit 2 greater fegmet ss the line CG, by the 8.0f the rhirtenth. Thos. pr apo- fitton after F: bufsas » forafinuch asthe {quare ofthe line A B is quintuple to ry of the 16 .of the thirtenth) .Wherefore the ti : ‘di which is de{cribed the coolest Sai ( by a oi dae veers ra gti extreante and meane proportion in the poynt X . And 2 ig Oy let the greater fegment thereof be M X . Wherefore the x line M X ts the fide of a decagon defcribed in the fame circle( by the corollary of the 9. of the thirtenth). And the {quare of the line M N:But the {quare of the li : ine B Ais treble tothe {quareof the line D G hs the co- rollary of the 15, of the thirtenth ).Wherfore three {quares of the line D G are equall to fiue {quares of the line M N." But as thre {quares of the line D G are to fi ue {quares.of the line. M N,fo are three {quares of the line CG to fine [quares of the line MX. Wherfore three {quares of y line C Gare equll to fine {quares of the line M X But fiue {quares of the line CG are equal to fiue {quares of theline MN cr to fine{quares of the M X. For (by thero.of the thirtenth) one {quare of 4 L the line C Gis equall to one [quare ofthe line MN & O . re fh of the ve a eno pony aeares of the line C G are equall to thre fquares : G and to three {quares of the line C G(as it is not hard to proue,marki r wn, 8 bene proued ) . But three {quares of the line DG; together ihe hee Wrst af a aa : 2 a ad to front ee of éW peace of the circle de{cribed about the | G (for it was before proued in the alfumpt put inthis ti Squares of : G and G C taken once,are quintuple to the a of the pied es iW, ie a ede(cribed about the pentagon C DEF G). And fine {quares of the line K Lareequall ‘ Sem fé Suche the sen famd ofthe.circle de{cribed about the triangle K LH. (For of the thirtenth,one [quare of the line L K is triplet he li from the centre to the circumference) . Wherefore eee an tene [quares of the line dra the centre to the circumference (of the circle which Ht 4 ; nappy ; cle ontayneth the pentagon C D EF G equall to fiftene{qnares of theline drawne from the centre to th Pt a which contayneth the triangle K LH) ::wherefor il ener banely Bune : | eone.of the [quares which is d; the centre to the circumference. of the one circle, 4 Se eae sungee : : cof the. cle, isequall to one of the, ich 4. a irae Bh sak i a Seeeafenence of the other circle. Aiton i equallto the atameter, wherefore one and t e felfe fame circle comprehendeth both th tagon of a.dodecahedron and the triangle of an Icofahedron epee: Re H of Thea, Theoreme. _..The.3 Propofition. If there be anequilater andequiangle pétagon,aud about i | ee in quiangle petagon aud about it be de[cribed “ss cle ees om the centre to one of the fides bedrawne a teeta ine that which is contayned ynder one of the fides and the perpendicular line of Euclides Elementes.. ~ Fol.4.18. Ane thirty timesyis equallto the fuperficies of the dodecahedron. V ppofethat AB CD.ke dm equilater and equiangle pentagon. And about the | [ame pentagon ,defcribe( by the 14.0f the fourth)a circle.And let the centre ther- of be the poynt F . And from the poynt F draw ( by the 12. of the firt ) unto the EEA CI line CD a perpendicular hue F CG. Now 1 fay that.that which is contayned vn- der the lines C.D and G F thirty times, ts equall ta 12. pentagons of the fame quantitie that the pentagon ABCD is. Draw thefe right lines CF and F D. Now fora{much as that which is. contaynea under the lines C D and FG is double to the triangle C D F ( by the.gi . of the firft) : therefore that which is contayned under the lines @ D and F G fine times is equall to ten of thofe triangles . But ten of thofe triangles are two pentagos,and fixe times ten of thofe triangles are all the pentagons. Wherefore that which is coutayned under the lines C D.and F G thirty times is equall to 12. pentagons But 12.pentagons are the [uperficies of dodecahedron. Wher fore that which is contayned under the lines C D and FG thirty times is equall to the {uperficies of the dodecahedron. In like fort allo may we prone that sf there be an equilater triangle as for example, the Conslruitign. Demon freq L26%e The 5, propo- triangle AB C,and about it be defcribed acircle, and the centre of the circle bethe point D, /tton after and the perpendicular line be theline DE; that whichis comtay~ ned vnder the lines BC and.D E thirty times,is equall to the [u- A perficies of the {cofahedron. For agayne forafmuch as that which is contayned under the lines D E and BC is double to the trian- gle D BC ( by the gz. of the firft) :therefore two triangles are e- quall tothat whichis contayned under the lines DE and BC, and three of thofe triangles contayne the whole triangle. Where- fore fixe {uch triangles as D B Cis, are equall to that which is® Ve, contayned under the lines D E and B C thrife.But fixe {uch tri- angles as D B Cis, are equall totwo {uch trianglesas ABC is. Wherefore that which is contained under the lines DE and BC thrife,is equall to two {uch triangles as A BC is. But two of thofe triangles také ten times contayneth the whole I cofahe- dron.Wher fore that which is contayned under the lines DE ¢ BC thirty times,is equall to twenty {uch triangles as the triangle AB Cis,that is,toy whole [uperficies of the 1 cofahedro. * Wherefore as the [uperficses of the dodecahedron 4s to rhe fuperficies of the leofahedron,fo ssthat which ts con- tayned Under the lines C D and FG to that which ss comtayned Vader the lines BC and D E. q Corollary. By this it is manifeft, that as the fuperficies of the Dodecabedron is to the fuperficies of the Icofabedron, fo is that which is contained ‘ynder the fide of the Pentagon, and the perpedicular line-which is drawen from the cen- tre of the circle deferibed about the Pentagon to the fame fide, to that which ts contained ‘ynder the fide of the Icofabedvon and the perpendicular lime which is drawen from the centre of the circle defcribed about the tri- angle to the fame fide : [0 that the Icofahedron and Dodecahedran be both defcribed in one and the felfe fame Sphere. | Q_ qj. q Tbe finfjas. Demonstra- tion. * This is the veafcn of the Corollary followin, 2- 4, Corollary which alfo Elufsas put teth ds a Co- rollary after the Shara tion in his order. * = —— : ——— a * i ° — = < . ame alp oa . — io. : = pe NE a Pm e . =" See — Se =s ——— ea pS ee — : 3 ee . : ¢ : af . . 5 = . 7 ee , a ane si etitiaeioreimadiadl Pere —— = \} {4 | god mn Ue | RO i \ ig ' ' ‘ey a ue te a ae ip fe | ee ay y Un Gt Fae: (0) yee BG be yi ‘ t ee —— The 6. popofitse after Flafjas. Couftrnttion. Demonuttra- £i0%60 * This 48 nov hard to prowe by thers .t6. and 19.0f the bet @. Tt Jz rhe Coro lary of the 17. of the thrrteth. * Here aga ine ss required the Aff] went wbich as afterward prowed tm thes 4 prepojisson, t But firft the Akampt fellow 682, f2E COn~ firattson whereof bere begsn meth, t ro be proucd, T he fourtenth Booke q The 4. 'beoreme. The 4. Propofition. This being done, now is to be prowed, that as the fuperficies of the Doe decahedron is to the fuperficies of the Icofahedron, fo is the fide of the cube to the fide of the Icofahedron. ° f So Ake ( by the 2. Theoreme of this booke)a circle containing both the penta- Ni gon of 4 Dodecahedron, and the triangle of an Icofahedron, being both de- ew) {cribed im one and the felfe {ame fphere,and let she fame circle be D BC. And Sy MOS 17 the circle D BC defcribe the fide of an equilater triangle, namely,C D, vee antl the fide of an equilater pentagon,namely, AC. And take (by the 1.0f the third ) the centre of the circle, and let the ame be E . And fromthe point E drawe vnto the lines D Cand AC, perpendicular limes EF and EG . And extend the line EG direttly to the point B. And drawe aright line from the point Bto the point C . And let the fide of the cube be ihe line H. Now Lfay,that as the {uperficies of the Dodecahedron is to the fuperficies of the Icofahedron, fo x the line H to the line CD . Fora{much as the ine made of the lines E Band BC added together (namely,of the fide of the hexagon,and of the fide of a decagon) is (by the 9.0f tue thirtenth) dinided by an extreme and meane proportion, and his greater ~ Segment is the line BE : and the line EG is alfo (by the +.of the foaretenth) the halfe of trae [ame line , and the line E F isthe halfe of the line B E ( by the Corollary of the 12.0f the thirtenth). Wherefore the line £ G being dinided by an extreme and meane proportion,” his greater feg- ment {hall be the line E F . And the line H alfo being diuided by an extreme ¢ meane proportion, his grea- ter fecment is the line C A, as it was prouedt inthe Dodecahedron.* Wherefore as the line H is to the line C A, fois the line E G tothe lineE F . Wherefore (by the 16.0f the fixt ) that which is contained under the lines H and E F, is equall to that which is contained under the lines C Aand E G. And for that asthe line H isto thelineC D, [ois that which is contained un- der the lines H and EF, to that whichis contained under the lines CD and E F (by the 1.of the fixt ) . But unto that which is contained under the linesH and E ¥ , is equall that which is contained under the lines C A und E G.Wherefore (by the 11.0f the fift) as the line H is to the line C D, fois that which is contained under the lines C A and E G, to that which is contained under the lines C Dand E F, that is (by the Corollary next going before ) asthe fuperficies of the Dodeca- hedron is to the {uperficies of the Icofahedron, [ots the line H tothe lineC D. An other demonftration to prone that as the fuperficies of the Dodecabes dron ts to the fuperficies of the Icofabedron, fois the fide of the cube to the fide of the Icofabedron. + | Zt there bea circle A BC. Andin it defcribe two fides of an equilater pen- tagon (by the 11.of the fift) namely, AB and AC: and draw aright line {rom the point B tothe point C . And (by the .of the third) take the centre at) of the circle, and let the fame be D. And draw a right line from the point AES A to the point D, and extend it directly to the point E,and let it cut the line _-« BCinthe point G . And let the line D F be hfe tothe line D A, and let the line aN i of Euchdes Elementes.. - Fol.419. the line G C be treble to the line H GC, by the 9.0f the fixck Nal faysthat that which ss'contairied Guder the lines A Fund B Hts eguall to the pentagon infcribediathecivcle ABS. Dyytya right line from the point B to the point D.. Now forafnzuch as the line A D.is double tothe line D F, there- fore the line AF is fe{quialter tothe line A Dw whites Againe,foralmuch asthe line GC is trebleto the line C H, therefore the line G H 1s double-to the - line CH . Wherefore the line GG is fe(quialter to the line H G . Wherefore as the line F A isto the line AD, {ois the line GC tothe line GH. Wherefore ( by the 16.0f the fixt) that which i contained under the lines A F ¢> H G, is equall to that whichis contained under the lines D A and GC .But the line GC is equall tothe line B G(by the 3.0f the third). Wherfore that which 48 contained vader the lines 'AD and BG; ise- guall to that which is contained under the lines A FandG H. But that whichis contained vn- | der the'lines A D and BG, is equall to two {uch triangles as the triangle ABD is'( by the gt.of the fir(t ). Wherefore that which is contained under the lines AF and G H,is equall to two {uch triangles as the triangle AB D is .Wherefore that which is contained under the lines AF and GH fiue times, is equallto ten triangles . But ten triangles are two pentagons. Wherefore that which is contained under the lines AF and GH fine times, is equall to two pentagons . And fora{much as the line G H is double tothe line H C, therefore that which is contained under the lines A F and G His double to that which ts contained under the lines A F and HC (by the 1.0f the fixt) . Wherefore that which is.contained under the lines A F and C H twife, is equall to that which is contained under the lines A F and G.H once. Take eche of thofe parallelogrammes fine times. Whereforethat which is contained under the lines A Fand HC ten times, ts equall to that which is contained under the lines AF & G H fiue times, that is, to two pentagons.Wherefore that which is contained under the lines A F and HC fine times, is equall to one pentagon . But that whichis contained under thelines A F and H C fine times, is equall ( by the t.of the fixt) to that which is contained under the lines AF and HB, for the line HB is quintupleto the line HC ( as it is eafie to fee by the con- firuttion and they are both under one cy the felfe [ame altitude ,namely,under A F .Wher- fore that which is contained vader the lines A F and B H, is equallto one pentagon. This being proued,now let there be drawne a Circle comprehending both the Pentagon of a Dodecahedron, and the triangle of an Icofahedron, being both de{cribed in one and the felfe fame Sphere. Vani

let the line C.K be treble to the ine CH,by the.o of the fixth. And fro the point G raifevp (by the.r1.of the firft) unto the line A F a perpendzcular lineG M:and extend the line GM directly to the point D.Wherforethe line M D is the fide of an equt- liter triagle, by thecorollary of the.x2.of the thirtenth:draw thee right lines AD and AM. Wherfore A D M is anequilater trianzle. And for as nsuch as that which is contained vn- der the lines AG and B His equalto the pentagon (bythe former alfumpt) and that which G ; ‘af - 9 24-4. Zs ; va The Affuinpty, which alfo Fluf [@ putteth as ats Afjump tufter the 6 .propefits Fe Demonftration af the Affumpt « Conftrudson pert#ining 66 thé fecond de~ monftratiin of the 4.propefittd, Second demon- fPration of the 4 prope Ez. B92 ae ee = —— i = ~ —— bid hit Hi —s. ' —— — pen: e< = 26 - a — ee yee * Dé: = = —— 5 ~ _. = ss > = Se rar : = s : 2 power’ 4 ee ESE ee Tie 7, prepo- fitton after Fluffase Conftruction. DewonStra- $82the ry, The fourtenth Booke is cotained under the lines AG andG Dis equal to the triangle: A.D M: therefore as that which is contained under the lines AG and H-Bisto that which is contained under the lines AG andG D, fossthe pentagon to the triagle.But as that which. #s contained under the lines B H Cy AG is tothat sphich is contained under the lines. AG andG D, fois theline B H to the line D G (by the.1.of the fixth ) wherefore (by the.rs.ofthe fifth ) astz. {uch lines as BH is,areto.20 [uch lines as D Gis, [0 are 12.pentagons to 20.triangles, that is the {u- perficies of the Dodecahedron,to the fuperficies of the Icofabedron. And 12.{uche lines as B His,are equall to tenne fuche lines as BC is (for the Ff line HB is quintuple to the line HC):and the line BC is fextupie to the line C H- Wherfore fix [uch lines as B H is,are equal to fine [uch lines as BC are: and tn the fame pro- portion are their doubles : and 20.fuch lines as the line D Gis, are equal to.ro.fuch lines as the line D Mis:for the line D M is double to the line D G. Wherfore as 10.fuch linesas BC is,areto 10.[uch lines as DM és,that is,as the line B Cisto the line D M,fo is the {uperficies of the Dodecahedron to the [uperficies of the Icofahedron. But the line B Cis the fide of the cube,and the line D M the fide of the Icofahedron: wherefore (by the 11.0f the fifth) as the Juperficies of the Dodecahedron is to the [uperficies of the I cofahedron, fois the line BC to the line D M,that is,the fide of the cube to the fide of the Icofahedron. Nowe will we prone that aright line being deuided by an extreme and meane proporttio,what proportio the line cotaining in power the fquares of the whole line and of the greater fegment hath to the line containing _ in power the [quares of the ‘whole line and of the fe Jegment the fame proportion hath the fide of the cube to the fide of the Icofahedron, being bath defcribed in one and the felfe fame {phere. | V ppofe that A B be a circle conta:ning both the pentagon of a Dodecahedron ey the tréangle of an Icofahedron de{cribed bothe in one and the felfe [ame [phere. a Take the centre of the circle,and let the fame be C.And from the point C extend piece to the circumference aright line at all aueatures,and let the fame BC. And (by the 30.0f the fixth )deuide the line B.C by an extreme and meane proportion in the point D, and let the greater feement therof te C D.Wherfore the line C D is the fide of a Decago de- fcribed in the fame circle(by the corollary of the 9.0f the thirtenth).T ake the fide of an Ico- {abedron,and let the fame be the line E,and the fide of a Dodecahedron, and let the fame be the line F and the fide of a cube cy let the fame be the line G. Wherfore the line E is the fide of an equilater triangle, and F of an equaliter pentagon defcribed in one and the felfe [ame circle. And the line G being denided by an extreme and meane proportion, his greater feg- ment is the line F, by the corollary of the 17 .of the thirteth.. Now forasmuch as the line E is the fide of an equilater triangle, but (ty the 12.0f the thirtenth) the fide of an equilater triangle isin power trebletotheline BC, (whichis drawne fromthe center to the circum- ference) therefore the fquare of the line Eis treble tothe {quare of the line BC : but the {yuares of the line BC and B D.are( bythe 4.0f thethirténth) treble to the {quare of the line CD. Wherfore as the {quare of theline E isto the {quare of the lineC B, fo are the {quares of the linesC B and B D tothe {quare of the line C D. Wherefore alternately (by - 16.0f | 7 the = CE of Euclides Elementes. Fol.420, the fifth) as the fquare of the line E és to the fquares of the lines C Band B D, fois the {quare. of the line CB to the {quare of the line C D.” But as the fquare of the line B Cis to the [quare of the lineC D, fois the fquare of the line G (the fide of the cube) tothe (< quare of the line F the fide of Dodecahedron. For the line F is the ercater feemet of the line G (as was before proued.)Wherfore( by the.r1. of the fift) as the (quare of the line E istathe fquaresC Band BD, fois the {quare of the line G, tothe {quare of the line F. Wherefore alternately (by the 16.0f the fifth) & alfo by conuerfion(Ly the corollary of the 4. Dy G of the fift)as the [quare of the line G,is to the {quare i F of the line E,fo is the [quare of the line F,to the See rE {quares of the lines CB cy BD. But vate the {quare . of the lineF are equal the {quares of the lines BC & CD. for the fide of pentagon cotaineth in power both the fide of a fixe angled figure,and the fide of « ten angled figure( by the ro.of the thirtenth.) Wherfore as the {quare of the line G,is to the Square of the line E, foare the [quares of the lines B Cand CD to the {quares of the lines C B and B D. But as the [juares of the lines C Band C D are to the {quares of the linesC B & B D,t fo (any right line what fo euer it be,being diuided by an extreme and meane proportion) is the line containing in ower the {quares of the whole line,and of the greater fegmét,to the line containing in power the fquares of the whole line,and of the leffe fegment:wherfore( by the 11.0f the fifth )as the fquare of the line G(the fide of the cube is to the {quare of the line E,fo(any right line being deuided by an extreme and meane proportion) is the line contatning in power the {quares made of the whole line,and of the greater fegmit,to the line containing in power the {quares made of the whole line,and of the leffe fegument: but the line G 1s the fide of the Cube, and the line E of the Icofahedron( by [uppofition.) If therfore aright line be deuided by an extreeme and meane proportion,as the line cotaining in power the {quares of the whole line,and of the greater fegment,ts to the line containing in power thefquares of the whole line and of the leffe fe gment: fo is the fide of the cube to the fide of the lcofahedron, being both defcribed in one and the felfe fame [phere. Now will we proue that as the fide of the Cube is to the fide of the Icos Jahedron fois the folide of the Dodecahedron to the folide of the Icofas hedron. | Orasmuche as equal circles comprehend both the pentagon of a Dodecahe- Se dron,and the triangle of an Icofahedron, being both defcribed in one and | Sea i the felfe fame [phere, by the 2.0f this booke:but in a phere equal circles are SONG equally diftant from the centre(for the perpendicular lines drawn from the i e centre of the [phere to the pou fuperficieces of the circles are equal,and do ee fall upon the centres of the circles. tWherfore ae. lines drawne fromthe centre of the (phere,to the centre of the circle comprehending bothe the triangle of an Icofahedron,and the pentagon of a Dodecahedron are equal : wherefore the pyramides, whofe bafes are the pentagons of the Dodecahedron,are of equal altitude with the piramides whofe bafes are the triangles of the Icofahedron. But piramids of equal altitude, are in that proportion the one tothe other that their bafes are (by the s.of thetwelfth) wherefore as the pentagon ts to the triangle, (0is the pyramis whofe bafeis the pentagon of the Dodecahe- dron and toppe the centre of the (phere,to the pyramis whofe bafe is the triangle and top the = Q.4.i4. centre ® Flere apaine ss required the Afsumt t af- terward pro- uedin thts fre proposition, T As may by the Afsumpe afterward im this tropofitss be plainely preued, The 8. profi tion after Finffas. TBy the Co- rollary added by Fluffas af= ter his Af- Jumpt put af- ter the 17. pro pofition of the 12. booke. a. _ 2 a OTT , * ee ~aor” Ae 7 . 4 me a alle oe wiercctts csi he. Corollary of the 3 after Flaffas. These Affismapt is the 3. propofitia after FlufSas, antis it which diuers times bath bene taken oS ST ARRLCAIB this booke and oe alfo s2 the Lift propofition of the 13 .boote: astwe baue be- fore noted, Demonftra- #i0n. T he fourtenth Booke centre allo of the {phere Wherfore( by the 1 5.of thé fifth as 12.petagons areto 20:triangles; fo are 12 pyramids hauing pentagons totheyr bafesto 20.pyramids hauing triagles to their bafes. But 12.pentagons ave the (uperficies of the Dodecahedron, and zo:triangles are the Super ficies of the 1cofahedron. Wherefore as the uperficies of the Dodecahedron is to the {ue perficies of the Icofahedron,foare 12. pyramids hauing pentagons to their bafes to 20. tyra mids, hauing triangles to taeir bales. But 12. pyramids hauing pentagons to their bales, are the (olide of the Dodecahedron, and 20.pyramidshauing triangles to their bafes are the fo- lide of the Ic ofahedron.W herfore ( by the 11 Of t he fifthe ) as the fuperficies of the Dodecahedron ts te the fuperficces of the lcofahedron fo is the folide of the Dedecahedron to the folide of the Icofahedron.But as the fuperficies of the Dedecahedron, is to the fuperficies of the Icofahedron, fo hane we proucd that the fide of the cube is to the fide of the Icofahedron.Wherfore, by the 11.0f the fifth, as the fide of thecube ts to the fide of the Icofahedron, foisthe folide of the Dodecahedron to the folide of the Icofahedron. | If two right lines be diutded by an extreame and meane proportion they [hall enery way be in like proportion: which thing is thus demonftrated. ecoe= Et the lime A B be(by the 30. of the fixth) diuided by an extreame and meane pro- We portion in the poynt C , and let the greater [egment thereof be the lineC A. And Likewife alfo let theline D E be dinided by an extreame and meane proportion in = ==" the poynt F and let the oveater fecment thereof be the line D F .T hen I [ay that as the whole line A Bis tothe greater fecment thereof AC, fo is the whole line D E to the grea- ter fecment thereof D F.F or forafinuch as that which is contayned under thelines A Band BC iswquall tothe (quare of the line AG (by the diffinition of a line dinided be an ex- treame and meane proportion):and that which is contayned under the lines D E and E F is alfo equal to the [quare of the line D F (by the fame diffinition ): therefore as that which is contayned vnder the lines AB and B Cis tothe {quare of the line AC , fois thatwhich is contayned vnder the lines D E and E F to the {quare of the line D F.. For in eche is the pro- portion of equalitie. Wherfore as that which is contayned under the lines A B and BC fower times, 13 to the [quare of the line AC, {ois that which iscontaynd. A ¢ B under the lines D E and E F ee ee : | fower times to the {quare of the D F E line DP (bythe rs.ofthe ffi}. 2 ee Wherfore by compofition ( by the i | 7 18.0f the fifth) as that which ts contayned under the lines AB and BC fower times, together with the fquare of the line A C,i3 to the {quare of the line A G, fois that which is contayned underthe lines DE and EF fower times,together with the {quare of the line D F tothe Square of the line DF .Wherefore.as the {quare which is made of the lines A B and BC ad- ded tagether aud made one line (which (quare by the 8. of the fecond is equall to that which as contayned under the lines A Band BC fower times together with [quare of the line AC) asta the {quare of the line AC, {ois the [quare made of thelines D E ¢y E F added together and made one line (which {quare is alfo,by the fame sequal to that whichis comtayned under the lines D E and E F fower times together with the [quare of the line D F) to the [quare of the line D F Wherefore alfoas the lines A B cy BC added together are to the line AC,fo are the linesD E & EF added together to the line D F (by the 22.0f the fixt).Wherefore by co- _ pofition( by. thes 8: of the fifth )as both the lines AB cr B C.added the one tothe other,toge- ther withthe line A C,that is,as twofuch lines as 4 Bis,areto the line AC, foare both the lines D E and F added the one to theother together with the line D F , thatis two fuch lines of Euclides Elementes. Fol.a2. lines as DE is to the line D F.And in the fame proportion are the halues of the antecedents by the 15.0f the fifth. Wherefore as the line A Bis tothe line AC, foistheline D E tothe line D F.( And therefore by the 19 .of the fifth,as the line A Bis to the line BC, [ois the line D F tothe line F E.Wherefore alfo by dinifion by the 17 .of the fifth,as the line AC is tothe line C B, (ois the line D F to the line D E). Now that we haue proued,* that any right line whatfoener being diuided by an extrcame and meane proportion,what proportion the line contayning in power the [quares made of the whole line and of the greater fegment added together, hath to the line contayning in power the fquares made of the whole line and of the leffe feement added together , the fame propor- tion hath the fide of the cube to the fide of the Icdfahedron: Now alfo that we haue proued, tthat as the fide of the cube is to the fide of the Icofahedron , {ois the [uperficies of the Dode- cahedron to the fuperficies of the Icofahedron,being both de{cribedin one and the felfe [ame {phere:and moreouer feing that we haue proued,tthat as the [uperficies of the Dedecahedron $s to the fuperficies of the Icofahedro, [01s the Dodecahedro to the Icofahedron, for that both the pentagon of the Dodecahedron, and the triangle of the Icofahedron are comprehended in one and the felfe fame circle : All thefe thinges fay being proued,it is manifelt , has if in ove and the felfe fame [phere be decribed a Dodecabedron,and an Icofakedron , t hey fhall be in pr oportion she one to the other ,as,a right line whatfoeuer being diuided by an extreame and meane pro- portion the line contayning in power the {quares of the whole line and of the greater fegment added tozether,is to the line containing in power the {quares of the whole line and of the leffe segment added together . For for that asthe Dodecahedron is to the Icofahedron, fois the [uperficies of the Dodecabedron to the fuperficies of the Icofahedron, that is, the fide of the cube to the fide of the Icofahedron : but as the fide of the cube ts to the fide of the Icofahedron.fo,any right line what foener being diuided by an extreame and meane proportion,ts the line contaynine in power the {quares of the whole line and of the greater fegment added together,to the line contayning iz pow- er the {quares of the whole line and of the leffe fegment added together. Wherefore as a Dodecahedron is to an Icofahedron defcribedin one and the felfe fame [phere, {0, any right line what fo ener being dinided by an extreame and meane proportion,is the line contayning in power the {quares of the whole line gy of the greater fegment added together,to the line contayning in power the {quares of the whole line and of the le{fe fegment added together. (? §) The ende of the fourtenth Booke of Euclides Elementes after Hypficles. Y) i gr} (Calan rat)’ a>" SP OI SS je) Lehr oy De x seit cates t RSF, nS Wi Md) s N 5 - —— =. = = a = —*< = ~ = _ y+ = = < = . = = Se ee, z Se pee ee re a oS tS eS = ee ———— iii intent aaah = TSS —- = - - = £8 ae = =~ . - et —-_ = a = — ie oe : ~— = = —- = = - = =3 — = Pers — ee eee Te es ers eee patra = > ws 2 =. : i “ ~ ea’ = = apa - —e - a — : 4 or = ¥ A ye ae ce = —_ - ase ieee ag ~~" = ie. = : - 3 — ~ —~- - tnt ae tweet ~ - = ‘ . - s -_ : -— -~- _ 4 -—— en > ~ ~~ a — ce >= — ~~ -- ~~ « i. amen “ =< = = = —_— ad oS oe = a a eer a — a eee 4 LS ~s = 7 a — Balter x Pe : Sg = eee = SO SST - —— ay £Z : next = as ~ = o—— = ,~ = _ ~ , - a an ~~ - e — ~ - - . = a = . == So : TI A I == rT en re at ~ Sa a = eer: = = ire a % a ee as “ RAEN : ee a en * J . At . b i = . ee ee ee tS re Lt. ~~ =a? np wore ae F 72 - £, ~ pears De SRF SL tS The firft pro- pofiston after Campane. Conftruction. Demonfira- tion, ~ @The fourtenth booke of Evuclides Elementes after Fluffas. Was Hypficles hath, are by him fomewhat otherwife de- S¢-xej monttrated, Ithought my labour well beltowed for g@.)) the readers fake to turne italfo all whole ,notwithftan- (era ~ ding my trauaile-before taken in turning the fame Feo. cA booke after Hypficles. Where note ye,that herein this ,“@. 14-booke after Fluffas, and in the other bookes follo- 4, ca? © \wing;namely, the 15.and 16. Thaue in alleadging of Rees the Propofitions of the fame 14.booke,followed the x \ De: alle er orderand nuinber ofthe Propofitions, as Fluffas hath ZF a A placed them. q I he first Propofition. A perpendicular line drawen from the centre of a circle, to the fide of a Pentagon infcribed in the fame circle: 1s the -halfe of thefe two lines taken together namely, of the fide of the hexagon, and of the fide of the decagon infcribed in the fame circle. Ake.a circle AB C, and infcribe in it the fide of a pentagon, which let be BC, and take the centre-of the circle; which let be the point D : and fré = SSS ‘poem ALDH ake itdraw vote the fide BC a perpendicularline DE: which produce to the EONSEES oint Ei Andvnto. the line EF putthe line E G equall . And draw thefe | WA if right lines €C G,C DjandC F . Then May that the right line D E (which = \ [ is drawen from the centre to B € the fide of the pentagon ) is the halfe NIN 4{) of the fides of the decagon and hexagon; taken together , Forafmuch as VO DAs NEG = the line D Eis.a perpendicular ynto the line BC : therefore the fections ; | Ss] BE and EC fall be equall (by the 3.ofthe third) : andthe line EFis ge Sowa ' €ommon vntothem both, and the angles F EC and FE B, are rightan- ls ES te Aly NDE gles, by fuppofition. Wherefore the bafes B Fand FC are equall (by the 4.0f the firft ) .Buttheline BC is the fide ofa pentagon, by conftruct- on . Wherefore E C which fubtendeth the halfe of the fide of the pentagon, isthe fide of the decagon infcribed in the circle A B C, Butvnto the line F C is, by the 4.0f the firit, equall the line CG, for they fubtend rightangles. E G,.and.C EF, whichfare, con- ' tained vnder equall fides. Wherefore alfotheangles ©-G E,and C £ F,0f the triangle C F G,are equall, by the’‘s.of the firft, And forafmuch as the arke F C is fubtended of the fide of a decagon, the arke C A thal! be quadruple to the arke CF > Wherefore alfo the angle CDA thall be quadruple to the angle C D F ( by the jaft of the fixt) . And forafmuch as the fame angle C D A, which is fet at the céntre,is double to the anglé @ PA, which is fetar the circumference,by the 20.of the third therefore the angleC FA, or CED, is double to the angle C D 8, namely, the halfe of qua- druple . But vnto the angle C FD or CEG, is proved equall the angle C G F : Wherefore the outward angle C G F; ts dotuble'to ~ the angle C DF . Wherefore the angles CD G and DCG, thall A be equall. For vnto thofe two angles the angle C G F is equall,by the 32.0f the firit. W herefore the fides G Gand G D, are equall, by the 6.of the firlt . Wherefore i. the of Euclides Elementesafter Flufvas. Fol.422 the line G D is equall to the line F C, which is the fide ofthe decagon . But ynto the right line F E is e- guall the line E G, by conftruction . Wherefore the whole line D E is equall to the two lines C F and ; F E..Wherefore thofe lines taken together.(namely, the lines D F and F C ) fhall be doubleto the line DE. Wherefore the line D E(which is drawen from the centre -perpendicularly to the fide of the pen- tagon)fhal be the halfe of both thefe lines taken together,namely,of D F the fide of the hexagon, and C F the fide of the decagon , For the line D F which is drawen from the centre, is equall to the fide of the hexagon, by the Corollary of the 15.of the fourth . Wherefore a perpendicular line drawen from the center ofa circle, to the fide of a pentagon infcribed in the fame circle : is the halfe of thefe two lines taken together, namely ,of the fide of the hexagon, and of the fide of the decagon infrcibed in the fame circle: which was required to be proued, % A Corollary. If a right line drawen perpendicularly from the centre of a circle to the fide of a pentagon, be dinided by an extreme and meane proportion : the greater fege ment fhall be the line which 1s drawen from the fame centre to the fide of an equilater triangle infcribed in the fame circle. For, that line ( drawen to the fide of the triangle ) is ( by the Corollary of the 12.0f the thirtenth ) the halfe of the line drawen from the centre to the circumference,that is, of the fide of the hexagon: Wherefore the refidue fhall be the halfe of the fide ofthe decagon . For the whole line is the halfe of the two fides, namely, of the fide of the hexagon, and of the fide of the decagon.But of the fide of a decagon and of an hexagon taken together,the greater fegment is the fide of the expen (by the 9.of the thirtenth) . Wherefore the greater fegment of their halfes fhall be the halfe of the hexagon,by the ry.of the fift: which halfe is the perpendicular line draw- en from the centre to the fide or the triangle, by the Corollary of the 12.0f the thirtenth. q Ube fecond Propofition. vd f two right lines be diutded by an extreme and meane proportion: they The». propos hall be dinided into the felfe fame proportions. fition after € ampane, vnto the line AB in the point C.So that let the line DE be to DC the greater part.as Demonfirats~ the greater part DC is to CE the leffe part,by the 3 definitio of the fixt.But(by fuppofiti- 9% leading to BR an impofsibi-~ rightline DE is diuided by an extreme A i—$——_—_——_+—________.B litic. and meane proportion in two pointes C | andZ . But the proportion of DE toDC :: CA E the leffe line, isgreater then the propor- tion of the fame D Eto D Z the greater line,by the 2.part of the 8.of the fift. ButasD E is to DC, fois D C.to.C E: Wherefore the proportion of DG to CE, is greater then the proportion of DZ to ZE. And forafmuch as D Z is greater then D C, the proportion of D Z to C E fhall be greater then the pro- portion of DC toC E, by the 8.of the fift . Wherefore the proportion of D Z to CE, is much greater then the proportion of D Z to Z E. Wherefore one and the felfe fame magnitude,namely, D Z, hath to CE the greater line, a greater proportion then it hath to ZE the leffe line, contrary to the fecond part ofthe 8.of the fift: whichisimpoffible. Wherfore the rightlines A B & D E, are not cut ynlike. Wher- fore they are cutlike,and into the felfe fame i oe alee the fame demonftration alfo will ferue,if the point C fall in any other place. For alwaies fome one of them fhall be the greater . If therefore two right lines be cut by an extreme and meane proportion : they fhall be cut into the felfe fame propor- tions : which Was required to be proued. q Lhe third Propofition. If in a circle be defcribed an equilater Pentagon : the fquares made of the Tp. i jrebes fide of the Pentagon and of the line which fubtendeth two fides of the fition afer | ; Rr. Pentagon, Campane, ae s\\Lbefourtenth Booke . Pentagon, thefe two fquares (fay) taken together, are quintuple to the Square of the line drawen from the centre of the circle to the cercuference. - ==] Vppole that in the circle BC G the fide ofa \j| Pentagon be BG : and ler the line B C fub- | tend two fides thereof, And let the line BG be diuided into two equall partes bya right line drawen from the centre D : namely, by the diameter C DE produced to the point Z. Anddrawe the rightline BZ. Then I fay,that the right lines BC and BG, arein power quintuple tothe rightline D Z, which is drawen fromthe centre to the circumference. Demonfira- For forafmuch as (by tlre 47. 0f the firft ) the fquares of the lmes'C Band BZ, are equall to the fquare of the! diameter CZ: therefore they are quadruple tothe {quareof the line D Z, by the 20.o0f the fixt ( for tHe line C Z is double to the line DZ}Wherefore the right lines CB,BZ,and ZDjarein power guintuple to the line Z D, But therightline B.G.con- - taineth in power the two lines B Zand Z.D,by the 10,0f.the thirtenth / ForD Z is the fide’of an hexagon, 8 BZ the fide | : ofa décagon. Wherefore thelines BC and B G (whole. powers are, equall to the powers of the lines ¢B,BZ,ZD )arein power quintuple to the line D Z.. Iftherefore in a circle be defcribed an equilater Pentagon : the fquares made of the fide of the Pentagon and of the line which fubtendeth two fides of che Pentacon, thefe two {quares (fay ) taken togeshier, are quintuple to the {quare of the line drawen from the centre of the circle to the circumference. Sas Conftre don, Star ne 2 ~ See = aS = — ee > Seat i £104. ge ~~ 2 ey meio Bae Ol em ~ ae ems wat team Mie ee eee ene rng - eT cen ve Som nals = By: iy” UES lie | eae | OR : hat Hf ha q, A Corollary. Ifa Cube anda Doderahedyon be contained in one and the felfe fame This Cortilae gods “giana AMS fees Cet ion ht ieee ry Canpane Sphere the fide of the Cube, and the fide tof the Dodecahedron, are in power alfobutteth quintuple to the line which is drawen from the centre of the circle which contats pe ee ip Ue th the Penta 1072 of the Dodecahedron . Forit was proued in the 17.0f the-thirtenth, his ase thatthe fide of the Gube fubtendethtwo fides of the Pentagon of the.Dodecahedron, where the fayd ep folides are contained in one and the felfe fame Sphere;..W herfore the fide of the Cube fubtending two fides ofthe Pentagon, and the fide of the {ame P entagonjare contained in one and the felfe fame circle. Wherctore, by chis bespo geen cy are in power quintuple tothe line whichis drawen from the cen- tre of the fame circle, which containeth the Rentagon,of the Dodecahedron. T he 4.Propofition..... Bp pesos ott ry Et tiat A rhe j-prope- One and the felfe fame circle containeth both the Pentagon of a Dodecas £409 apter | hee capes i. 4 ° bedron and the triangle of an Icofabedron defcribed in one and thefelfe (xmpane. Const; ion. m4 drawne fromthe centres of thofe circles to. the circumferences be LNand OK. ThenI PemonSire- PiGNe the circle, which containeth five angles of the Icofahedron, and the fide of the pentagon deferibed ; mas in a? of Euclides Elementesafter Flufas. Fol,4. 23. in that circle D Z G,namely the line ZG is fide ofthe Icofahedron defcribed in the Sphere,whofe diameter is the line A B: therefore the right lineZ G,is equal to the line MN,which was put to be the fide of the Icofahedré, or of his triagular bafe. Moreouer,by_the 17.0f the thirtenth, it was manifelt that the right line Fu(which fubtertdeth the angle of the pentagon of the Dodecahedron infcribed in the fore- fayde fphere) is the fide of the Cube, in- {cribedin thefelf fame fphere.(For vpon che angles ofthe cube,were made the an- gles of the Dodecahedron.) Wherefore the diameter A B is in power triple to F- H,the fide of the Cube (by the rs. of the thirrenth). Butthe fame line AB is (by {fuppofition) in power quintuple to the line C G. Wherefore fine {quares of the line C G,are equal to thre fquares of the line EH : (forecheis equal to:one and the felf fame fquare of the line AB), And forastnuche as EG the fide of the Deca- gon, cuttech the right line C G by.an-ex- treme and meane proportion (by the co- Ty roilary of the potas thirtenth).: Like- G wife.the line H.K,cutteth theline F H; E: the fide of the Cube by an extreeme and meane proportion (by the Co- | rollary of the 17.0f the thirtenth) : therfore the lines C G and F H, are deuided into the felf fame pro- portions, by the fecond of this booke:and the right lines C Tand E G,which are the greater fegmentes of oneand the felfe fame line C G,are equal: And forasmuche as fiue fquares of the line C G are equal to thre fquarts.of the lines FH : thetefore fiue {quares of the line GE, are equal tothre [quares of the line EK (for the lines G E and HK are the greater fegméts of the lines C G and F H); Wherefore fiue {quires ofthelines CG.& GEare equalto thre fquares of the lines FH & HK, by the 12 of the fifc. Bue vnto thefquaresof the linesC G and GE,jis equal thefquare of theline Z G, by the ro.of the thirtéth : and vito the line Z G the line M N was equal: wherfore fiue fquares of the line M N,afe equall to three {quare’s ofthelines F H,H K,.Burt the fquares of the lines F H and H K,are quintuple to the {quare of the line O K (which is drawne from the centre) by the third of this booke. Wherfore thre fquares of thé lites F Hand H K make rs.{quares of the line O K . And forasmuchas the {quare of the line M N is triple to the {quare of theline L N (which is drawne from the centre)by the 12.0f the thirtenth, ther- fore fiue {quares of the line M N are equal to 15. fquares of the line LN.But fiue {quares of the line MN are equal ynto thre {quares of the lines FHandH Kk. Wherefore one fquare of the line LN is equall to one fquare of the line OK (being eche the finetenth part of equal magnitudes) by the 15.of the fifeld, Whetfore the lines L N and O'K,which are drawne from the centers, are equal. Wherefore alfo the circles N R M.and F K H which aredefcribed of thofe lines, are equal. And thofe circles contayne (by fuppofition) the bafes of the Dodecahedron and of the Icofahedron defcribed in oneand the felfe fain {phere:Wherfore one and the felfe fame circle.8c. asin the propofition: which was required to beproucd. T he 5.Propofition. >. Ef in acirele be inferibed the pentagon of a Dodecahedron, and the triane son 1 gle of an Icofahedron ,and from the centre to one of theyr fides, be drawwne ~ a perpendicular line: T hat which is contained 30.times pnder the fide, 7 the perpendicular line falling vpon it, ts equal to the uperficies of that fos lide wpon whofe fide the perpendicular line falleth, 7A Vppofe that in the circle AG E,be defcribed the pentagon of a Dodecahedron, which lerbe AB G D E,and the triangle of an Icofahedron de{cribed in the fame fphere, which i étbe A EF H.And lerthe centre be the poyntC. From which draw perpendicularly the line. C Ito the fideot the Pentagon,and the lineC L to the fide of the triangle. Thent Gil fay chat the re@angle figure contained vnder the lines C I and G D 30.times, — ta RRr.ij. the This is the 68 and 7.propo~ fitions afier Campane. Conflructions Demon/jira- $60Me ' timesis equal to two pentagés. Wherfore that which This Covolla- ry Campane aifo addeth after the 7. propoftionét bis order. The 5+ pYopo~ fation after Campane. Coustrullion. Demonflra- bi0%. T he fourtenth Booke the fuperficies of the Dodecahedron : and that that : XC siovio Ra which is cétained ynder the lines C L & AF 30. times is equal to the fuperficies of the Icofahedr6 defcribed in the fame fphere.Draw thefe right lines CA,CF,CG and C D.Now forasmuchas thar which is cérained vn der the bafe G D & the altitude IC, is double to the triangle GCD, bythe 41.of the firft: And flue triangles like and equal to the triangle G C D do make the pen- tagon A B G DE of the Dodecahedron: wherfore that which is contained ynder the lines G Dand1C fiue is contained vnder the lines GD andIC 30: times is equal to the 12.pentagons, which containe the fuperfi- cies of the Dodecahedron. Againe that which is contained vnder the lynes C Land AF, is double to the triangle A C F: where- fore that which is contained vnder the lines C Land A F three times is equal to two fuche trianglesas A F- H is, which js one of the bafes of the Icofahedron (for the triangle A C F,is the third part of the triangle AF- Ha it is eafie to proue,by the 8.& 4.0f the firlt.) Wherfore thac which is eétained vnder the lines CL and A F,30 times times, 1s equall to 20.fuch triangles as A F H is; which containe the fuperficies of the Icofahedron . And forasmuch as one and the felfe fame phere containeth the Dodecahedron of this pentagon,and the Icofahedron of this triangle (by the 4.cf this booke : ) and the line CL falleth per- pendicularly vpon the fide of the Icofahedron, and the lire CI vpon the fide of the Dodecahedron : that which is 30.times contained vnder the fide, and the perpendicular line falling vpon it, is equal to the fuperficies of that folide,vpon whofe fide the perpendcular falleth. If therefore in a circle &c.asin the propofition: which was required to be demonftrated. A Corollary. Phe fuperficieces of a Dodecahedron andof an Icofabedron defcribed in one and the felfe Jame {phere , are the one to the other ,as that whichis contained 'vn der the fide of the one and the perpendicular line drawne ynto it from the cene tre of bis bafe, to that which is contained vnder the fide of the other, and the perpendicular line dravwne to it from the centre of bis bafe. Foras thistye timesisto thirty times,fo is once to once by the 15.0f the fifth. T he 6.Propifition. The fuperficies of a Dodecabedron, is to the Juperficies of an Tcofahee dron defcribed in one and the felfe fame fphere, in that proportion, that the (ide of the Cube is to the fide of the Icofabedron contained in the felf fame fphere. a Vppofe that there beacircle A B G,& init ( bythe 4.of this boke) let there be infcribed the | eg Gg. fides of a Do decahedron and of an Icofahedroncontained in one and the {elfe fame fphere. pap)! And let the fide‘of the Dodecahedron be A G,ad the fide of the Icofahedron be D G. And —e=.* let the centre be the poynt E:from which drawynto thofe fides,perpendicular lines EI and EZ.And produce the line E I to the poynt B, and draw the line B G. And let ibe fide of the cube con- tained in the felffame {phere be G C. Then I fay that the faperficies of the Dodecahedron is to the fu- perficies of che Icofahedron,as the line C G,is tothe lineG D. For forasmuche as the line EI being diuided by anextreme and meane proportion, the greaterfegment therof fhall be the line E Z, by the corollary of the firtt of this booke: and the line CG beirg diuided by an extreme and meane pro- panes » his greater fegment is the line AG, by the corollary of the 17. of the thirtenth: Where« ore the tight lines EX and CG are cut proportionally by the fecond of this booke. Wherefore as the line C G,is to the line A G, {0 is the line Eltoche line EZ. Wherefore that which is con- tained vnder the extreamesC Gand EZ, is equall to that whichis contayned ynder the meanes A Gand E I.(by the 16,0f the fixth.) Butas that which iscontained vnder the lines C Gand EZ isto that which is contained vnder the lines D G and E Z »{0(by the firit of the fixth) is the lineC G to the line of Euclides Elementesafter Flufas. Fol.ara. {ine DG, for both thofe parallelogrames hanconeand the felfe fame altitude,namely the line EZ. Whuirfore as that which is contained vnder the lines E land AG (which is proued equalto that which is contained vider the lines C Gand EZ) is to that which is contained vnder the lines D G) and\E/Z, fo is the-line.C G to neline DG. Butas that which is contained vnder the lings Eland AG is to that which is contained vader-the lines } Gand Exz, 2 | fo (by the corollary of the former propofitien) is the fu- perficies of thé Dodecahedron, to the fuperfties of the I- cofahedran..Wherfore as the-fuperficiés of thr Dodecaheé- dron is to the fuperficies of the Icofahedron,o is C G the fide of the cube,to G D the fide of the Icofaledron. The fuperficies therefore of a Dodecahedron is te the fuperfi- cies.&c. asin the propofition,; which was required to be proued, | An affumpt. Ihe Pentagon of a Dodecahetron,is equall to that-~which is contained Yne der the perpendicular line whrh falleth ypon the bafe of the triangle of the Icofabedron and fiue fixth paites of the fide of the cube , the fayd three fo- lides being defcribed in one and the felfe fame {phere. Suppofe that in thecircle A BE G,the sentagon of a Dodecahedron be AB CIG, and let two fides thereof A B and A G be fubtended of tle right line BG. And let the triangle of the Icofahedron infcribed in the felfe fame {phere , by the 4.olthis booke,be A FH. And let the centre of the circle be the poynt D,and let the diameter be AD E, citing F H,tne fide of the triangle in the poynt Z,and cut- ting the line B Gin the poyntK.. And draw che rightline BD. And fromthe rightline K G cut ofa third pare T.G;by the 9.of the fixth.Now the: the line BG fubtending two fides of the Dodecahedron fhalbe the fide of the cube infcribed in the fane {phere, by the 17. of the thirtenth: and the triangle of the Icofahedron of the fame {phere fhalbe A? H by the 4. of this booke . And the line A Z which paf feth by thecentre D fhall fa!i perpendiculary vpon the fide of the triangle.For forafmuchas the anges G A E 8 B A Eare equall ( by the 27, of the third , foirhey are fee vpon equall citcumferences): therefore the bfes BK and K Gare(by the 4.of the firft equall. Whereore the line B T contayneths.fixth partes.of the line B G. Then I fay that that which is contayned vnder the line; A Zand B- Tis equall to. the pentagon A.B.C 1G, For hrafmuch as theline A Z is fefqnialter to the line A.D (fo: the line D- E is diuided into.two equall partes in the poynt Z, by the corollary ofthe 12. of the thirtenth ) . Likevife by con- ftruction,the line K G is fefquialter to the lincK T: there- fore as the line:A Z is to the Jine A D, fois thrlinéK Gro the line KT .. Wherefore that which is contiyned vnder the extreames A Zand KT,,is equali to that which is con- tayned vnderthe meanes A D and KG, by tle 16. of the fixth. But vngo the line K G is the line B K preued equall. Wherefore that which is contayned vnder tle lines A Z and K T is equall to that which is contayned ‘nder the lines A D and BK . Butthat which is contayned wnder the lines A Dand B Kis ( by the qr. of he firlt ) double to thetriangle ABD . Wherefore thar which is contayned vnder the lines A Z and I Tis double to the fame triangle A B D . And forafmuch as the pentagon AB CIG contayneth fine rianglesequallto the triangle A BD, and that which is contayned vnder the lines A Z and K T contarneth two {uch triangles: therefore the pentagon ABCI Gis duple fefquialter to the reftangle paralldogramme contayned ynder the lines A Z and K T. And forafmuch as by conitrution the line B T is duple fefquialter to the right line KT : but by 1. of the fixth that which is céteyned vnder the linesA Z and B T is to that which is contayned ynder the lines AjZand K T,as the bafeB Tis to rhe bafe K 1: therefore that which is contayned vnder thelines A Z and B T is duple fefquialter to that which is iontayned vndertheline A 7 & KT. But vnto that which is contayned vnder the lines A Z and K T th: pentagon ABCIG is proued duple fefquialter. Wher fore the pentagon A B C1G of the Dodecaledron is equall to that which is contayned vnder the per- 2 RRr.iii. pendiculas This Afumpt Campane aljo hath after the 8. propefition, in bas crder. ConSrucion. Demonfira- ¢40%. chk Phe fourtenth Book, 0: The 9. prope- fition after 5 Be : : Campane. hath to the line contayning in power the whole andthe leffe segment : the: PSE a = ae oe raz Confirnition. 3 a pz H :and(by the fecond of the fame)an equilater triangle ABI. Andlet the centre theteof be 3B Dmone proportion in the poynt D( by fh 39,.of the fixth ). And ler the line M L contaynein eoment B=" B D/(by thecorollary of the 13.0f the tenth ).And draw the rivht line BEvubséding che angle of the\penragon, which ghall be the fide of the cube-( by the coroilary.of tne 12. . . ofthe thircenth) ': and the ine B Pthall'be the fide UF tle? Lcofahedron., andthe line RZ the fide of the Dodecahee. . Ay? \ «.' dron by the. of this booke . Then I fay that B E the fide & = ofthecube isto BI the fide-of the Icofthedtony as‘thes ‘W's | line contayning in power fags? BG : Pa is cies line.contayning,in power,.the lines G Band BD, Fortor- _ SD sara G Dersomftre- ree ae ee Uby tie 1asok the thitonth ) the line BEis ia | FOB > powertriple to theline BG ,and.( by the 4. of the fame) | che {quares of the line GB &ZBD are triple to the fquare It EE ise \ ofthe line GD... Wherefore ( by the rs. oftheffth) the . A fquare of the line B Lis to.the {quares of thelines GB & é BBok \ ‘ BDnamely,triple to triple) qs the fquare of the line BG E a Siiiadios Actor C 0h) is.to Shedquare of the line G D(namely,as one isto one). er ede Butas the {quare of the line BG 1s to the fquare of the DAR Yohei 30s) 28) io soto msts GD, fois the fquare ofthe line B E to the {quate of the ‘lp 3s Sar | 7 het line BZ. ( For the lines BG, GD, andBE, BZ arein oneand the rage sale » by the fecond oF this booke. Eor B Zis the greater fegment of the line B E,by the corollary of the 17. of the chirtenth }. W herefore the fquare of the line B E is to the fquare of the line BZ, as the fquare of the line B listo the {quafesof thedines B G andB D.. Wherefore alternately the {quare ofthe line BE isto the fquare of the fine B I,as the fquare‘of the line B Z is to the {quares of the lines'G B and B D. But the fyuare of the line B Z is equall to the fquares of the linesB G and GD ( by the ro.0f the thirtenth ) . For the line B G is equallto the fide of the hexagon,and the line GD to the fide of the decagon,by the corollary of che g,0fthefame . Wherefore the {quares of the lines B Gand GD, areto the fquares of the lines GB and & D, 4s the fquare of the line B E is to the fquare of the litieBT . But the linezB contayneth in power the linesB GandG D,and the line M L contayneth in power the'lines G B and B D by conftruc- tionk! Wherefore as theline Z B (‘wnich contayneth in power the whole line B G and the greater feg- ment G D)is to the line M L ( which contayneth in in power the whole line G Band the feffe fegment BD )fo isBE the fide ofthe cube to B I the fide of the Icofahedron, by the 22.of the fixth “Wherefore a right line diuided by an extreame and meane proportion : what proportion the line contayningin owerthe whole lineand the greater fegment, hath to the line contayning in power the whole line and theleffe feement: the fame atk the fide of the cube to the fide of the Icofahedron cérayned in one andthe fame {phere -which was.required to be proued. iG fen ytt) PSS es Ss is = = wee eS Se ——— . = ies, a ee —s as = = == : “eee eae ks ae SS ee a : i ~ oa ee gn a — ne eae v- - - ~ = ; ees eee _ —~~ sy ease eg OF bare Peon ANON: wi, : . Forafmuch of Euclides-Elementes.after Fluffas. Fol.425. & R24» Orafmiich as in the 4.0f this booke, it hath: bene proued, thatoneand the {elf famecir- Ex) he NA cle containeth both thésriangle of an icofahedron, and thé pentagon of a Dodecahe~ i. =“N tes =| RAL a “sp ey ae 4 Von we dron defcribed in one and the felfe fame Sphere : Wherefore the circles, which cétaine “Wi PRA yy) thofe bafes, being equall,the perpendiculars alfo which are drawen from the centre of Melt ite the Sphere to thofe cireles,thall be equal! (by the Corollary of theAffumpt of the 16 of OS Soe") the rwelfth ) .And therefore the Pyramids fet vpon the bafes of thofe folides haue one "SS and the felfe fame altitude: For.the alritudesof thofe Pyramids concurfé tn the centre. Whterefore they are in proportion as their’bafes are, by thes and 6.of the twelfth . And therefore the pyramids which compofe the Dodecahedion, are to thepyranuds which compofe ‘théIcofahedron, as the'bafes are, which bafes are the fuperfisiects of thof¢ folides.. Wherefore their folides are the one to the orher,as their fuperficieces. are . But the fuperficiés of the Dodecahedron is to the fuperficies of the Icofahedron, as the fide of the cube is ro the fide of the Icofahedron, by rhe.6:of this booke. W her- fore by the 11.0f the fifth, as the folide of the Dodecahedron is to the folide of the Icofahtedron, fo is the fide of the cube to the fide of the Icofthedron, all the faid folides being infcribed in one and the felfe fame Sphere. Wherefore the folide ofa Dodecahtedron is to the folide of an Icofahedron : as the fide of acubé is to the fide of an Icofahedron, all thofe folides being deferibed in one and the felf fame Sphere which was required to be prowed. ss A Corollary. T be folide of 4 Dodecahedron is to the folide of an Icofahedron, as the faz This Corelle perficieces of the one are-to the fuperficieces of the other being defcribed in one "I¥*709- propofition and the felfe fame Sphere > Namely, as the fide of the cube isto the fide of the Icofahedron, after Cam-~ as was before manifeft : for they,avé refolued into pyramids-of one and the felfe fame altitude. ‘pane. qt he 9. Propofition. If the fide of an equilater triangle be rationall: the fuperficies [hall be irra- The ik: pee tionall, of that kinde whichis called Mediall. pofition after Campane. =S 5) Vppofe that AB G be an equilatertrianglesiand from the point A draw vnro the fide BG \X jl a perpendicular line A D : and let the line\A B be rationall , Then I fay, that the fuperfi- Con fruttions | cies AB G is mediall . Forafmuch as the line AB is in power fefquitettiato theline AD ~ Mo S20 \ 71] (by the Corollary of the 11.0f the thirtenth ),4 of what parses thelline AB.containethin qs, KS YX) power 2, of the fame partes the line, AD containeth in powér_9: wherefore the refi- Demonftra~ ve due B D containeth inpower of the famé partes 3. (Eor the line AB c6tainethin power 70%, the lines A D and B D, by the 47.0f the firft) . Wherfore the lines A Dand DB are rationa!l and, commenfurable. to the rational line fet A B, by the ¢. of the tenth , Bud forafmnch A as the power ofthe line A D isto the power ofthe ling DB in that proportion that 9.a fquare numbers to 3.2 number not{quare : therfore thcy are notin the proportiai of fquares § num bers, by the Corollary of the 25.of the eight. And ther- fore they are not commenturable in lenéth, by the 9. of the tenth . Wherefore that which is contained vnder thelinés AD and DB, which are rationall lines commenfurable in power onely, is mediall, by the 22. of the tenth . Butthat which is contained vnder the ines A D and DB, fs double to the triangle A B D, by the 41.of the firit. Wherefore that which is contained ynder the lines AD and D B,is equall to the whole triangle AB G ( which is-double to the triangle ABD,by the 1.0f the fixt). WhereforethetriangleABG 5 is Mediall . lf rhertore the fide ofan equilater triangle be ra- , | : tionall : Soest cics fhall be srrationall, of that. kinde which is-called Mediall ‘ which was required to be proued. 3 G wy * A Corolfiry. ? If an Offohedron and a Vetrahedron ‘be inferibed ina Sphere whofe dias The1z.prope- meter is rational : their fuperfi cieces fha fi be medtall:, For thofe fuperficieces con fifte fition after of equilater triangles, whofe fides are. commen{urable tote diameter which is rationall, by the 13 2nd Campane, 3 14.08 pofition after Campanee Demon fra- Eon of the first part. Dewar Tretia oi of the fem cond part, ‘The 17.proe pofition after © Campane, Fir partof the consirne= thor, Firkt part of she Demon- firations fubfefquioGauato. the fame fide. ThéI fay,that the Tetrahedron ACD ference ) is triple to the line HT: and therefore the whole line AT is T he fourtenth Booke r4.0f the thirtenth, and therefore they are rationall But they are commenfurable in power Onely to the perpendicular line, and therefore they containe a mediall triangle, as it was before manifett. q Ibe 10. Propofition. If aT etrabedron and an Offohedron be infcribed in one and the felf fame S phere: the bafe of the Tetrahedron [hall be fefquitertia to the bafe of the Oétohedron and the fuperficteces of the Offohedron [hall be fefquialtera to the [uperficieces of the I etrabedron. | } so- Otaimuch as the diameter of the Sphere is in power fefquialtera to the fide of the Tetrahe- SS %0, dron (by the 13.0fthe thirtenth) and the fame diameter is in power duple to the fide of the | Lie Otohedron (by the 14.0f the fame booke) : therefore of what partes the diameter contai- fl \>a®, Y> neth in power fixe,of the fame the fide of the Tetrahedron contayneth in power 4,and of the MOLES fame the fide of the O@ohedron containeth in power 3. Wherefore the power of the fide of the Tetrahedron, is to the power of the fide of rhe Oftohedron in the fame proportion that 4.is to Pe which is fefquitertia . And like triangles (which are the bafes of the folides) defcribed of thofe fides, fisall haue the one to the other the &me proportion rhat the {quares made of thofe fides haue.For both the triangles are the one to the other, and alfo the {quares are the one to the other, on of that in which the fides are, by the 20.0f the fixth . Wherefore of what partes one bafe ofthe Te- trahedrou was 4: ofthe fame are fower bafes of the Tetrahedron 1¢ : likewife of what partes of the {ame one bafe of the O@ohedron was 3: of the fameare 8.bafes of the Oétohedron 24, Wherfore the bafes of rhe O&ohedron are to the bafes of the Tetrahedron,in that proportion that 24.is to 16: which is fefquialtera . Iftherefore a Tetrahedron and an Otohedron be infcribed in one and the felfe fame Sphere : the bafe of the Tetrahedron fhall be fefquitertia to the bafe of the O@ohedron ,and the fuper- ces of the Tetrahedron: which was re- in double proporti- ficieces of the OGtohedron hall be fefquialtera to the fuperficie quired to be proned. q Lhe 11. Propofition. A LF etrabedron is toan Offohedron infcribed in one and the ‘felfe fame Sphere, in proportion as the reffane gle parallelograme contained bnder _ the line, which containeth in power 27. fixty fower partes of the fide of the I etrahedron, bnder the line which is [ubfe/quioltaua tothe fame fide of the Tetrahedron, is'to the J quare of the diameter of the Sphere. e~gEt vs fuppofea Sphere, whofe diame~ y Wy terlet be theline AB,andlet the cen- Aa Pex A& tre be the point H . And init let there be inferibed a : P f V4 Tetrahedron ADC, and'an O@ohedron AE KBG. = } Ealen (ed Andlet the line NL containein power — of AC the (PAIN fide of the Tetrahedron.Andlet the line ML bein légeh ee is to the Oftohedron AEB, as the rectangle parallelogramnte contay- ned vnder the lines N Land LM, is to the {quare of the line A B.Foraf- much as the line drawen fré the angle A by the centre H perpédicular- ly vpon the bafe of the Tetrahedron, falleth ypon the céter T of the cir- cle which containeth that bafe,and makerhhe right line H T the fixth part of the diameter'A B (by the Corollary of the 13.0f the thirtenth) : therefore the line H A ( which is drawen from the centre to the circ tQ { 4 a <4 : a wen 4 of Euclides Elettentes after Flifias. Fol.4r6. to the line.A Hyas 4,is to 3. Letthe Tetrahedron A D C be cut bya plaine G H K pafling bythe certre H, and being parallel vnto the bafe D T C, by the Corollary of the 15.0f the eleuenth. Now. then the triangle A D C of the Tetrahedron, fhall' be cut by the right line K G, which is parallel to the litte D €, by the 16.0f the eleuenth: Whetfore asthe line AT is to the liné A -H;,fois the line A C to the litie AG (by the 2:0f the fixth) . Wherefore the line A C isto the line A G fefquitertia, tHatis,as 4. to 3. And forafmuch as the triangles ADC, A G;and the rett whith are cut by the plaine KH G, ate like the one to the other; by the 5.of the fixth:! the pyramids A D'C and AK G; fhall be-like the one to the o- ther, by the7.definition of the eleuenth? Wherefore they are in triple proportion of that in which the fides AC aud A Gare, by the 8. of the twelfth . But the proportion of the fides A C to A G is, asthe proportion of 4.to 3 . Now then, if, by thez .of the eight, ye finde out 4.0f the left numbers in continu- all proportion, and 1 that preportion that 4.is to.3.: whichthall be 64.48.36. and 27 ¢ it is manifett, by the 15. definition of tlic fifth, that the extremes 6q:to 27. afe in triple proportion of that in which the proportion geulen 4.to 3.is 1 Or the quantitie of the proportion of 4. to 3. (which is 1.and — being twife multiplied into it felfe, there fhall be produced the proportion of 64.to 27. Wherefore the Pyramis or Tetrahedron A D Cis to the pyramis AK G, as 64.is to 27. :.which is wiple'to. the propor- tion of 4.to 3. And foraf{much as the line A Cis vnto the line A G in length fefquitertia: of what partes the lire AC containeth in powers4 : of the fame partes doth the line A G containe im power 36. For (by the 2.of the fixth) the proportion of the powers or {quares, is duple to the proportion of the fides which are as 64.isto 48. Now then ypon the line R $ which let be equall to the line’A G y let there be an equilatersriangle Q RS defcribed(by the firft of the firft),And from the angle Q_, draw to the bafe RS a perpefidicular line Q T.And extend the line R S to the poynt X.And as 27.1s to 64. (fo by the corollary of the 6.of the tenth )let the line RS be to the line R X. And diuide the line R X into two equall partes in the poynt V, and draw the line Q V . And forafmuich asthe lire RS 1s equall to the line A G, of what partes the line AC contayneth in power 64. of the fame part the line RS contayneth in power 36.for it is proued that theline A G contayneth in power 36. of thofe:partes : And of what partes the line RS contayneth in power 26 of chefame partes the line: Q Tscontayneth in power 27,-by the corollary.of the «2. of the thirtenth. Wherfore of what partes the line AC contayneth in power 64, of the fame parts the line Q- T contayneth in power 27 . Wherefore theright line Q T thall be.equall to the right line'L N by fup- pofition.Agayné forafmuch as the line R'S is put equall to the line A G: and of what partes the line R- 5 contayneth in length 27. of the fame partsisthe line RX put tocontaynein length 64. and of what partes the line R Xcontayneth in | length ¢4. of te fame the lineAx 7 o C(which is -in length fefquitertia to the line A G or: R$) contay- neth 36. Wherefore the lineR V ( which is the halfe of the line R- X)containeth in légth of the fame partes 32. of which the line AC contayned in length 36. Where- fore the line RV isto theline A- ° C fubfefquioctaua: and therefore iS pT ee the line RV is equall to the line ry > fr LM-which is alfo fubfefquiottana | = to the fame line AC . And forafmuch as the line NL is equall to the line Q T , and the line LM to the line R V (as before hath bene proued:) the reG@angle parallelogramme contayned vnder the lines Q. F and RV, fhall be equall tothe reCtangle parallelogramme, contayned ynder the line N L whichis in power = to thefide A C,and vnder the line L M,which is in length fub{efquiottaua to the fame fide A- C.But that which is contayned vnder the lines Q T-and R Vis double tothe triangle Q V R by the 4. of the firit:and tothe fame triangle QV Ris the-triangle Q.X R duple by the firftofthe fixth. Where« fore the whole triangle Q XR is equall to that which is contayned vnderthe lines Q TandRV, and therefore is equall to the parallelogramme M N, And forafinuch as the line R X by fuppofition céntay- neth in length 64.0f thofe partes of which the line RS contayneth 27: and the triangles Q R X,and Q- RS arc, by the firft of the fixth,in the proportion of their bafes, that is,as64.1s to 27: but as 64.is to 27. fois the pyramis or tetrahedron AD C to the pyramis A K G: wherefore as the parallelogramme N M or thetriangle Q R X,isto the triangle QRS, fo is the pyramis A DC tothe pyramis AK G. And fors afmuch as the femidiameter A H ‘is the altitude of the pyramis.AK G yandalfo of the two equall and like pyramids of the o€tohedron which haue their common bafe in the fquare of the o¢tohedron ( by the corollary of the 14.0f the thirteenth): therefore as the bafe of the pyramis_ A K G (whichis the tri- angle Q RS)is to two {quares of the odohedron, that is,to the fquare of the diameter A B,which is e- qualt to thofe fquares(by the 47.0f the firft),fo is the pyramis AK G to the o¢tohedron A EB, by the 6. of the twelfth.And forafmuch as the parallelogramme M Nis to the bafe Q R S,as the pyrantis A DC isto the pyramis A KoG,and the bafe Q KS is to the (quare of the line BE , as the pyraniis AK Gis to the oétohedron A E B: therefore by proportion of equality taking away the meaues (by the 22. of thé fifth as the parallelogramme N M is to the {quare of the line B E , fo is the pyramis ADC as ofte- $Ss.1. edron Second pats of thecun- ° firnét tou. Second park of the De- 1; Ons ratiONe ~ & ee - - / = ~ ro | é ~y o ~ > See TS ee ee 4 = = Sa =e > ee Pe a =e ee See = ce ‘ : = r : : rsa > = : _gund + _ 4 - ee — i See ee pr me 3 ——_—— “> — a - ~ - - > ee gre - . aS. ae ans re =e nn om er : SP a = —~ ~ ~ < a: -— ae -_ set al. a rw - a . = —: oe i ET het ene =? a : Of ae AE = ns A ———— — : — - = ee ~ Sn AA “ - _—— - —_— = 3 a ; ——e “ —- 3 = ae eras ere i = r ne Ra . The 18. pro pofition after Campane. Demo sétre- fon 6 f the jorst part, Demeufirs- sion of the fecond part, The Corofla- ry of thes, propofitzo after Cam- Panes a", ™ T be fourtenth Booke Wedron A EB inferibed in oné and the felfe fame {phere . But the parallelogramme NM is contayned vider the line N L which by fuppofition is in power to AC the fide of the tetrahedron A DC , and vnder the line L M which is alfo by fuppofition in length meoeiauiortees to the fame line AC, Wher- fore a tetrahedron & an oCtohedron inicribed in one and the felfe fame {phere, are in proportid,as the teGangle parallclogramme contayned vnder the line, which contayneth in power 27.fixty fower parte of the fide of the Tetrahedron,and ynder the line which is fubfefquiogtaua to the fame fide of the Te- erahedron,is to the {quare of the diameter of the {phere: which. was required to be proued. q Ube 12.Propofition. If a cube be contayned in afphere:the fquare of the diameter doubled,is e quail to allthe {uperficieces of the cube taken to gether. And a perpendicue tar line drawne from the centre of the [phere to any bafe of the cube , ise gualito halfe the fide of the cube. 7) Or forafmuch as(by the 15 .of the thireenth) the diameter of the Iphere is in power eS 1 a Foe triple to chefide of the cube: therefore the {quare of the diameter doubled is fexa oY ¥| tuple to the bafe of rhe fame cube. But the fextuple of the power of one of the fides NS As contayneth the whole fuperficies of theicube . For the cube is compofed of fixe SS aN \{quarefaperficieces( by the2z1,diffaition ef the eleaenth) whofe fides therefore are YSN equal: wherefore the fquare of the diameter doubled is equall to the whole fuper- acs Ss ficies of the cube . And forafmuch as the diameter of the cube , and the line which falleth perpendicularly yp6 the oppofite bafes of the cube, do cur the one the other inte two equall partes in the centre of the {phere which containeth the cube (by the z.corollary of the 15.0f the thirtenth)and the whole right line which coupleth the centres of the oppofite bafes,is equall to the fide of the cube by the 33 .of the firit,for it coupleth the equall and parallel femidiameters of the bafes : therefore the halfe thereof fhall be equall to the halfe of the fide of the cube by the 1s. of the fitch if therefore a cube be contayned in a {phere : the {quare of the diameter doubledis equall to all he fuperfcieces of the cube taken together. Anda perpendicular line drawne from the centre of the Sphere to any bafe of the cube,is equall to halfe the fide of the cube: which was required to be proued. @ A Corollary. If. two thirds of the porwer of the diameter of the [phere be multiplyed into the perpendicular line equall to halfe the fade of the cube there fhall be produced afolide equallto the folide of the cube.For itis before manifett that two third partes of the power of the diameter of the {phere are equall to two bafes of the cube. If therefore ynto eche of thofe two thitds be applyed halfe the altitude of the cube , they thall make eche of thofe folides equall to halfe of che cube, by the 3 1-ofthe eleuenth:for they haue equall bafes. Wherefore two of thole folides are eqnall to the whole cube. i _.. ‘You thall ynderftand( gentle reader)that Campane in his 14. booke of Euclides Ele- mentes hath 13, propofitios with divers corollaries following of them.Some of which propofitions and corollarics I haue before in the twelfth and thirtenth bookes added out of Piuflas as corollaries(which thing alfo [have noted on the fide of thofe corol- laries, namely, with what propofition or corollary of Campanes 14. booke they doo agree). The reft of his 18. propofitions and corollaries are contained in the twelue for- mer propofitions and corollaries of this 14. booke after Fluflas: where ye may fee on the fide.of eche propofition and corollary with what propofition and corollary of Casipanes they.agree. But the eight propofitions following together with their corol- jaries,Flufias hath added of him fifo, as he him felfe affirmeth. aust = of Euchdes Elementes after Flapas. Fol.427. The: 3.Propofition. One and the elf fame circle containeth both the [quare of acube, and the 4 triangle of an O¢gohedron defcribed in one and the felfe fame [phere. CO Vppofe that there bea cube AB G, and an O&ohedron D EB deferibed in ore and the lelte fame{phere whofe diameter let be A Byor DH/And let the lines. drawne: from the x cétres(that is the femidiameters of the circles which'¢téaine the bafes of thofe folides) jy, fbe CA and ID.Then I faythac the linesC A and 1 Dare equal .“Forafmucheas A B the eee g diameter of the {phere which containeth'the cube, is in power triple to B G the fide of the cube (by the 15.of the thirtenth) vnto which fide,A-G the diameter of the bafe of the cube, isin Dersin tras power double (by the 47.0f the firft): which line AG 'is‘alfo the diameter of the circle,which cétai-: *on. neth the bafe (by the 9. of the fourth: )therfore A B the diameter of the {phere is in power fefquialter to the line A G:namely,of what partes the line A B,containeth in power 12.0f the fame the line A G fh al containe in pow- srs er. And therfore the _A D right line A C whiche a is drawn from the cé- treof thecircle tothe circumference,contel- neth in power of the fame partes 2. Where- fore the diameter of the {phereis in power fextuple to the lyne whichis drawne from the centre to the cir- cumference of the cir- a cle whiche containcth B the fquare of the cube But the Diameter of ’ the'félfe fame Sphere whych containeth the H OGohetees , is one ‘ dnd the felfe fame with the diameter of the cube, namely, DH, is equa ; diameter is alfo the diameter of the {quare which is made of thé fides of ee Site the faide diameter is in power double to the fide of the fame OGohedron, by the 14.0f the eebith; Butthe fide D Fis in power triple to the line drawne from the centre to the circumference of the cir cle:which containeth the triangle of the oftohedron(namiely to the line I D)by the r2.0f the thirtenth Wherfore the felfe fame diameter AB or DH, which was in power fextuple to the line drawne fom the centre to the circumference of the circle which containeth thefquare of the cube; is alfo fextuple to the line I D drawne from the centre to the circumference of the circle, which containéth the nie ea gle of the Octohedron : Wherefore the lines drawne from the centres of the circles to the citcumfe rences _— = an bie bafes os —— and of the oftohedron are equal. And therfore the Birrlae are equal, by the firl nition ofthe third. Wherfore onéan ; in the propofition: which was required to be proued. tem pabeet ee-n yi ee _ _ a TS aes. Pree —— - ee ee J SS SA Sa = A eRe ee <= : ; Sot Ss = —-—- Aa ; } } il t a eS a oe a ee a, Sa ow o-- -— — ae ow SS

SS = a = z = — = ~ bens — - a anor ye SPS. “s Se 2 ¥ > - — = = : = tah 2; oe = me . > = y= a * + = — C. J ™% - . ~<= = i-<—- — ee aa = . — = — : = ——s = o _ ns qe > 2 r- a S —— 2 --—— —— + Con fraltion. Demon ra- EEGs Conflraction. Demaonftra- . tion. Sep \ |. DbefourtenthBooke soyned together by an extremeand méane proportion:the fide of the Icae —. fahedron contained in the fame [phere yfralbe duple to the lefs e fegment. Ec A B the fide of the O&ohedron AB G contaitie in power the two lines C and H,which let haue that proportion thatthe whole hath'to the greater fegment ( by the corollarye of the fit propofition.added by Fiuja after the lat propofitié of the fixth booke) And let the Icofahedron contained in:the fame {phere be D EF, whofe fide let be D E, and let the right eZ) line fabtending the angle of the pentagon made of the fides of the Icofahedron be theline EF.Then I fay that the fide ED is in power double to the line H the lefle of thofe fegmentes. Foraf- much as by that which was demonttrated in the 15 of this booke, itwas manifeft that E D the fide of the Icofahedron is the greaterfegment of the line E F,and thatthe.diameter DF. containeth in power the two lines EDand E F,namely, the wholeand the greater {egment: but by fuppofition the fide AB cétaineth m po- / pe wer-thetwo lines la cals of C & Hioined to- tins getheria the felf fame proportid. Wherefore the | ry So ] And f E ine H. Anadfor- | afmuche as the | | | SEE, Hi line D F containethin power the two lines ED and E:Byand theline A Bcontaineth in power the two lines C and H®: theretore the {quares of the lines B Fand ED are to the {quare of the line D F, as the line EF isto the line) ED, :as: the 4quares of thelines Cand Hto the {quare of the line A B.And alternately, the {quares of the lines EE and ED,are to thefquares of thelines C and H,as the {quate of theline D F is to the pe of the line line ED,is to the line Cyis to the - line H, by thea. of thisbekesAnd alternatly bythe a6io¢the fiueth, the:line EF is to theline-C,as-the A B.But DF the diameter is (by thé 14.0f the thirtenth) in powerdouble to A B the fide of the oGo~ hedron inicribed (by fuppofition) in the fame{phere. Waesiiete the {quares of the lines EF “a f D are dowble to the fquares of the lines C and:-H.And therfore one {quare of the line ED is double to il {quare‘of the line H by the 12.0f the fifth. Wherfore ED the fide of the Icofahedron is in power duple to the line H,which is the leffe fegment. If therfore the power of the fide ofan oGohedron be ex ref- fed by two rightlines ioyned together by ah extreme and smeane proportion : the fide of the Icofahe- dron contained in the fame fphere,fhalbe duple to the leffe fegment. | The 17/Propofition. Ifthe fide of a dodecahedron, and the right line, of whome the faid fideis the leffe fegment, be fo fet that they make a right angle: the right line which contarneth in power halfe the line /ubtending the angle, is the fide ofian O¢fohedron contained in the [elfe [; here. of ned in the felfe Jame fbhere 4 aati Ae ree Vppofe that A B be the fide of a Dodecahedton, andlet the rightline of which that fide is the léflefeptnentbe A G,name- ly which coupleth the oppofite fides of the Dodecahedron, by the 4.corollary of the 17.0f the thirtenth : and Jet thofe lines 4) befo fet that they make aright angle at the point A.And draw Spas the right lineB G. And let the line D containein powerhalfe the line B G (by thefirft propofition added -by Fhujfas after the latte of the fixth). ThenT fay thatthe lineD isthe fide of an O@oliedron contayned itt the fame {phere . Forasmuche as the line«A'‘G maketh the greater fegment G C the fide of the cube contained in the fame {phere (by the fame 4.corol- lary of the 17.0f the thirtenth ):and the {quares of the wholeline A G. and of the leffe fegment A B are triple to the {quareof the greater egment G C, by the 4.of the thirtenth: Moreouer the diameter of the fphere, is in power tripie to the fame line G C the fide, of the cube(by the 15.0f the thirtenth: = Waerefore " = eS of Euclides Elementesafter Flufias. Fol.429¢ Wherfore the line B G is equalto the diameter, Forit containethin power the two lines A Ban (by the 47.0f the firft,) and therefore it containeth in power the triple of theline GC. But the ee the O&ohedron coutained inthe fame fphere,is in power triple to halfe the diameter of the {phere by the rg.ofthethirtenth. And by fuppofinon theline D containeth in power the halfe.ef the line BG. Wherefore the line D Sites po in power the halfe of the fame diameter) is the fide of an oGohe- dron, If therfore the fide of a Dodecahedron and the right line of whome the faid fde is the lef: ° feg- ment,be fo fet that they make aright angle : theright line which containeth in power halfe the line fubtending the angle, is the fide of an. Ocohedron contained.ia the felfefame {phere: Which -was re~ quired to be proued. | % A Corollary, Vnto what right line the fide of the Ocfohedronis in power fe/qualter:‘ynto the fame line the fide of the Dodecahedron infcribed in the fame phere, is the greater ‘fegment,For the fide of the Dodecahedronis the greater fegment of the fegment C G,vyn- to which D the fide of the OGtohedron is in power fefquialter, that is, ishalfe of the power of the linc B G,which was triple ynto the line C G. . Patch, power of the line q Ube 18. Propojition. If the fide ofa Tetrahedron containe in power two right lines ioyned to- gether by an extreme andmeane proportion : the fide of an Icofahedron defcribed in the felfe fame Sphere, is in power fefquialter to the leffe right line. ee into the lines A G and GB, ioyned together by an extreme and meane proportion: name- Po SAG ly, lecit be diuided into\A.G the whole line,and GB the greater fegment (by the Corolla- © UG i ry of the firlt Propofition added by Flufflas after the lait of the fixth ). Andlet ED be the (k= Sey fide of the Icofahedron E D Fcontained in the felfe fame Sphere .’And let the line which ~——"~——~ fubrendeth the angle of thePentagon defcribed of the fides of the Icofahedron be EF. Then I fay, thatED thefide of the Icofahedron is in power fefquialter to thelefie line G B.- Forafmuch as ( by that which was demonftrated in the 15. 0f this booke ) the fideED isthe greater fegment of the line E F which fubtendeth the angle of ‘ the Pentagon But as the whole line EF is ‘to the greater feg- ment E D, fois the fame grea- ter feement to the lefle (by the 30.0f the fixth ) * and by fuppo- fition ,A G,was the whole th Vppofe that A B C bea Tetrahedron, and let his fide be AB, whofe power let be diuided AX 1, . and GB the greater fegment : WhereforeasE F is to E D, fo is AG to G B, by the fecond of the foure- tenth . And alrernately, theline E F isto the line A G, as the line E Disto the line GB. And forafmuch as(by fuppofition) the line A B containethin power the two lines AG and G B : therefore ( by the 48. of the 7 the angle A GB is aright angie . But the angle DE Fis a right angle, by thar which was de- monitrated in the 15 .of this booke . Wherefore the triangles A'G B and FED, are equiangle,by the &. of the fixth. Wherefore theirfides ate proportionall: namely,as theline E Dis to cheline GB, fo is the line F D to theline A B, by the 4.of the fixth . But by that which hath before bene demonftrated, F D is the diameter of the Sphere which containeth the Icofahedron.; which diameter is in power fefqui- alter to AB the fideof the Tetrahedron irfcribed in the fame Sphere, by the 13.0f the thirtenth. Wher- fore the line ED the fide of the Icofahedron, is in power fefquialter to G B the greater fegment or leffe line. If therefore thefide of a Tetrahedros containe in power two fightlines ioyned rogether by an ex- treme and meane proportion : the fide ofan Icofahedron defcribed in the felfe fame Sphere,is-in pow- er {efgwialrerito.the leffe night line; q The Coustrnttione DemonSira- 10M. * mnie er”, ~The fourtenthBooke ) q The ry:Propofition. ~ g scibads siahiosl v7 | The fuperficies of aCubeis to the fuperficies of an Oobedron infcribed | in one and the felfe fame Sphere,in that proportion that the folides are. Couwtrufion. S| rege thatA BC DE bea Cube, whofe fower diameters let be the lines AC,BC,DC, EC produced on ech fide. Let alfo the O¢tohedron infcribed in the felfe fame 4 | Sphere be F GH K: whofe three diameters let be FH,G K,and ON. Then I fay, thac i ‘| the cube ABD is to the O&tohedton F'G H, asthe fuperficies of the cube is to the fuper- ficies of the O@ohedron . Drawe from the centre of the cube to the bafe A B ED, a per- pendicular line C R . And from the centre of the Octohedron draw to the bafe GN H,a Demonftra- perpendiéular line LL « And forafmuch as the thrée: diameters of the cube do pafle by thecentre C, tien. therefore, by the2 Corollary of the 15.0f the thirtenth, there fhall be made of the cube fixe pyramids, as thys pyramis ABDEC, equall to the whole cube : Forthere are in the cube fixe bafes, vypon which fall equal perpendiculars from the.centre, by the Corollary of the Affumpt of the r6.of the twelfth, for the bafes are contained in equall circles of the Sphere". Butin the O¢tohedron the three diameters do take vpon the 4. bales, 8. pyramids, having their toppes in the centre, by the 3.Corollary of the 14.0f, the thirtenth . Now the bafes.of the cube and of the O&ohedron are contained in equall circles of the ’ Re NOTES —_,. ZN! ’ - s 7 ~. ad - : - : - - - ’ esti ; - Sphere, by the 13-0f this booke . Wherefore they fhall be equally diftant from the centre, and the per- pendicularlines C Rand L, fhall be equall, by the Corollary of the Affumpr of the 16. of the neath ~ Wherefote thepyrainids of the cube fhall be-ynder one and the felfe famealtitude with the pyramids, of the Oftohedron, namely, vnder the perpendicular line drawen from the centre to the bafes. Wher- fore fixe pyramids of the cube, are to 8.pyramids ofthe O@ohedron being vnder one:and the fame al- titude, in that Pr oportion that their bafes are, by the 6.of the twelfth : thatis,one pyramis fet vpon fixe bafes of the cube, and hauing to his altitude the perpendicular line, which pyramis is equall to the pyramids,by thefame 6.of the twelfth, is to one pyramis fet vpon the 8.bafes of the O@ohedron, being. equall. to.the OGohedron, and.alfo vnder oneand the felfe fame altitude, in that proportion; that bafes of thecube, which containe the whole fuperficies of the cube, are to 8.bafes of the Oftohedron, Which con taine the whole fuperficies of the O@ohedron . Forthe folides of thofe pyramids are in pro- portion the one ro the other, as their bafes are, by the felfe fame 6.of the twelfth . Wherefore the fu. perficies of the cube is to the fuperficies of the O@ohedron inferibed in one andthe felfe fame Sphere, in that proportion, that the folidesare : which was required to beproued.. ‘ ete .strsitine qT be 20. Propofition. * ya Tf a Cube and an O&tohedron be'contained in one cx the felfe fame Sphere: they fhall be in proportion the one to the other, as the fide of the Cube isto =e | | the of Euclides Elementes after Fluffas. the femidiameter of the Sphere. Fol.430. KE Vppofe that the Octohedron AE CDB be infcribedin the Sphere ABCD; and let the Or | cubeinfcribed in the fameSphere be F G HIM : whofe diameter letbe H I,, which is e- XS} quall to the diameter'A C, by the 15.0f the thirtenth: let the halfe-of the diameter be AE.Then' fay,that the cube FGHIM is tothe O&ohedron AEC DB,as the fide | M G is tothe femidiameter A E . Forafmuch as the diameter A C isin power double to BK the fide of the O@ohedron ( by the 14.0f the thittenth ).and is in, power triple to M G thefide of the cube (by the 15.0fthe fame ) : therefore the fquare BK D L fhall.be fefquialter to FE M the fquare of the cube . From the lineA E cut ofa third pare AN;.and fré.the line M G cut of like- wife athird part G O, by the 9.0f thefixth . Now then the line EN fhall be two third partes of the line A E, and fo alfo fhall the line M O be Of the line MG. Wherefore the oar weligipedon fer vpon the bafe B K DL, and hauing his altitude the line E-A, is triple to theparallelipipedon fet vpon the fame bafe, andhauing his altitude the line-A.N, by the Corollary of the 31-of the eleventh: but it is alfo eri- ple to the pyramis A BK D L which is fet vpon the fame bafe,andis vnderthe faine altitude (by the fe- cond Corollary of the 7.of the twelfth ) . Wherefore the pyramis ABKD L is equall to, the parallelipi- edon, Which is fet vpon the bafe'B'K D/Lyand hdet to his altitude the line A N.Butvnto thatpa- rallelipiped6,is double the parallelipipedon'which is fet vppon the fame bafe BK DL, andhath to his altitude a line double to'theline EN, by the Corollary of the 3 1.0f the firfts and: vnto the pyra- misis double the O&ohedron ABK LDC; by the .Corollary of the 14.of chethirtenth: Where- fore the O@ohedron AB KDLC is equall to the parallelipipedon fet ypon the bafe. BK L D, & ha- ving his altitude the line EN’ (by the 19. of the fifth). But the parallelipipedon fet vpon the bafe BK DL, which isfe(quialter to the bafesF M, and hauing to his altitude theline MO; which is'two third partes of the fide of the cube .M G, is equall to the cube E.G: by the 2.part of the 34-0f the ele- uenth .( Forit was before proued-that the bafe BK DL is fefquialter to the bafe FM) .Now then thefe two parallelipipedons,namely,the paralleli- ipedon Which is fet vpo the bafe B KDL (which 1s (efquialter to the bafe of che cube) and hath'to his altitude the line M.O (which is two third artes of M.G the fide of the cube), which paral- (linia is proued equall to. the cube, and the parallelipipedon fet Y Ss the fame bafe BKDL, and hauing his altitude the line EN ( Aes Jelipipedon is proued equall to the Octohedron): thefe two parallelipipedons (I fay) ‘are the one to the other, as the altitude M O,1s: to the altitude EN: ¢ by the Corollary of the 31.of the-eleuenth). W herefore.alfo asthe altirude M O, is to the alti- tude EN, foisthecube F GHIM,to the O¢to- hedron AB K DLC, by the 7.of the fifth . But as theline’-M’O ‘is tothe line EN, fo isthe whole line MG to the wholeline E A, by the 18.of the fifth . Wherefore as MG the fide of the cubc,isto BA the fcinidiameter; fois the line F @H'EM tothe O&ohedron A BK DLC infcribed in one & the felfe fame Sphere « If therefore a cube and an Ottohedron be contained in one and ‘the: felfe fame Sphere: they thal! be in proportion the one to the other,as the fide of the cube is to the femidiameter of the ‘Sphere; which was required to be. demonitrated, | re | 1 A Corollary... DistinE#ly to notefie the powers of the fides of the fine folides by the power of the diameter of the [phere. | The fides of the tetrahedron and ofthe cube doo cut hep Ones ofthe diameter of the fpherein- to two {quares which are in proportion double the one to the other. The odtohedron cutteth the =~ TTtJ. power Conflruction. Demon Straq- ti0n. =— =o >>. <> = SSS a = > practife in aG, it is very hard to fe and conceiue the con- tes (Oy ftrudions and demontftrations of the propofitions of [Wige— IXY this booke,vnles a man haue avery depe, fharpe;& fine (3 Te pimaginatign, Wheskase I would wifh the diligent ftudét x yin this booke,(to make the {ftudy thereof more pleafant TIN [ysl vnto him ) to haue prefently before his eyes,the bodyes “4 ke formed & framed of pafted paper (as I taught after the diffinitions of the eleuenth booke.) Andthen to drawe and defcribe the lines and diuifions, and fuperficieces ,according to the conitructions of the propofitions.In which defcriptions ifhe be wary and diligent, he fhall finde all things in thefe folide matters,as clereand as manifeft vnto the eye, as were things be- fore tanghtonly in plaine or fuperficial figures. And although I haue before in the twelfth bokeadmonifhed the reader hereof, yet bicaufe in this boke chiefly that thing is required, I thought it fhould not be irkefome ynto him, againe to be putin minde thereof. ‘arther thisis to be noted,that in the Greke exemplars are found in this 15 .booke only 5.propofitions, which 5,arealfo only touchedand fet forthe by Hypficles ; vnto which Canipane addeth 8 and fo maketh vp the number of 13. Campane yndoubted- ly although he were very well lerned,and that generally in all kinds of learning, yet af- {uredly being brought vp in a time of rudenes, when all good letters were darkned, & barberou{nes had onerthrowne and ouerwhelmed the whole world, he was ytterly rude and ignorant in the Greke tongue, fothat certenly he neuer redde Exchde inthe Greke,nor (of like) tran{lated out of the Greke: but had it tranflated out of the Ara- bike tonge. The Arabians were men of great ftudy, and induftry, and commonly great Philofophers, notable Phifitions, andin mathematicall Artes moftexpert, fo that all kinds of good learning flourifhed and raigned amongft themina manner only. Thefe men turned whatfoeuer good author was in the Greke tonge (of what Art and know- ledge fo euer it were) into the Arabike tonge. And fro thence were many of thé turned into the Latine,and by that meanes many Greeke authors came to thehandes of the Latines,and not from the firft fountaine the Greke tonge, wherin they were firft writ- ten.As appeareth by many words of the Arabike tonge yet remaining in fuch bokes;as are (enith nadir ,helmuayn ,helmuaripbe,and infinite fuche other. Which Arabians alfoin tran{flating fuch Greke workes,were accuftomed toadde,as they thought good,& for the fuller vnderftanding ofthe author,many things:as is to be fene in divers authors, as,namcly,in T heodofius de Sphera,where you {ee in the olde tranflation (which was vn- doubtedly out of the Arabike) many propofitions,almoft euery third or fourth leafe. Some fuch copye of Exuchde,mott likely ,did Campanus follow, wherein he founde thofe propofitios,which he hath more & aboue thofe which are found in the Greke fet out by Hypficles:and that not only in this 15.boke,but alfo in the 14. boke,wherin alfo ye finde many propofitions more thé are founde in the Greeke,fet out alfo by Hypficles. Likewife in the bookes before, ye fhall finde many propofitions added, and manye in- uerted,and fet out of order farre otherwife, then they are placed in the Greeke exam- plars. Fluffas alfo a diligent rettorer of Euchde, amanalfo which hath well deferued of the whole Artof Geometrie, hath added moreouer in this booke(asalfo in the former TTt.ij. 14.boke m, 2, aX net i a Sel er ot = ee is arty - | » “ ss os 4 Lo : P Sacdteclin The fiftenth Booke us 14. boke he added 8, propofitids)9.propofitios of his owne, touching the infcription, and circumf{ctiption of thefe bodies,very fingular yndoubtedly and wittye-All which, forthat nothing fhould want to the defirous louer of knowledge,I haue faithfully with | no fmall paines turned. And whereas ¥/ufas in the beginning of the eleuenth booke, i namely,in the end of the diffinitions there fet, putteth rwo diffinitions, of the infcrip- tion and circumf{cription of folides or corporall figures, within or about the one the other,which certainely are not to be reiected:yet for that yntill this prefent 15 .boke, there is no mention made of the infcription or circum{¢ription of thefe bodyes, I echought it not {fo conuenient there to place them, but to referre thé to the beginning ofthis 15.booke:where they areinmaner of neceffitie required to the elucidation of the Propofitidns anddemonftrations of thefame. The diffinitions are thefe. .... Diff inition. +. EE ka = Pe TET en re Sr = —— 3 —— = Hw an — om - 3 3 a eae A folide ficwre, is then Jaid to be infcribed in a folide figure, when the ane gles of the figure infcribed touche together at one time, either the angles of ‘the froure-circum|cribed ,or the fuperficieces or the fides. oe . =. —. ——_————— Tifa = ——s ‘9, 7 re . , Diffmition.z. A folide figure is then Jaid to be circumfcribed about a folide figure : when together at one time exther. the angles or thefuperfacteces or the fides of the . ficure civcum/cribed touch the angles of the figureinfcribed. ae 74 ; ™ a7 7k N the fourth booke in the diffnitions of the infcription or circumfcription of playne | 7) & rectilinefigures one with in’or aboutan other, was trequired that all the angles of the 5) at 95 figure infcribed sfhould atone time touch all the fides of the figure circum{cribed: bur SP gin the fie regular folides (to whome chefely thefe ‘two ‘diffnitions pertaine) forthat i : Zz mn i Hii y t | r. ry! Mi | f y i 1 f re ey a | W al , a whee | , tel “ iy | [- ae ‘ ee! i l te .. Bae q “ - 4 : wae *! ae rl ; > I ae Ae , “ } : : aa) ae a) ) : Py s 7s : 74 @ wat . 0 7 i zt 4 ; (en ia ao: ba! r E | { y y ee ' y ‘ yi t 4 Me : - ; & > a. Aon — gS one — 2 rw. fy ZS og ~t 2 > oS -* oe Ma + as YS a or id “> O« ~~’ =e ae ew he a sf on ray ES, re i UO x, E oa i, 2.0 © = en en tine ) epee - — a = ¢ ‘ > -» - SSS ra = = = —= — Sa - —- = = A 1 eA * —— ———s 1 — > * J T hefiftenth Booke one of the three equal {quares -which diuidej Offohedvon into two equall partes and perpendicularly . Forthethree diameters of thofe {quaresdo in the centre cut the one the other into two equall partes and perpendicularly, by the third Corollary of the 14. of the thirtenth, | which fquares, as for example, the {quare EX L I, do diuide in funder the pyramids and the prifmes, * namely, the pyramis KL TD and the — KLTEIA from the pyramis EK ZB, and the prifme EK ZILG, which pyramids are equall the one to the other,'and fo alfo are the prifmes equall be one | to the other : by the 3.of the twelfth. Andin like fort do the reft of the {quares, namely, K ZIT and | ZLTE: which {quares, by the fecond Corollary of the 14.0f the thirtenth, do diwide the O&ohedror into two equall partes. | q The 3.Propofition. — The 3. Probleme. In a cube geuen, to defcribe an Ofohedron. 9 (Ake a Cube, namely, AB CD EF GH. And dinide ewery one of the Constructions. 4s ay fides thereof into two equall partes . Aud drame rightJines coupling toeether : the fections, as for example, thefe right lines, P 2 and RS, which foal be td REA equall unto the fide of the cube ( by the 33 .of the firft ) and fall dinide the “ one the other wto two equall parts in the middeft of the diameter AG in the pointt (by.the Corollary of the 34.0f the | Gin, isa right angle ( by the ro. of the ele- : uenth, for the linesI P and P L are pa- vallelstothelinesRAand AB). And © : the right line IL {ubtendeth the right angle I PL, namely, it fubtendeth the - halfe fides of the cube which containe aa she right angle IP L, and likewife the right line IM fubtendeth the angle sia Tae 1 QM whichis equall tothe famean- hiags Say. 3 gle IP L, and is contained under right eg | | lines equall to the right lines which containe the angle! P L. Wherefore the right linel M is equall to the right line I L ( by the 4.of the first ).. And by the fame reafon may we prow, that euery one of the right lines M 0,0 L,K 1,K L,K M,KO,NI1,NL,N M,and NO, which fubtend angles equall to the felfe fame angle 1 P L,and are cotained under fidesequall tothe fides which containe the angle 1P L, are equallto theright line 1.L. Wherefore the triangles KL1,KLO,KM1,K MO, and NLI,NLO,N MI,NMO, are equilater and equall : and they containe the folidel K LO NM .Wherefore1 K LON Mis an OGfo- hedron; by the 23 definition of the elenenth . And forafimuch as the ‘sd thereof do alto. gether in the pointes I,K,L,0,.N, M, touch the bafes of the cube which containeth jt, it followeth that the Octohedron is infcribed in the cube ( by the firft definition of ehis booke) . Wherefore in the cube gentn,is defcribed an O¢tohedron ; which was required go be done. ees : | KL,KM,KO,NI,NLNM,eoNoO. Demoniire- And now forafmuch as the angle IP L f Yi Zi rN fae ee = - : S48 = = t- A 9 = —:-= SS = == ————— _ —-- - pS Sain alae = oi 5S: rt << "1 = a —s a eS ee * = =e - m —— ~- Se. E - — a es : = eh eet ee => celetgeo;' = =e sua = abs 55; *S = Steen ee Fee omen —— _ 53 pen = —es oe i < & =, — Ss => a aanean — ——— - MES : — = — ~ ~ ss a - > . P > ———E “= —— as —_—— a ee ee te Pt < = se r+. - > = — ———— a te = = - =a i sil = ancl = a . - — - mn “ = - —- = — z. . sa - —s a, *; Fo ag oo = = —- =. —— ~-! s — — : - —— = a oi - Ae EEE Mae : - — oe = “ SS st a“ a So eo: “ ~~ - “ Ls S&S # —_ —_+ ee -_ . cree me tm -- — ~~ — - =a = 22a — —poe +s ee = ae - < < —— —e a we ~~ = ns . L — re ars £5 NS ae aT ~ eo ee Ste =. nnn _———s P - Cae ee ~~ =e a — = > - o - ~ ee = ed -. x > = -* ees ipeemen ane = —_> = -_— ~ : = : Sa ee we t in mre iaehcar ~*~ - ~ aa es . . a. ~ - =< = = _— = —— — SS a paates i “ _ =_ : a oe on Pe < = a — ere ct < Soke oe > = 7 eae ao - 2 = oa - — —- ae e <3 : eel ne eS es = = = ai — = ™ = * a . 2 : = SS —— > a —= —— ‘ i c = me a q A Corollary 1 of Euchdes Elementes. Fol,4.33. q A Corollary added by Fiaffas. Hereby itismanifeft; that vight lines ioyning together the centres of the oppofite bafes of the cube,do cut the one the other intotwo equall parts ,and perpendicularly, in the centre of the cube, or in the centre of the Sphere which containeth the cube. | For forafmnch as the right lines LM ana 1 0 which knit together the centres of the oppofite bafes of the cube, do alfo knit together the oppofite angles of the ottobedron infcri- bed in the cube, it followeth (by the 3 Corollary of the 14.0f the thirtenth ) that thofe lines LM and10, do cut the one the other snto two equall partes ina point . But the diameters of the cube do alfo cut the one the other into two equall partes,by the 3 9.0f the eleuenth . Wher- fore that point fhall be the centre of the {phere which containeth the cube . For waking that point the centre, and the (pace [ome one of the femidiameters, deferibe 4 {phere, and it hall pA i[fe by the angles of the cube : and likewife making the fame point the centre, and the (pace halfe of the line L M, defcribe a [phere, and it [hall alfopaffe by the ancles of the ocfohedron. q I'he 4. Propofttion. T he 4. Probleme. In an Oobedron geuen,to defcribe a Cube. i r pale that the octohedron.geuen be ABG.DE Z. And let the two pyramids “thereof be ABG DE,and ZBG DE. And take the centres of the triangles ConStruflion. NY of the pyramis A BG D E, thatis, take the centres of the circles which containe SESE thofe triangles : ani let thofe centres be the pointes T,1,K,L . And by thofe centres let there be drawen parallel lines-to the fides of the = fquare BG D E : which parallel right Gneslet be. MT N, i | NLX,XK0O,G01M. And forafmuch as thofe parallel Demonfira- right lines do ( by the 2. of the fixth ) cut the equall right s10Mte lines AB,AG,A D,and AE, proportionally, therfore they concurre inthe pointes M, N,X,0 . Wherefore the right lines N,N X,X 0, and O M,which {ubtend equall an- 4s i O gles fet at the point A,cr contained underequall right lines, ® Ald. E areequall ( by the 4.of the firft) . And moreouer, feing that Po 2 they are parallels unto the lines BG,G D, D E,E B; whith Ki ace make a {quare, therefore MNXO is al{o.a {quare, by the 10.0f the elenenth. Wherefore alfo, by the 15.0f the ame;the {quare MNXO = tothe {quare BG DE . For all the right lines touch the one the other in the pointes of their ¢ a /\ D fections . From the centresT ,1,K,L, drawe thefe right lines ; TIIK,KL,LT . And drawe the right line AIC. And : forafmuch as Tis the centre of the equilater triangle A BE, * therefore the right line A I being extended,cutteth theright line B E into two equall partes (by the Corollary of the 12.0f & — oe = at . =x — << - = —_—— ——— = = = oe ~ >= Sere oe se t a = — = rz = BS a~ . Sar o- >. = =n Y Wh x - aul ra a ’ ~ ee mens = 7 = = = ott = ee =e" mtiows = SS el > = — a = ~ -- on ~ _— -— = ~ ~ = ez _ — > * ae, = in ae Pre : - ad — - o —+ er : = = So om ir = SS ee the thirtenth) . And fora{much as M 0 is aparallelto B E, therefore the triangle _A10 is like to the whole triangle ACE (by the Corollary of the 2.0f the fixth). And theright line M10 is diuided into two equall partes in the point I (by the 4.of the fixth) . And by the [ame reafon nzay we proue, that theright lines MN, NX; X 0, are divided into twoequall partes se . . a” li T befiftenthBooke» é partes in the pointes TL ,K. Wherefore alfo againe;the bales T 1,1 K,K L, LT which [ub- tend the angles (et at the pointes M,O,X,N,which angles areright angles, and are contained | under eqnall fides thofe bales, 1fayyarcequall .. And forafiouch as T IM is aw I[ofcéles tri- ‘ dugle, therefore the.angles fet atthe bale, namely ,the angles MT I and UIT ,are equal (by the s.of the firft,) . But the angle M is aright angle : wherefore eche of the angles MIT ana MT I, isthe halfe of aright angle . And by the fame. realon the aneles OLK & 0 KI, are cquall .Whereforethe angleremayning, namely, T TK, isavightangle ( by the 13..0f thé firit).. For the right lines | ? T land LKavefet-vpon the line M O..And by the fame rea- ‘fon may the rest.of the angles, namely, IK L,KLT,LT I, be proved right angles, and they are in- one and the felf fame plaine [uperficies oamely,M NX O (by the 7.of the eleuéth). AI O Wherefore the right lines which ioyne together the centres of .. the plaime {uperfictall triangles which-make thefolide angle. B A,do make the {quarelT K L. Aud by the fame reafon may be proued, that the plaine fuperficiall triangles of the reft of the fiue folide angles of the Octohedron Jet at the pointes B, af PS G,Z,D,E, doin thecentres of their bafes receaue {quares, Sothat there are in number fixe {qnares, for enery Oohe- dron hath fixe folide angles : and thofe {quares are equall: for their fides do containe equall angles of tuctinations con- tained underequall fides, namely, under thofe fides which ; are drawen from the centre to theyide of the equall triangles AL OY: : (by the.2. Corollary of the 18 of the thirtenth ) . Wherefore IT KDRPV'S isacube ( by the 21. definition of theele- | wenth ) and hath his angles in the centres of the bafes of the Otohedron, and therefore is inferibed init ( by the firft de- | finition of this booke ) .Wherefore in an Octohedron genen & 45 defcribed a cube : which was required tobe done... ee | D The 5. Propofition. The 5. Probleme. “In an Icofabedron genen,to defcribe a Dodecahedron. ph ttl Akean 1cofahedron sone of whofefolide angles let be Z..Now foralmuch as Conitrattions 4 te WL (by thofe thinges which hane bene proved in the16 of. the thirtenth) the ba» j Xen) [es of the triangles which contayne the angle of the Icofahedron doo make & 4 DE RECS pentagon in{cribed in acircle, let that pentagon be ABGDE, which ss @ va made of the fiuebafes of the triangles , whofe playne Juperficiall angles re- mayning make the folide angle geuen,namely, Z.« Andtake the centres of the circles which contayne the fore{aid triangles which centers let be the poyntes 1,1 ,K,M,L- and draw thefe Demanfira- right lines 1T,T K,K M,M L,& 1. Now thena perpendicular line drawne from the poynt tion. Z to the playne [isperficies of thepentagon AB G D E,fhall fallupon the centre of the circle which contayneth the pentagon A BG DE(by thofe thinges which haue bene proucd in the _ felfe fame 1630f the thirtenth).. Moreoner perpendicular linesdrawne from the centre to the fides of the pentagon AB G-DE fhallin the poyntes C,N, 0 where they fall cut the right lines A B,B G,G D into two equall partes( by the 3.0f the third).Draw the[e right lines CN and N.Q.And forafmnnch as the angles CB Nand N GO are equall,and are contained un- der equall fides therefore the bafeC N is equall to she bafe NO (by the 4.0f the firft).More- sive Guer t. tga 20 n6e fetta met 554°" " aN cage: Ais: ~ ew of Euchdes Elementes’ Fol.434.. ouer perpendicular lines drawne from the poynt Z tothe bafes of the pentagon ABG DE, [hall ikewife cutte the bafes into twoeguall partes( bythe 3 .of the third) . For the perpendi- culars pajfe by the centre ( by the corollary A of the 12.0f the thirteth):Wherfore thofe perpendicular lines [hall fal vpo the points C,N, 0 . And now fora{much as the right lines Z1,1C are equall tothe right lines ZT ,TN, cy alfoto the rightlnes ZK,KO (by reafon of the likenes of theequall tri- angles). therefore the line IT is a parallell to the lineC Nand foalfoisthelineT K to the line NO( by the 2.0f th: fixt).Wher- forethe angles 1T K,and CN O.areequal (by the 11.0f the eleuenth). Agayne foraf- much as the triangles CBN ,and NGO are Ifofcels triangles , therefore the angles BCN and BNC are equall ( by the $. of the firjt). And by the fame reafon the ; angles GN O,andG O N areequall . And morcouer theangles BC Nand B NC are equall to the anglesG NO, and GON, for that the triangles CB Nand NG O are equalland like . But the three angles B NC,C.NO,0O N G,are equall to two right angles (by the 13. of the firit) :for that upon the right line B G are fet the right lines CN & O N.And the three angles of the triangle C BN , namely, the angles B N.C,B C Nor G NO (for the angleG- NO is equall tothe angle BC N_as it hath bene proued ) and N_B C are alfoequall to two right angles( by the 32.0f the firft). Wherefore taking away the angles BNC c GN_O, the angle remayning,mamely,C NO ts equall to the angle remayning,namely, to.C BN. Wher- fore alfothe angle 1 T K (whichisproned to beequall tothe angle C N O)}is equall to the an- gleC BN:Wherefore1T K is the angle of a pentagon. And by the fame rea[on may be pro- ued thatthe rest of the angles,namely theanglesTIL1ILM,LMK,MKT ,areequall to the reft-of theangles namely toB AE,AE D,EDG,DGB Wherefore IT K M Lis ane- guilater and equiangle pentagon (by the 4. of the first) F or the equall bafes of the pentagon IT K ML doo-f{ubtend equall angles fet at the point Z,and comprehended vnder-equall fides .- Moreouer it is manifest that the pentagon IT K.M L-isinone and the (elfe [ame playne fuperficies. For forafmuch as the angles ON Cand NCP are in one and the felfe fame playne{nperficies namely inthe fuperficies A B G D E: But wnto the fame playne [uper- ficies the playne fuperficieces of the angles KT LandT I L are parallels (by the 15.0f the ele- nenth ). And the triangles KT LanaT IL concurre : wherefore they arein one and the felfe fame playne fuperficies( by the corollary of the 16.of the eleutth). And by the [ame reafo fo may we proue thatthe triangles I LM, LM K,M KT arein the felfe fame playne {uper- ficies wherein-are the triangles KT I andT 1L.Wherefore.the pentagon IT KM Lisin ope and the [elfe fame playne {uperficies:. Wherefore the [olide angle of the Icofahedron name ly the. folide angle at the poynt Z [ubtendeth an equilater and equiangle pentagon plaine fu- perficies, which pentagon hath his plaine fuperfictall angles in the centres of the triangles which make the folide angle Z «And in like fort may we proue that the other elenen [olide angles of thelcofabedron , eche of which eleuen [olide angles are equall and like to the folide angle Z (by the 16.of the thirtenth) are [ubtended unto pentagons equall, and like , and in like fort fet to the pentagon 1T K M L. And fora{much as in thofe petagons the right lines, which ioyne together the centers of the bafes,are common fides , it followeth that thofe 12. pentagons include a folide which folide is therefore a dodecahedron (by the 24. diffinition of she elenenth):and is,by the firft diffinition of this booke,de{cribed in the Icofahedron , fine PV V4. fides #. or ~ ¢ % * ey aN Ea RIM ig ~ wee a alls” nt SX. -_ I - Ae ‘ hy . . Con & a oF a Me T hefourtenth Booke » fides whereof are Jet vpon the pentagon ABG DE : Wherefore in an icofahedron geuen is in{cribed 4 dodecahedron: which was required to be done. - An annotation of Hyp/icles. . T his is tobe noted that tfa man fhorld demaund how many fides an Icofahedron hath, di we may thus anfwere : Itis manifeft that an Icofabedron is contayned under 20. triangles, | and that enery triangle conjifteth of three right lines . Now then multiply the 20. trian- gles intothe fides of one of the triangles and [0 [hall there be produced 60. the halfe of which is 30. And [0 many fides bathanIcofahedron. Andin like {ort ina dodecahedron , foraf- much as 12./pentagans makes doaecahedron,and enery pentagon contayneth 5. right lines, multiply 5:into 12.and there fhall be produced 60.the balfe of which is 30. And [o many are the fides of a dodecahedron. And the reafon why we take the halfe, is,for that euery fide whe- ther it be of a triangle or of 4 pentagon,or of a{quareasin a cube,is taken twife . And by the fame-reafon may you finde out how many fides are in.acube , and inapyrmis, andin an ottobedroné | Ssh UN Butnom acayne if ye will finde out the number of the angles of encry one of the foltde ficures;whenye hane done the fame multiplication that ye did before, dimde.the fame fides, by the number of the plaine '[uperficieces which comprehend one of the angles of the folides “As for example fora{much as s.triangles contayne the folide angle of an Icofahedron,dinide 60.by s.anid there will come forth 12:and-{o many folide angles hath an Icofahedron . Ina dodeczhedron.forafmuch as three pentacons comprehend an angle,dinide 60.by 3 .and there will come forth 20-+ and [omany are the angles of wdodecahedron.. Aud by thefame reafow many you finde ont Trew many aneles avetn eche of the-reft of the folide figures. “ Fp he required to be knowne; how one of theplaines of any of the fiue foltdes being ge- That w p ich -gen,thereimay befoind out theintlination of the fayd plaines the one to the other, which con =? wee ‘tayne eche of the folides. T his (as ‘fayth Ifidorus our greate master ) is found out after this inclinationof Wweber.Itis manifest that in'acube,, the plaines which contayne it , doo cutthe one the other the plaines of “by a7i¢ht angle. Butin a Tetrahedron, one of the triangles being geuen , let the endes of one the fine folides, of the fides of the fayd triangle be the centers,and letthe foace be the perpendicular line wasbefore = “eawye'fiom the toppe of the triangle to the bale’, and deferibe circumferences-of a circle, ts 8 es he : ‘which, all cutte the one the other: and from the interfection to the centers draw right lines, after 1 Fe Sie which fhallcontaine the inclination of the plaines totayning the T etrahedron.in an Oétohe- manerjout of 470n,take one of the fides of the triangle thereof and vponit defcribe a {quarejand draw the Fluffasinthe “diaeonallline;and making the centres}the endes of the dingonall line, anid the [pace likewife latter ende of the perpenditnlar line drawne from the toppe of the triangle to the bafe jaefcribe circumfe- therz.booke. “pences : andl acayne from the common fection to the centres draw right lines , and they Ml contayne the inclination fought for. In an Icofahedron,vpon the fide of one of the tri- angles thereof de{cribe a pentagon, and draw the linewhich fubtendeth one of the angles of the fayd pentagon and making the centres the endes of that line , and the {pace the perpendi- ‘cular line of the triangle , de{cribe circumferences and draw from the common interfection of the circumferences,unto the centres right lines + and they fhall contayne likewife the incli- nation of the plaines of the icofahedron. In a dodecahedron,take one of the pentaguns , and draw likewife the line which fubtendeth one of the angles of the pentagon and making the centres the endes of that line,and the {pace , the perpendicular line drawne from the. fection into two equall partes of that line to the fide of the pentagon’, which is parallel vnto it , de- fcribe circumferences:and from the point of the inter fection of the circumferences draw un- tothe centres right lines:and they hall alfo containe the inclination of the plaines of the do- decabedron .T hus did this moft incular learned manreafon , thinking the demonftration in enéry one of them tobe plaine ‘and cleare. Bur to make the denonfirationof ee g * i ae | 7 - 5 1) var. | "4G | fe A i) aaa — a oe m2 rl 5 Fe ae! —_—_ = = S, a a i Se Sb. ee > = = et tpee> 2 seer ae eran * — =e ee ~~ — nae = os te ee - =. of Euclides Elementes: Fol.435. felt think st good to declare and make open his wordessand firftinaTl etrahedron. Suppofe that there be a Pyramis or T etrahedrt ABCD | catained under 4.equilater triangles: ¢> let the toppe ther. 4 of be the point D.And (bythe 10.0f the fir ft) dinide the fide AD into two equall parts in the point E: Cy draw the lines BE and EC. And fora{muchas.a-D Band ADC arees quilater triangles , and the line.A-D-is. diuided into two e- quall partes, therefore the lines B E and EC fall perpendi- cularly upon the line A D,by the 8. of the firft. Now] fay that the angle BEC is amacute angle..For forafmuch as the line AC is double tothe line A E.( for by construction B co the line AD, which is equal te the line A C,is dinided into tmoequall partes in the point E ) : therefore the fauare of the line AC is quadruple to the Square of the line A E(by the corollary of the 20.0f the fixt). But thefquare of theline AG is equall to the {quares of the lines .A E.and E C(by thez 7 .of the first) :and the {quare of the line AC 1stothe {quare of the line CE ( fefquitertia) as 4.t0 3: ( for the {quare of the line ACis pans quadruple tothe {quare of the line AE.) : wherefore the {quare of the line B C (which is equall to the {quare of the line AC) is leffethen the [quares of the two perpendicu- lars BE & E C(for it is untathemin [uble{quialter proportio,namely,as 4.to 6.or 2.to 3.) Wherefore( by the 13.0f the fecond the angle BEC isan acute angle.Now forafmuch as the line A Dis the common inter[ection of the two plaines ABD, and ADC, andin either of thofe plaines to one point of the common fection are drawne perpendicular linesB E and EC which containe an acute angle B E G; therefore( by the $.diffinition of the eleuenth ) the an- gle BE Cis theinclination of the plaines , and it is genen. For the line B C,which is the fide of the triangle, being geucn, and any one of the lines B E or EC, whichis the perpendicular of the equilater triangle, being allo gemen: make the centres the poyntes Band C, that is, the endes of one of the fides and the {pace the perpendicular of the triangle,and defcribe circum- ferences and they fhall cutte the one the other in the poynt E. And from the poynt E draw to the centres B and C right lines,and they fhall containe the inclination of the plaines : and this is it which Ifidorus before fayd . And now that making the centres the poynts Band C, and the {pace the perpendicular of the triangle, the circles decribed fhall cutte the one the other, st is manifeft,*for either of the lines B Eand E C is ereater then halfe of the line BC. Now if the centers were the poynts Band C,and the {pace the halfe of the line B C,the circles defcribed fhall touch the one the other But if the [pace be leffe then the halfe,they fhal neither touch nor cut the one the other :bus if it be greater, they fhall undoubtedly cut. Againe {uppole that upon the (quare A BC D be fet a pyramis, hauing his altitude the poynt E,and let the triangles which containe it,be : equilater: wherfore the pyramis ABC DE fhalbe A * the halfe of the octohedron (by the 2 corollary of the 14.0f the thirtenth.) Deurde by the 10.0f the» firft) one fide of one of the triangles, namely, the line A E,into two equal partes in the poynt F:and draw thelines B F and D-F.: wherefore the lines BF andDF are equal and fal perpendicularly upon the line AE (by the 4.and 8.of the first.) - Then I fay that the angle B F D,is an obtufe an- gle.F or draw the line B D.And fora{much as AC ; 4s a {quare,and the diameter is B.D :thereforethe B G Square of the line BD is doubleta the{quare of the | hh line D A )by the 47.0f the firft.) But-the{quare of theline D A isto the [quave of the line PR uy. D F The veafon of thisyou hell moft plainely See i thar which is ade d out of Proclus after the 22. propo- Sition of the frrft booke. %. ge ae ee... ~ q ; > i } & ’ “noo ‘ Po tyes ” te “ oh si n y » > Ve } } r ’ \ . : ig ‘ 3 . ~~ Na i NTI scree ~ wT » Bed ene tees tel tomlin The fiftenth Booke “= D F,as g9t03.) a5 wasin the former proued.) Wherefore the fquare of the line D B is to 4 the [quare of the line F D,as 8.15t03. (namely ,as 2.t01.and g.to 3.added together ),batthe | ’ line D F ts equal to the line F B.. Wherefore the {quare of the line D B,is greater then the [quares of the lines D F and F BY for tt isto them,as 84s to 6.) Wherfore the angle B F D, i és an obtufe angle (by the 12.0f thefeconad:) And foralmuche asthe line A E is the common fection of the twoplaines ABE and ADE cutting the one the other and in either of thofe plaines to a poynt in the common fection are drawne perpendicular lines,B F and DF ,con- taining an obtufe angle B F D:wherforethe angle B F D (contained of the right lines B F and D F )is the angte of the inclination of the plaines ABE and AD E. If thereforethe angle BF D be geuen,the {aide inclination alfois geuen . For forafmuch as the triangle of the ocfobedron ts genen and one of the fides of the Ottohedron is the line A D, and upon it is de{cribed the {quare A C,and BD the diameter of that {quare being geuen, and the lines BF and F D are.the perpendiculars of that triangle : wherefore alfo the angle B F Dis ge- uen. Now then if vponthe fide of the triangle be defcribed a fquare:as the fquare AC, and the diameter B D be drawne, if alfo making the centres the poyntes B anid D and the teh she {aid perpendicular of the triangle,wedefcribe circles, they [halt cut the one the other in the poynt F And the right lines which are drawne from the centres to the poyat F {hal con- taine that inclination which ts comprehended under the angle BF D, which is the angle of the inclination of thofeplaines. And itis manifeft that either of the lines B F and F Dis greater then the halfe of theline.. For for that by the demonstration, it was proued that the [quare of the line B D 1s to the {quare of the line F D,as 8 is to 3 therfore the (quare of baif the line B Dis tothe {quare of theline F D,as 2.is to thre (forthe fquare of halfethe line BD is the fourth part of the {quare of the whole line BD by the 4.of the fecond) Wherefore either of the lines B F and F D,is greater then the line B D : wherfore the circles which are de{cribed by thofelines.B F and F D,and haning their centres the poynts Band D hall-cut the one the other..Anid thus much touchine the oltohedron. sieht As touching the Icofabedron, Sappale aneguilater petagon ABCDE, G vpon it let there be fet a pyra- mis hauing his toppe the poynt.F: and let the trian. — gles which cotaine it, be equilater.Now thé the pyra mis ABCDEF ;fhal be a part of the Icofabedro.Let F C one fide of one of the triangles be deuided into zwo equal partes in the poynt G.And draw the lines BG ¢ GD ,which fhal be equal cy fal perpedicular- - dy upon the line FC (as itis io to fe by the demo- A » ftratia of the former). The 1 fay that) angleBGD © isan obtufe angle : which thing is manifest. For drawing the line B D,it fhall{ubtend the obtufe an- gle BC D of the pentagon (whichis oh that avhich was demonstrated in the ende of thefirit cox’ es a ‘vollary of the 18.0f the 13 .booke:) But the angle BG D és greater then the ancle BCD, for the lines B G and G Dire lefve then the lines BC and C D: wherefore likewife asin the for- mer the angle B G Dis the inclination of the triangles BF Cjand C F D.Wherfore the an- gle BG D being genuen,the inclination alo of the plaines of the tcofabedronfhall be geuen. For if upon the fide of the triangle of the Icofahedron be defcribed am equilater pentagon, and then be drawne the line which fubtendeth two fides of the pentagon,as in this figure the line B Dif alfo the perpendiculars BG andG D of the triangles be dratwne,the angle BCD fhalbe genen. For if ye make the centres the enides of the line which {ubtendeth two fides of the pentagon,as the poynts Band D,and the fpacethe perpendicular of the triangle, and fo defcribe circles they fall cut the one she other in the poynt Gand from the poynt of the in- ) ee | ter[ection 4 f of Euclides Elementess Fol.436. terfection G ,drame unto the centres B.and D right lines, and they shal containe the anele of the iuclination BG D.And it is manifest by the defcription of the figure, that either of the lines B G and G D ts greater then the lineBD Which thing may alfothusbe proued. Sup- pofe an equilater H K Land upon K-L (one of the fides thereof ) defcribe an equilater pen. tagon K MN X Land draw thelineM L.And dinide (by the ro.of the firft)tbe fide K L into twa equal parts in the poynt Oc draw the line H O,which fhall be the perpendicular of the triangle HK L (ly the 8.of she firft.)T hen 1 fay that the line H ois greater then half of the line M L,which {ubtendeth the inclination of the plaines. F or from the poynt K draw (Ly the 1 2.0f the firft) unto the line M Lu perpendicular line K P : and fora{mucheas the angle K L Pis greater then thethird partof a right angle, that is, then the angle K HO (For the angle K L.M is two fiveth partes of a right angle, | by the 4.0f the firft,and by the afsumpt put after the firft coc rollary of the 18 .of the thirtenth booke,and the angleKHO >. is onethird part of one right angle, forthe whole angle K~. HL ,wherof the angle K HO is the half, by the g.of the firft, 1s one third part of two right angles,by the 3.2.0f the firft-) upon the line ML, and at the poynt L put unto theungle K HO anequalangleP LR (by the.23 of the fir fh.) Wher. fore the triangles P LR OH K,fhalbe equiangle, by the corollary of the 32.0f the first. Wherefore alfothe line PL fhalbe the perpendicular of the equilater triangle deferibed upon the line R L. Wherefore (by the corollarye added by Fluflas after the 12.propofition of the thirtenth booke) the line R Lis in power fe[quitercia, that is,as 4.is to 3.t0 the perpendicular LP. But the line K Lis greater then theline L R(by the 19.0f she firit. Wherfore y [quare of the line K- L hath to the {quare of the ine LP a greater proportion thé hath 4.to 3 :but it hath to the {quare of the line H O that proportion that g.hath to 3.Wher- fore the line K L hath to the line L P a greater proportion then it hath to the line HO. Wherfore the line H O is greater then the line L P by the 10.0f the fifth. As concerning a Dodecahedron. T ake one of the {quares of the cube wheron the Dode- cahedron is de{cribed (by the 17.0f the thirtenth):and let the fame be 4 BC D:and let the two plaines of the Dodecahedron fet vponitbe AE BF GyandGFDHC.Then1 fay that here alfo is geuen the inclination of the two pentagons. Diuide (by the 10.0f the firft) % — the fide F G into two equal partes in the poynt K. And from the poynt K draw unto the line F G ineither of the plaines AE B- yy L a E F G and GF DHC perpendicular lines K Land K M (by the 11.0f the firft.) And “ draw the line M L. Firft 1 fay that the an- gleM K L is an obtufeangle. For,by the D B difcourfe of the demonftration of the 17. propofition of the 13 .boke, where is taught the de{cription of the Dodecabedron,it is mani- feft, that the line drawne perpendicularly from the poynt K to the {quare ABC D,ts equal to halfe the fide of the pentagon.Wherefore it is leffe then halfe of the line M L. Wherefore the angle M K Lis an obtufe angle. Moreouer by the former di{cour{e of the 17 .propofition of the 13 .booke,it was manifeft that the {quare of the line K L is equal to the (quare of balf the fide of the cube,and to the {quare of halfe the fide of the pentagon. And foraf eke as VV vy. the > oes ~ 7 a T he fiftenth Booke i she lines K Leand:K .M are equal,and are eche greaterthen balfe of the line M Liwherfore : the angle M.K L being gewen, there fhall alfo te genen the inclination of the two plaines of the Dodecahedron. F or forafmuch as the fide of the{quare A BC D fubtendeth two fides of the pentagon geuen,the pentagon alfois geuen,and therefore alfois geuen the line M L. But there is alfogeuen either of the lines. M K and K L:for they are dravene perpendicularly fre the feciton into twe equal partes of the line A B, which fubtendeth two fides of the pentagon unto the fide of the pentagon, which is a parallel unto the line A B : namely,to the fide F G. Wherfore there is geuen the angle L K M,which is the angle of the inclination fought for. And now touching Uidorus wordes,he fayeth,that the pentagon being geuen,we muft draw the line which {ubtendeth two fides of the pentagon, which line ts equal to the fide of the cube:and making the centres the endes of that line, and the fpace the perpendicular line, which is drawne from the fection of the fame lineintotwo equal parts to the fide of the pen- sagon which is parellel to the {aid line asin the former defcription the line K L, or the line K M,defcribe circumferences and from the poynt of the interfettion of the circumferences draw unto the centres right lines whith fhatl containe the angle of the inclination: For by that which was fayd before,namely touching she Icofahedron, it is mmanifelt that the perpendicular K L,is greater then halfe of the line M L or C D,which is equal unto it. And therefore the cir- cles def{cribed by tb perpendiculars and haning to their centres theend of the lineC D, fhall cut the one the other, as was be- fore proued. See he ee The ende of the ficererich Booke - of Euclides Elementes after Aypfielen Mbp a * ss *% * " on . ~ >4 * . - SS of Euclhiles Elementes. Fol.437. q he. Propofrtion. The 6. Probleme. In an Oétohedron geuen,toin{cribe a trilater equilater Pyramis. Vppofe that the O@ohedron wherein the Tetrahedron is required to be infcribed, be AB GD EY . Take fower bafes of the OGohedron,that is, three which clofein the loweft triangle EG D, namely, AEG,B ED, IGD: and let the fourth be A I B,which is oppofite ro the loweft trian- | gle before put,namely, toEGD. Andtake the centres of thofe fower bafes, which let be the pointés H,C,NJL. And vpomthe triangle HCN mi erecte a pytfamis HC NL . Now forafmuch as thefe two bafes of the NS ‘ OGohedron,namely, AGEand ABI are fet vpon the rightlines EG and Bl which are oppofite the oneto the other, in the {quare G E B I of the Octohedron , from the point A drawe by the centres of the bafes, j namely, by the centres H,L, perpendicular lines A HF, ALK, cutting thélines EG andB I into.two equall partes in the pointes F,K ( by the Corollary of the 12.of the thirtenth) .W herfore aright linedrawenfr6 the point F to the — K, fhallbe ‘a parallel and equall'to the fides of the Oétohedron,nainely to EB and GI (by the 33.0fthe firft) . And the rghtline HL which cutteth the equall fides A F,A Kproportionally (for AH and AL are drawen from the centres of equall circles to the -circutnferences) 1s a pa- rallel tothe right line F K (by the 2.of the fixth) and alfo té the fides of the Oftohedron,name- ly, to EB andi G (by the 9. ofthe eleuenth ).. Wherefore as the line AF isto the line AH, {o isthe line F K to the line H L (bv the 4.ofthe fixth ) : Forthe triangles AF K and. AH Lare like (by the Corollary of the z.of the fixth).But the line A F is in fefquialter proportion to the line A H: (forthe fide EG maketh H F the halfe of the rightline AH, by the Corollary of the 12. of the thirtenth).Wherfore FK or GI the fide of the O@ohedron, is fefquialter to the right line H L.And by the oe may we sae # des of the OGtohedron are fefquialter © 0 is ugly Se of the right lines which mae the pyramis HN C Lynamely, to the right lines H N5N C,C Ly LN,andC H: wherefore thofe right lines are equall,and thereforethe triangles which are defcribed of them,namely, the triangles H CN,HNE,N C LandiCHL,j which make the pyramis HN C L, are equalland equilater . Andforafmuch as the angles of the fame pyramis,namely,the angles H, C, Ny Ly do end in the centres of the bafes of the Odtohedron, therefore it is infcribedin the {ame Otiohedron, by the firft definition of this booke . Wherefore in an Oégtohedron geuen, is inferibed a trilater equila- ter pyramis which was required to'be done. eee & A Corollary. T he bafes of a Pyramis infcvibed in-an O€fohedron, are parallels to the bafes of the 0 Gohedr on. For forafmuch as the fides of the bafes of the Pyramis touching the : e rhi he other, as for he other. are parallels co the fides of the OG@tohedron which alfo touch the one t A a A H L was proucd to bea parallel toGI, andLCtoDTI, therefore, by the 15. ofthe eleuenth, the plaine fuperficies which is drawen by the lines H L and LC, isa parallel to the plaine fuperficies drawen by the lines GI and D1 . And folikewife of the reft. + Second Corollary; “A right line igyning together the centres of the oppofite bales of the Otto- hedron, is fefquialter to the perpendicular line drawen from theangle of the ine Jeribed 7 Construttions Demonfira- ti0Me lal " on ti ui heb { Uy .” bh Un H ot ‘ M4 * ME J a. % eof es By a wi ., is gi a J : V2 | = “ml A 4 ei : Tr y ak | > 4 a Pi ft) iva 7 WY €. : « OF r, : a) ta et | Aw? OY ; el rg ot j MS) A Me / if ry i) : * rc t+ , al ‘ & : ¥ ‘ft mf | a \ R! bi he {Gs : “i, . By 4 ie i 7 | yi rile 7 ; Construction. Demon ftra- $304. Coufiruction. es * T he fiftenth Booke feribed pyramis.to the bafe thereof . For forafimuch asthe pyramis and the cube which containeth it do in the felfe fame pointes end their angles ( by the 1.0f this booke ) : therefore they {hall both be inclofed in one and the felfe fame Odtohedron ( by the 4.0f this booke ) , But the diame- ter of the cube ioyneth cogether the centres of the oppofite bafes of the O@ohedron , and therefore is the diameter of the Sphere which containeth the cube and the pyramis infcribed in the cube ( by the 3, and 14. of the thirtenth) : which diameter is fefquialter to the perpendicular which is drawen trom the angle of the pyranus to the bafe thereof: for the line which is a te from the centre of the {phere to the bafe of the pyramis, is the fixth part of the diameter ( by the 3..Corollary of the 13.0f the thir- tenth ). Wherefore of what partes the diameter containeth fixe,of the fame partes the perpendicular containeth fower. qFher.Propofition. The 7.Probleme. In a dodecahedron genen, to infcribe an Icofabedron. MM @ Vppofe that the dodecahedron geuen ,be AB CDE. And let the centres of the circles sO N |W hich cétayne fixe bafes of the fame dodecahedron be the points L,M,N,P;Q _,O.And draw thefe rightlines OL,OM, ON, OP; OQ, and moreouer thefe right lines LM, MN N P,P Q,QL. And now forafmuch as equall and equilater pentagons are contay- sen! ned in equall circles , therefore perpendicular lines drawne from their centres to the fides fhall be equall(by the 14. of the third),and fhall diuide the fides of the dodecahedron into two ¢- quall partes(by the 3.of the fame) . Wherefore the forelayde perpendicular lines fhall concurrein the point of the feGion, wherein the fides are diuided ee 4 | into two egtia!l partes, as LFandM Fdoo. And they alfo containe equall angles,namely, the in- clination ofthe bafes of the dodecahedron, ( by the z.corollaryof the 48.of the thirtenth).Wher- fore the right lines LM;M N,N P,P Q ,QL,and the reftof the rightlines whichioyne together two centres of the bafes, and which fubtende the equal angles contayned vnder the fayd equall perpendicular lines, are equall the one to the o- ther(by the 4.of the firft ). Wherefore the trian- glesOLM, OMN,ONP,OPQ,OQL,and the reft of the triangles whichare fet at the cen- | tres of the pentagons, are equilater and out ms Now forafmuch as the 12.pentagons of a dodeca- hedron containe 60. plaine fuperficiall angles , of which 6o.euety thre make one folide angle of the dodecahedron,it followeth that a dodecahedron hath 20.folideangles: but eche of thofe folide an- les is fubtéded of ech of the triangles of the Ico- jahedron ,namely,of ech.of thofe triangles which ioyne together the centres of the pentagés which ral | make the folide angle , as we haue before proued . Wherefore the,20. equall and. equilater triangles which fubtende the 20. folide angles of the dodecahedron , and haue their fides which are drawne from the centres of the pentagons common,doo make an Icofahedron ( by the 25. diffinition of the e- Jeuenth) :and it is infcribed in the dodecahedron geuen(by the firft diffinition of this booke) for that the angles thereof doo all at one time touch the bafes of the dodecahedron. Wherefore in a dodecahe- dron geuen,is infcribed an Icofahedrons which was required to be done. % >> we i The, Pr opofition, ) The 3.Probleme.. In a dodecahedron geuen,toincludea cube. G2 Efcribe(by the 17.0f the thirtenth)a dodecahedron.And by the fame,take the 12. fides Ben \( pot the cube,eche of which fubtend one angle ofeche of the 12. bafes of the dodecahe- aX * \ dron 3 for the fide of the cube fubtendeth the angle of the pentagon of the dodecahe- oy NG dron,by thez.corollary of the x7. of the thirtenth . If therefore in the dodecahedron : ais (i deferibed (by the felfe {ame 17. propofition) we draw the 12. rightlines fubtended vn- Sere ier the forefayd 12 angles , and ending in 8. angles of the dodecahedron , and concur- ring a a of Exclides Elementes, ring together in'fisch ‘fort that they bein'like fort fieuate,O bor O Mion shies weed ns. asit was plainély:proued in'that prdpofition , then fhall- +) = +. 127 NOIbSdersbob sso sata it bemanifeft ;thattheright lines drawneé in this dodew!:\ : cahedron from the forefayd 8-.angles thereof doo make»: theforelayd:cubeswhich therefore isintcluded.inthe do- docahedron:; forthat the fides.ofithe cube aredrawtein’. « -- thepfides: of the-dodecahedrany andethe angles ofthe! ss.ic famecubeare fer im theanglesof the faid. dodecahedron vise As for.example: take 4. pentagons.-of a dodecahedtonjs-:2:5 » namely &GDBO.BHCNO,@KED Nand-D F AQ4s >: NiAnd draw cheferightlines AIB;B G3@D,D A-Which. fowerright lines make afquares: forshateche of thefe right lines doo fubrend equall angles. of equall penta- § ~ gons,& the angles which thofe 4.right lines cétaine ares, right angles,as we proued in the-eontftruétion of the do-%- decahedron, in the 17 .propofitié beforéalledged. Wher-~»; »» fore the fixe bafes being {guares, do makea cube (by the fanios- 21.diffinition of the eleuench ) and for rhat the 8.angles nc) of the fayd cube are fet in’8. aneles of the dodecaheeron, 2s :::! = therefore is the fayd cnbe infcribed in the dodecahedron - »- (by the firit diffnition of this booke )-.. Whereforeina |» dodecahedron is infcribed a cube : which was required to be doone. | | q Ihe. Propofition, TE be § Probleme | Conttrufion. nao “Sse! take thes, fides which are eppofire the one tothe other; thofe-vfides,I faye whofe fections wherin they*are denided in- to two equal partes, are coupled by three right lines whichin the centre of the {phere, wherin the Dodecahedron is contained,doe cut the one the othen perpendicularly}, And let the poyntes wherin the forfayde fides are ¢ cut into two equal partes be A,B,G,D,C,.. \ And let the forefaid threright lines iaytirig’ %* together the faide feftions be AB, GDan Cl. And ler the centre of the Sphere De \ Ne Now forafmiich as (by the forefaid correla" Eo eG ry )thofe thre richtliaes are equalit falow- >’ fF oe Paie ec vedus fic YE ps (by the 4.of rhe firft) that the tight lines, — piace haba +9 ntti dus te ubteding the rgtt angles which’ they make par eer ate Cri oe ubteding the nght angles which they make : -adead bediisis at the centré BF cheipheie, Wwhicke AGN ae eo cea atast « gles are contained vnder the halues of the faid three right lines,are equal the one to the other: that is; the right lines A G,G B.B D,D A,;CA,CG,CB,CD, andI A,1LG,1 B,.D are equal the one to the o- ther.Wherfore alfowhe §.triangles QnA G,CGB,CBD,CD ATA GI GBSIBD &IDA are equal and equilater.And therefore A G B D C Tis an Odtohedron by the23.definition of the eleuéth,) And the fayd Octahedron is included in the dodecahedron (by the firft definition of this booke: ) for thae all the anglées\thercofidoe anone tinte rads the:fides'of the dodecahedron. Wherefore itt the dodeca- hedron geucn,is included an O@ohedron: which was required tobedone, — ME ela ails 2004 I'he s0,Propofition. The ro.Problemey 0 2 Demon Tra- . 820M 20 La. Dodecahedyon genen; toinfcribe an-equilater trilater Pyramis; &, Vppofe that the Dodecahedron geuen,be A B C D,of which Dodecahedron take thre bafes : AG meting at the poynt S,namely thefe thre bafes ALSIK, DNSLEandSIBRN:and of Construction. WWF thofe thre bafes take the three angles at the poynts A,B,D: and draw thefe right lines AB, WANG BD and D A:and let the diameter of the {phere containing the dodecahedron be $ f and XXX.1. cn T he fiftenth Booke ) chen draw thferight lines A O,} O and D O: Now forafmuch as (by the 17. of che thirtenth ) the an Demont gles of the dodecahedron are fein the fuperficies ofthe {phere defcribed abour the Dodecahedron + EMOnsTa= —cherefore if vpon the diameter ‘ O,and by the poynt A,be defcribed a femicircle,itdhall make thean- £50 Ke geSAOan ht angle (by the :1.0f the third.) And likewife if the fame femicircle be drawne by the poyntes D and B, it fhall alfo mae the angles $B O,and $ D Orright angles. Wherefore the diameter S O containeth in power bothethe lines SA,A'O,or the lines $ B,B O,orels SD,D O, but thelines S A,S D,S B are equal the one 0 the other, for they eche fubtend one of the angles of equal pentagos, Wherfore the other lines remaiing ,namely,A.O,B O,D Oare equal the one to the other:And by the fame reafon may be proued tat the diameter H D which fubtendeth the two rightlinesH A,AD, containeth in power both the hid two right lines,and alfo concaineth in power bothe the right lines H BandB D,which two right Ines italfo fuhtendeth And moreouer by the fame reafon the diameter A C,which fubtendeth the rigkt lines C B and B A,containeth in power both the faidright lines CB and BA. But the rightlines HAH Band CB are equal the one to the other,for that eche of them alfo fubtendeth one ofthe angles of equal pentagons: wherfore theright lines re~ maining,namely,A D, B D,andB A are equal the onc to the other. And by tle fame reafon may be proued that eche of thofe right lines AD,B DandB A is equal to ecle of theright lines A O,B O and DO. Wheefore the fixe right lines AB,BD,DA, AO,BO, & DO are equal the one to the other.Andtherefore the triangles which are made of thi, namely, the triangles AB D,A O B,A O DindB O Dare equal and equilater: which triangles therfore : do make a pyramis ABDO, wiofe bafe is A- B D and toppe the poynt O. Ehe ofthe. an- gles of which pyramis,namely, the angles at the pointes A,B,D,O,doe inthe felfe fame pointes touche the angles of heDodecahe- dron. Wherforeé the faid pyranis isin{cribed in the Dodecahedron, (by thefirft diffinition of this boke.) Wherefore in a Dodecahedron geuen, is infcribed a trilater quilater pyra- mis: which was Tequired to bedone. gy The 11 Propofition. > The 11.Probleme. at ll 9 Be: man Icofabedion genen to infcribe acube. T was manifeft by tle 7.ofthisbooke,that the angles of a Dodecahedron are fet in the cen- tres of the bafes of he Icofahedron.And by the 8.of this boke,it was proued, that the angles of acube are fet in ne angles of aDodecahedron. Wherefore the felfe fame angles of the cube, fhall of necefitie be fet in the centres of the bafes of Icofahedron. Wherfore the cube albe inf{cribed in the Icofahidron(by the firft diffnition of this boke..) Wherfore in an Icofahedron geuen,is included a cube: whrh was required to be done. gq Ube 12. Propofition. T he 12.Probleme. - Inan Icofabecron genen to infcribe a trilater equilater pyramis. = eer. ¥, a DY the former propdition it was manifeft,that the angles of a cube are fet in the centres of Ps ‘4 the bafes of the Icdahedron And(by the firft of this booke )it was playne that the foure an- a % } gles ofa pyramis ar fet in foure angles ofa cube. Wherefore it is euident, by the firlt diffini- ‘3 be-—mS1tion of this booke,hat a pyramis defcribed of right lines ioyning together thefe foure cen tres of the bafes of the Icofatedron,thalbe infcribed in the fame naihedron . Wherefore in an Icofa- dron geuen,is infcribed an ecuilater trilater pyramis: which was required to be done. The iw a ee 4 ~ se Rat ee we <3 >= "7 —— -“ 5 of Euclides Elementes: Fol.439. q Lhe 13: Probleme Lhe 13.Propofition. In a Cube genen, toinfcvibe a Dodecahedron. Akea Cube A D F L. And diuide euery one of the files therof into two equall partes & b pz in the pointes T,H,K,P : G,L,M,F: and pk Qf. dnd drawe theferight lines T K, | [nes GF,p Q,Hk, Pi, and M : which lines againe ditide into two equall pattes in the mesg pointes N,V,Y,1,Z,X. And draw thefe right lines N Y,V X,/and1Z: Now the three S Sd lines N Y, V X,and IZ, together with the diameter of the cube, fhall cutthe one the 7 Sas other into two equall partes in the centre of the cube by the 39. of the eleventh ; let that centre be the point O. And norte ftand long about the demonttration, vn- derftand all thefe right lines to be equall and parallels to the fides of the cube and to cut the one the other fight angled wife, by the 29. of the firit . Let their A halfes,namely, F V,G V,H Land kI,and | the reft {uch like, be divided by an ex treme and meane proportion, by the 30. of the fixth: whofe greater fegméts let be J ee BP the lines FS,G B, H C,andk E,&c. and drawe thefe right lines GI,G E,B C,and \ 4 BE. Nowforafmuch as the line GLis ] equall to the whole line G V, which is k ts the halfe of the fide of the cube: and the F line LE is equalltothe line BV; thatis, to the leffe fegmét : therfore,the fquares ‘ of the lines GI and IE, are triple to the fquare of the line GB, by. the 4.of the Xi NS thirtenth.; Butvnto. the {quares. of the RE lines Gi-and) LE, the fquare of theline K G E is equall, by the. 47.0f the firft.:, for \ the angle G1 Eis aright angle. Where- fore the {quare of the line GE is,triple to the {quare of the line GB. And foraf- much asthe line FG is ere&ed perpen- dicularly to the plaine AGKL, by the 4.0f the eleuenth ; for it is erected per- pendicularly to the two lines AGand GI: therefore the angle B GE is aright angle : for the line GE is drawen in the plaine AG kL. Wherefore the line BE, containing in power the two lines BG and GE, by the 47. of the firlt, is in power quadruple to the line G B (forthe line G E was proued tobe in power triple to the fame line GB): Wheretore the line B E isin length double to the line B G, xy the 20.0f the fixth . But( by conftruction ) the line C Eis double to the line IE + Wherefore the hales G B and I E, are in ropor- tion the one to the other, as their doubles B Eand C E : by the rs.of the fifth. Wherefore the fine CE is the greater fegment of the line B E diuided by an extreme and meane proportion . And forafmuch as the felfe fame thing may be proued touching the line B C : therefore th: lines B Eand BC, are equall, making an Ifoiceles triangle. Now let vs proue that three angles of the Iencagon of the Dodecahedron are fet at the pointes B,C,E : and the other two angles are fer betwene he lines B CandBE. Forafmuch as the circle which containeth the triangle B C Ecircunfcribeth the Pentagon whofe fide is the line C E, by the 11.0f the fourth : Extend the plaine of the tringle B C E, by the parallel lines dBand HE, cutting the line A D, namely, the diameter of AD the bife of the cube in the pointI. and letit cutthe line Ah the diameter ofthe cube in the point m. Aid by the point1 drawe in the bafe A D,a parallelline vnto theline Ad: whichletbeI1. And forafmich as from the triangle AH N is, by the parallel line 11, taken away the triangle AJ I,like vnto the whole triangle A H N,by the Co- rollary of the 2. of the fixth : the lines Al, and 11, fhall be equall. But a:the line H A is to the line Ad, fo (by the 2.0f the fixth) is the line H| to the line I I,or to the line 1 A,which is equall to the line1I. And the greater fegment of the line H Aw which is halfe the fide of thecube ) is, as before hath bene proued, the line Ad, that is, theline GB, whichis equall to theline Ad (by the 33.0f the firft). XXx.ij. Wherefore First part of the con/iruts 120%. Firl pave of the demon Jitucidiss Second part of the con- firudtion. Second pare of the De- mon firatiote _ , P| a qi » ve 1 i ie = - 3 ee : —— was A ——— a 8 - . = « —— - gna T he fonrtenth Booke Wherefore the greater fegment ofichedine H1 is che linelA And as the wholeline H 1 is to the grea- ter fegment, fo fhall the fame greater fegment H1 be tothe leffefegment 1A, by the 5. of the thir- tenth . Wherefore the line H A is di- | uided by an extreme and meane pros portion in the point 1. But in the tri- 4 d angle AHN, the line NA, which is drawen fr6 the centre of the bafe A D, isin the point’I cut like vnto the line AH, “by the parallel line 1 1(by the fame fecond of the fixth ) : for the lines HN and 11; are parallels, by conftruction. Wherefore the line. N Aisin the point I diuidedsby aw extreme and “meanesy.. proportion by the fuperficies dBEH. And forafmuch as the line YON which coupleth the centres of the oppofite bafes, isa parallel to the line HE: A plaine firperficies extended by the line Y ON, parallel wife to the plaine dB- EH : thetwo plaines fhall cut the lines A Oand AN{ the femidiameter of the cube, and the femidiamerer of the bafe AD ) inté the felfe fame proportions in the pointes m and{, by the 17. of the elenéth. But the line AN is in the point I diuided by an extreme & meane pro- portion : Wherefore the femidiameter of the cube is in the point m dinided by an extreme and meane proportion by the plaine ofthe triangle BCE. And A ' forafmuch as the reft of the triangles de- Third part of the confirut SiC Third part of tie en70n - firattote {cribed in the cube after the like maner, may by the fame reafons be sproned to be in aplaine which cuttcth the femidiameter of the cube by an extreme and meane proportion :; it is manifeft that threé plaines of the Dodecahedron fhall-ynder euery angle of the cube concurre in one & the felffame point Of the femidiameter being cut by an extremeand meane proportion . Nowrefteth: to proue that the right lines which couple that point of the femidiameter with the angles of the triangle BE C, are e- quall : whereby may be proned that the Pentagons are equilaterjand equiangle. 3 Take the two bafes of the cube. Whereon are fet the triangle BCE, namely , the bafes AFand A k, take alfo the fame diameter of the cube that was before, namely,Ah : and let the fide ferat the poynt n, of the fect- F on of the diameter by an extreame & meane proportion,be the lineC n or Bn: and letthe centre of the cube be as before the point O-And extend the lite C'nto the line B djandletit ¢on- curre with icin thepoint a.And foraf- | bs: much as the plaine which paffeth by the line HC Eand the centre O (cut- by conftruction * es pt that by the | aplayne fuper- yy | ting the cube into two equall partes) is parallelto AF the bafeofthe cube poynt n, be extende fictes parallel to the former parallel playnes, which fhall cutte the femidi+ ameterO A & che line C a,proporti+ i: G onallyinthe point n, by the 17. of the | eleuenth : For thofe lines doo touch the extreame parallel plaines exten- ded by the lines H EandE O , and by the lines Ad anddB. Butitis proued that the line O A is diuided by an ex- treame aud meane proportion in the poynt of Euchdes Elementes. Fol.4.60, poyntn : wherefore the line C a,is alfo diuided by an extreaime and méane proportion in the poyntn. Agayne forafmuch as B C Eis an Ifofcels triangle , and it is proued that the line BI cutteth the bafe C E into two equal! partes in the poynt [, che angles B I-C and B I E. fhall be rightangles , Lmagine by the line B Land the centre O aplaine to-paife (cutting the cube into two equall partes ) parallel to the bale AD. And ynto thofe plaines let there be imagined an other parallel plaine: paffinie by the poynt n: whichlet bene: which fhall cutte the femidiameter A O.and the halfe fide of the cube ; namely, the line 1H, like,in the pointes n and e by the 17. of the eleuenth . Wherefoxe the like IH isin-the poynt ¢ diuided by an excreame 8a meane proportid. Wherfore the line H eis equall.to the line C I or TE:namely,ech are lefle fegméts.And forafmuch as thé line Ie is to.the line I C( which is equallto the line E H)as rhe whole is to the greater fegment , take away from the whole line Ie the greater fegmét 1C:there fhallremayne the lefle fegmrent'C e’by the 5 .of the thirtenth . Wherefore the line Leis diui- ded by an extreame & meane proportion in the point C. Againe vnto the fame playnes imagine an o- ther playne to patie by the point a,parallel wife, and let the fame be a g.Now then (by the fame 17. of the eleuenth ) the lines Caand C g are in like fort cut in the pointesnande. Bur the line Ca wasin the point n cutte by an extreame and meane proportion , wherefore the line C g fhall be cutte in the poynt e,by an extreame & meane proportion.But the line I C is to the line C c,as the greater legment is to the lefle:wherfore the line C e, is to the linee g,as the greater fegment to the lefle:and therefore their proportion isas the wholeline IC isto the greater fegment C e,and as the greater fegment Ce is to the leffe fegment eg: wherefore the whole line C e g which maketh the greater fegmentand the leffe,is equall to the whole lineI C or E . And forafmuch as two parallel plaine fuperficieces (namely, that which is extended by I O B and that which is extended by the lineag ) are cutte by the playne of the triangle B C E,, which paffeth by the linesa gand1B,their common feétions a g and 1B fhall be paraliels(by the 16.0f the eleuenth ) . Buc the angle BI E or BI C isa rightangle, wherefore the angle ag C isalfoa right angle(by the 29, of the firft)and thofe right angles are contayned vnder equall fides, namely,the line g C is equal to the line CLand the line ag to the line B Iby the 33.0f the frit : wher- fore the bafes C a and C B are equall, by the 4.0f the firit.But of the line C B theline C E was proued to be the greater fegment : wherefore the fame line C Eis alfo the greater fegment of the line Ca: but cn was alfo the greater fegment of the fame line C a. Wherefore vnto the line C E,the line on which is the fide of the dodecahedron., ands fet atthe diameter,is equa!l.And by the fame reafon the reft of the fides which are fet at the diameter may be proued equall te lines equal! to the line C E. Wherfore the pentagon in{cribed in the circle where in 1s contained the triangle B C Eis, by the 11, 0f the fourth equiangle,and equilater. And foraf{mch as two pentagons , {et vpon euery one of the bafes of the cube doo make a dodecahedron, and fixe bafes of the cube doo receaue twelue angles of the dodecahedron: and the 8.femidiameters doo in the pointes where they are cutte by an extreame and meane proporti- on receaue the reft: therefore the 12. pentagon bafes contayning 20. folide angles doo infcribe the do- decahedron in the cube: by the 1. diffinition of this booke. Wherefore in acube geuen isinfcribeda dodecahedron: which was required to be done. Firft Corollary. The diameter of the Sphere which containeth the dodecahedron, containeth in power thefe two fides namely, the fide of the Dodecahedron. and the fideo [ ? JB) - the cube wherein the Dodecahedron ts infcribed. for in the firtt figure a line drawne from the centre O,to the poyntB the angle of the Dodecahedron,namely the line O B, containeth in pow- er thefé two lines O V'the halfe fide of the cube,and V B the halfe fide of the dodecahedron,by the 47. of the firft. Wherefore by the 15.0f the fiueth,the double of the line O B, which is the diameter of the {phere containing the Dodecahedron,containeth in power the double of the other lines O V and VB, which are the fides'of the cube,and of the dodecahedron. q Second Corollary. The fide of a cube dinided by an extreme and meane proportion, maketh the loffe fegment the fide of the dodecahedron infcribed init: and the greater feg- ment the fide of the cube inftribed in the fame Dodecahedron:Forit was before pro- wed, that the fide of the dodecahedron is the greater fegment of BE the fide of the triangle B EC: but the fide B E(which is equall to the lines G B and S F)is the greater fegmét of G F the fide of the cube: which line B E(fubtending the angle ef the pentagon) was(by tae 8.of this booke)the fide of the cube infcribed in the dodecahedron. XXx.il- Third se << Ss --.- Demonjira- tide ~~ . ) q : eh ry ff I a fl : 4 } : - ; 7 } - ‘ i i ‘ib | ' | i \ BR | : i} ( | ey 1 , 4 : a { \ ; ! ; : a4 , i - 1, ' : | i Mail ; 1; } ) 1 } : } i { } ; ! ‘ i \ Hal | we Ta ii - i Vay ) { i i 0] f 4 i ] i ide ft} wii) ra 7) i | i} in. i , ‘ i ty , hid { hat Uf) ay a ae if i ’ ; 1} ie imo ZZ ‘a } : { iP ih Ay ims | oe ; 4 | ae } it H ' ‘-) my Riel i i oy) Of ' {| a ti : ' 4 i rity * ae} y) ia )| ie ay | i We 4 ie ; am | q he : ’ A. 4 | bbe iB ; A. | ' , ; oe ab ‘ ; 4 a oP | a) 1) fie i) fen >> : 4 4) * Wt Aa ly d aA . 5 : : A | Sen } ry . | q vali ne 4 ¥ 3 ‘a. ete ¢ 2) r 4 me ¢ > 1 | : u, “4 , 4 : { | 7 ‘ es : ae : 1, i ’ 7 2a Lui to i i] iy ae i — = =x —_— —st age * ~ a hm greys a a A a ee PE a as aA 2 Sa: ” hi “a ee ee . <= Ss tt eee ios T hefiftenth Booke Third Corollary. The fide of a cube, is equalto the fides of a Dodecaledron infcribed in it, and cercum|c vibed about it,For it was manifert by this propofition,that the fide of acube ma- keth the leffe fegrment,the fide of a Dodecahedron infcribed in it, namely, as in the firft figure the line BS the fide of the Dodecahedron infcribed,is the leffe fegmét of the line GF the fide of the cube.And it was proued in the 17.0f the thirtenth,that the fame fide of the cube fubréceth the angle of the penta- gon of the Dodecahedron circumferibed:and therefore it maketh the greater fegment the fide of the Dodecahedron or of the pentagon, by the firlt corollary of thefame. Wheeforeit is equal to bothe thofe fegments. 3 “d a is , Ihe 14.Probleme. I he 14.Propofiuon. In a cube genen, to infcribe an Icofahedyon, aah, Vppofethat the cube geuen ye be AB C, the Centres of fee whofe bafesiet be the points Fvrn* D,E,G,H,1.K : by whiche ‘poyntes draw in the bafes vnto the o- ther fids parallels rot touching the one the other.And denide the lines drawn from the centres, as the line DT. &c. by an extreme and meane pro-! portion yn the poyntes A, F: L, M: N,B:P,Q:R,S :C,O: by the 30.0f Sy fixth:and Jet the greater fegmentes be about the cétres. And draw thefe right! lines,A L,A G,A M,and T G.And for-| afmuch as the lines cut are parallels to} the fides of the cube : they fhall make! right angles the one with the other by the 29.0f the firft : and forafmuche as they are equal: their fections fhall be e- qual,for that the fections are like by the 2.of the fourtenth. Wherfore the line T G is equal tothe line DT, for they are eche,halfe fides of the cube.Wherfore the {quare of the whole line TG, and of the lefle fegment T A, is triple to the fquare of the line AD the greater feg- ment(by the 4.of the thirtéth).But the line A G containeth in power the lines | AT &T G,forthe angle A T G isa right angle. Wherefore the {quare of th: line A G is triple to the {quare of the line A D.And forafmuch as the line M G Lis erected perpendialarly to the plain paffing by the lines A T,,& which is parallel to the bafes of the cube(by the corollar’ of the 14.0f the eleuéth) therfore the angle A G Lisa right angle But the line LG is equal to the lineAD, for they are the grea- ter fegments of equal lines : Wherfore the line A G (which is in power tripe to the line A D)is in po- wer triple to the line L G. Wherefore adding vnto the fame fquare of thelne A G, the fquare of the line L G,the fquare of the line A L,which (by the 47.0f the firlt) containethin power the two lines A- G and GL, fhalbe quadruple to the line AD or L G. Wherefore the line A 1 is double to theline A D (by the 20.0f the fixth : )and therfore is equal to the line A F,or to theline 1M. And by the fame rea- fon may we proue that euery one of the other lines which couple the next {tions of the lines cut, as the lines A M,P F,P M,M Q_and the reft are equal. Wherfore the triangles 4 LM;A PF,A M P,PMQ_ and the reit {uch like,are equal,equiangle,and equilater, by the 4.and eigth of thefirft. And forafmuch 2s vpon euery one of the lines cut of the cube are fet two triangles, as the triangles A L M,and BLM, there fhalbe made 12.triagles.And forafmuch as vnder every one of the 8.argles of the cube,are fub- tended the other 8.triangles,as the triangle A M P..&c.of 12.and 8.trianglesfhall be produced 20.tri- angles equaland equilater cétaining the flide of an Icofahedron,by the 25 .cifhinition ofthe eleuenth, which thalbe inferibed in the cube geuen A B C by the firft diffinition of thi: booke. The inuention of the demonttration of this dependeth of the ground of the former. Wherfort in a cube geuen,we haue defcribed an Icofahedron sthich Was required to be done. Firft > a4 of Euchides Elementes Fol.4 4.1. 3p Firft Corollary. T he diameter of « [phere ‘which containeth an Icofahedron, containeth two fides namely the fide of the Icofahedron, and the fide of the cube which contaie neth the I cofa Ledron for if we drawe the line A B, it fhall make the angles at the poynt A right angles:for that it isa parall:lco the fides of the cube : wherfore the line which coupleth the oppofite angles of the Icofahedron,it the poynts F and B,cétaineth in power the line A B (the fide of the cube) and the Jine A F (the fide o'the Icofahedron) by the 47.0f the firft. Which line F B.is equal to:the dia- meter of the {phere, whichcontaineth the Icofahedron,by the demonftration ofthe 1¢.0f the thirtéth. x Second Corollary. T he fix oppofite ides of the Icofahedron deuided into two equal parts:thetr fe&tions ave coupled ly three equal right lines, cutting the one the other into two equal partes, and pupendicularly in the centre of the phere ‘which containeth the Ic ofahedron. For tlofe three lines are the three lines which couple the centres of the bafes of the cube, which do in fuch: fort in the centre of the cube,cut the one the other,by the corollary of the third of this booke,and thirfore are equal to the fides of the cube.But right lines drawne from the cé- tre of the cube to the anglis of the Icofahedron, euery oneof them fhall fubtend the halfe fide of the cube,and the halfe fideof he Icofahedron (which halfe fides containe a right angle) wherefore thofe lines are equal. Wherby it s manifeit that the forefaid centre is the centre of the [phere which contai- neth the Icofahedron. x. Third Corollary. | T he fide of a cur denided by an extreme and meane proportion, maketh the greater fegment thelide of an I cofahedron defcribed in it.For the half fide of the cube maketh the halfe of the fice of the Icofahedron the greater fegment : wherefore alfo the whole fide of the cube,maketh the whoe fide of the Icofahedron the greater fegment by the 15.0f the fifthe, for the feétions are like by the 2.0 the fourtenth. @ Fourth Corollary. T he fides and bafes of the Icofabedron , which are oppofite the one to the o ther ,are parallels. Foafmuch as euery one of the oppofite fides of the Icofahedron,may bein the parallel lines of the cube, namely,in thofe parallels which are oppofite in the cube + and the triangles which are made of paralle lines,are parallels, by the 15.0f the eleventh: therfore the oppofite triangles of the Icofahedron,as alft the fides,are parallels the one to the other. q Uhe 1s. Probleme. The 15. Propofition. In an Icofahecron geuen, to infcribe an O€fobedron. Vppofe tha: the Icofahedron geuen be AC DF: and by the former fecond Corollary; Jet there betaké the three right lines which cut the one the other into two equall partes | perpendicuarly, and which couple the fections into two equall partes of the fides of \ | the Icofahelron : which let be B E,G H,and K L, cutting the one the other in the point ANS SEX | 1. And drwe thefe right lines BG, GE, EH,and HB. And forafmuch as the an- gles at the point I are (oy. conftruction ) right angles, Demon ra- and are contained vnder quall lines : che bafes GB and $30Me HE shall make a ey the 4.0f the firit . Likewyfe ynto thofe bafes fhall beequall the fines drawen from the pointes K and L,to tuery one of the pointes B,G, E,H: And therefore thetriangles which make the Py ramis BG E.H XK, fhall bcequall and equilater . And by the fame reafon fhall the ‘eft of the triangles which make the other pyramis B GEHL vpon the fame bafeB G- EH, be equall and equiater . Wherefore BGEH KL {hall be an O&ohedron : by the 23.definition of the ele- uenth): And fhall be infcibed in the Lcofahedron,by the firft definition of this bocke . Wherefore in an Icofahe- dron geuen, is inferibedin O@ohedron « which wasre- quired to be done. qi he ConStratlier. is 1 4 af ‘ ul . ‘ ‘ ; a bY . nt » Pe ‘al eH 2 ri yt) 1 4 4 i] Fy y) i 44) , De .; t 157 +h: a4 on ™ # : , : fi ; (a : N : -_ 5 : a vy a : A 2 i) : > ie ee) , a 7 a ; iy a | ot , ‘* - aa a 3 : : “ xd | it} we a ¥ . V7 ie ue q Wy re tin! ‘ ,) - al » aa ; - - fh } ".. | >}! . ’ oe na re ; - Z ye 5 fs) he ile Bk - ug a q 4, 4 of + at hh aes “Hi ie ae + =) fh » _ \ h oD ¢ 4 , ” BY - - t ~ (Re « ‘Lei AY \ . ia. $ - if BY ‘a ae, if 4 q p ‘ } % rue a ConfEruttion, Demonftra- ton. qy Ube 16. Probleme. 10 ‘Phe 16. Propofition. > & . » etree A \wdaun 0 Gobedron wenensto inferibe an leofahedron. Etthere be taken ‘an O&ohedron, whofe ¢.angles, let be A,B,C,FJP3L. And draw thé lines AC, BF PL jcutting the one the ochet perpendicularly in the point R (by the = 5 Corollary of the 14.0f the thiirtenth )'sAndletenery one of the i2.fides of the O@ohe- ” jron be divided'by amextréme and meane proportion,in the pointes H,X,M,K-D;S.N “A 35V3E,Q ,T. And let the greaterfegmentes be the lines B H,B X,FM,FK,AD,A Q; CS,CT,PN,PG,LV,L E+ And drawethefe lines HK,XM,GE,N V,D$,Q T. Now “otc sat) Peceftuch asia the rrianglesAB Fythedides ate cut proportionally; namely, as the line B H 1s to the line HA, fois theline F K to the line K A ( bythe2.of the fourerenth ) : therefore the line H K thallbe aparallelto the line BE (by thez.of the fixth) “And forafmuch as the line A’C’ cutteth the yr line. X in the poiie.Z, and the line Z K isa parallel ynto,thedine: RE, the line R Avihall be'cut by.an'éx- treme and meane pr Opertion in the poing Z : by the 2. of the fixth : namely, fhal] be cut like ynto the line B.A saad then:.- IGALW 2303 +3 ut 9: mx } greater feamér ther. : of fhall bethe Jing’ ZR Nato.the line ZK puctheline RQ... equall, by the. 3. of _. the firit: and drawé the line KO: now then, the line KO fhall beequalktothe line Z R, by the 33.0f the fifft’. Diaw -the ines Gy K Eyand KI... Andforafmsich » as the triangles A R- F,and A ZK ,are e- AZ and ZK,fhallbe equail the one to the: ether sby, the. 4.e€ thetys 50. fixth, for phe,fidegris schmsis : ds AR and RF, are e- guall. Wherfore the kine ZK fhallbe the «, lefle fegment of thes) line RA. But ifthe peu fegment RZ be.diuided by an ex- treme & meanepro-,., portion .the greater ©, fegment,therof thall. . e iy bes, 3, CQ ine | be theline ZK. which wasthe lefledegment of the wholelinéR A, by the'y.ofthe thirteenth. And for- afmuch as the two lines F Eand FE G,are equalltothe two lines A H and A K, namely, echare leffe { mentes of equall fides of the O@ohedron, andthe angles HA‘ and E FG are equall namely,are righ angles, by the 14.efthe thirtenth : the bafes K and G F fhall be equall,'by the 4.0f the frit Ah by the fame reafon vato them may be proued equall the lines XM, NV,DS$,and Q T . And forafinuth as - lines A C,BF and P-L;do cutthe one theothér into two equall parts,and perpendicularly, by con- Sage : the lines H KandG E (vhich fubtend angles of triairgles like ynto the triangles whofe an- gles the linesA C.B Band P L fubtend ) are cut into two équall partes in the'pointes ZantI, by the. of the fixth, fo alfo are the otherlines N VX M;D'S,Q T (which are équall vnto the lines H K & G E) cut in like fore, and they thall cut the lines AG; B F,and PL like . Wherefore the line K O ( whichis _ to RZ i thall make the greater feomentthelineR O,,which is €quall to theline Zk (for the erea- Whole line RT eset line ZK ) : and therefore the line © Tfhall bethe leffe feoment,when se the wee “ ine RY a equall to the whole line R Z Wherefore the fquares‘of the wholeline K O,and of the st e Be ys - are triple to the {quare of the greater fegnrent R O,yby the 4.0f the thirtenth. Wher- fore the line KI, Which containeth in power the two lines K O'and © T,isin power triple to the line R oO by the 47.0f the firit) for the angle K O Lis aright angle. And forafmuch as thelines FE and FG (which are the lefle feomentes of the fides ofthe OGtohedron ) are equall : and the line F K is c6- mon B | 4 4 al “f of Euchdes Elementes. Fol.4.42, mon to them both ; and theangles K E.G and K FE (of the trianglesof the @@ohedron ) are equall § the bafes K G and XE fhall (by the 4 of the firft ) be equall*: and therefore theahgles KIE and KIG which they fubtend, are equall ( by the 8.of the firft )}: Wherefore they are rightangles,by the 13 .0f the firft. Wherefore the right ae KE (whith containeth impower the two lines KT and LE, by the 47.0f the firit ) is in power quadruple to the line R O (orl E ) * for theiine K Lis proned to be in power triple to the fame line R O: Bur the line G Eis double to the line IE: Wherfore the line G Eisalfo in power quadfup!é to theline LE (by thezo.of the fixth’) . Wherefore.thetwo lines KE and G E are equall. And by the farne reafon, may the reft of the lines,namely; HK, HN, NV, VX, XS, andrheother lincs whichcouple the fections of the fides of the Octohedron. be provedequall to, rhe fame lines K Eand GE. Wherefore the triangles defcribed of them,namély,GEK,G K D,GDS,G $ M;G M Ejthall be e- qualfand équilater, by the 8.of the firft, making a folide angle atthe point G': which is therefore the angle'ofan Icofahedron,by the s6.of the thirtenth, andis fet in thefe@ion G of the fide PF. And by the fame reafon may be proued,that theyreft of the eleuen folide angles of the Icofahedron,-ate ferin thefections of cuery one of the fides of cae O&ohedron,namely, in thedpointes E;N,;V,H,K,;M,X,D,S;, Q,T . Wherefore there are 13. angles of the Icofahedron . Moreouer, forafmuch as euery one of the bafes of the Octohedron, do eche containe triangles of the Icofahedron, as im the pyramis AB C+ FP (which ts the halfe of the O&tohedron ) thetriangle F C P receaueth in the fections of his fides the triangle GM S : and the triangle C PB containeth the triangle N XS : and thetriangle BA P. contay- neth the triangle H N D : and moreouer the triangle P F containieth the triangle KD-G,and the fame may be proued in the oppofite pyramis A B.C F L : Wherefore there fhall be eight triangles . And for- afmuch as befides thefe triangles, to euery one of the folide angles of the O@ohedron are fubtended two triangles,as the tridngles K EG and MEG, tothe angle F : and the triangles HNV and XN Vy, to the angle B valfo the triangles ND Sand GDS, tothe angle P : likewifethe triangles DHK and HK, to the angle A = Moreouer the triangles EQ.-T and V Q T, to the angle L: and finally the tri- anglesSX¥M and IT XM, to the angle GC’ thefe rz. triangles being added to the 8.former triangles, fhall produce 20. triangles equalland equilater coupled together : which fhall make an Icofahedron,by the 25.definition of the eleuenth : and itfhall be infcribed in the OGohedron geuen AB CF P L ,yby the firft definition of this booke : forthe 12, angles thereof are fet in 12. like aions of the fidesof the Octohedron. Wherefore in an Oétohedron geuen, isin{eribed an Icofahedron. @; Firft Corollary. The fide of an equilater triangle being diuided by an extreme and meane proportion: a right line fubtending within the triangle, the angle which is cone tained pnder the greater fegmentand the le/Se: is in power duple to the leffe fege ment 0 if the fame fide . Fortheline KE,which fubtendeth the angle K FE of the triangle AFL, which angle K F E is contained vnderthe two fegmentes K F & F E,was proued equall to the line H K, which containeth in power the two leffe fegmentes H A and AK, by the 47. of the firft, for the angle HA Kis aright angle. Wherefore the line K E or H K; is in power duple to the line AK, x Second Corollary. The bafes of the Icofahedron are concentricall ( that is, haue one and the felfe fame centre ) with the bafes of the O&ohedron which contayneth it, For fuppofe that A B G be the bafe ofan O@ohedron contay- ning E C D the bafe of an Icofahedron : and letthe centre of the bate AL G bethe point F. And drawe thefe right lines F A,FB, FC,and F E. Now then the rwo lines F Aand)A E:thall be equall tothe two lines FBand BC : for they 4re linesdrawen from thecentre, and are alfo leflefegmentes : and they contayne the halfes ofequall angles . Wherfore ( by the 4.0f the firft) the bafes FC and FE are cquall : and by the fame reafon vnto them fhall - be'equigif the other line FD. Wherefore niaking the centre the point F* and th@fpace FE deftribe a circle and it fhall be cir- cum{tribed aboutthe triangle C ED : and fo fhall the point F the centre of the bafe of the OGohédron be the centre of CED the bafe of the Icofahedron. T he fiftenth Booke q Tbe 17. Probleme. T he 17. Propofition, In an O&ohedron genen, to infcribe a Dodecahedron. Conktrafion. {Sa >2= 05) Vppole that the O@ohedron geuen beABGDEC : whofe 12.fideslet be cut by an ex- (XK treme and meane proportion, as in the former Propofition.. It was manifeft that ofthe aN right lines which couple thefe fe@ions,are made 20.triangles,of which 8. are concentri- call with the bafes of the Oétohedron, by the fecond Corollary of the former Propofi- tion. If therefore in euery one of the centres of the 20.triangles be infcribed (by the 5.of this booke ) euery ene of the 1a.angles of the Dodecahedron, we fhall finde, that &, an- Demonsira- gles of the Dodecahedron A $16 %- are fet in the 8. centres of the bafes of the O@ohe- dron: namely ,thefe angles ¥,a, &, O, M,2a,P,and X: and of the other 12: folide angles there are two in the centres of the two thian- gles which hane ‘one fide common vadér euery one of the {olideaigles of the Oégtohedron : namely, vn- der the folide angle A, the two folide angles;K,Z: vn- der olide angle B, the » ewo folide angles H, T: vnder the folide angle G, the two felide angles Y,V: vider the folide angle D, the rwo folide angles F,L: vnder the folide angle E, the two folide angles $,N: vider the folide angle C, thetwofolideangles QR: and forafmuch as in the Gaoliedromare fixe folide angles, viiderthemfhallbe. : if | fubtended 12-folide angles © ~~ ) Oe ee ee , of the Dodecahedron: and | G. | HRY fo are ne 20, Paes an- é om aay} oataho't df ~ les compofed of r2.equall and equilater fuperfici ntagons (4s it was manifelt, by the ¢.of this ke) which therefore Satie Soda dias. definition of the ede . And itis infcribed in the OGtohedron (by the 1.definition of thisbooke ) : for that euery one of the bafes of the ab Bags do receaue angles cherof. Wherefore in an Oftohedron geuen, is infcribed a Dodeca- Caron. # = ies 4 * a = z 4 Pe eT q Ibe 18. Probleme. ; The 18. Propofition. a: AS Ina trilater and equilater Pyramis, to infcribe a Cube. < es SSS Vppole that there bea trilater equilater Pyramis,whofe bafelet be AB C,and toppe the Re pitas D.Andletitbe ceniprenenden in A Sees by the 13. of the oe r ag th. Andles SY ithe centre of that Sphere be the point E. And from the folide angles A,B,C,D, draw righe BO lines pafling by the centre E, vnto the oppofite bafes of the pyramus, and they fhall — perpendicularly vpon the bafes, and fhall alfo fall vpon the centres of the circles which containé the bafes, by the Corollary of the 13.0f thee irtenth . Letthe centre of the tri- angle A B C, be the point G, and ler the centre of the triangle A D C be the point H, and of the erian- gle A DB let the point N be the centre, and finally,let the point F be the centre of the other triangle DBC. Andlet the rightlines falling vpon thofe cétres be D E G,B EH,C EN,& AEF. And by thofe centers G,HN,F,let therebedrawen sae the angles to the oppofite fides thefe rightlines, AGL, | DHK; Construltion. ; i wy 7 l | 1 f ‘ Bs * it i ) + ' H, ; ea N H { 4 th a) : : = 7 (i t { d 7 \& : 8, y : ' : 4 - : % ; if ‘i iy \ | : ’ - bie : o } { 1 - oa ] er —— 4 ek eee Se SB —- -— la oNe ee = ee -— 7 es + 2. Se ee er t of Euclides Elementes.. Fol.4.43. DHK,BN M,andD-E L, which fhall fall perpendicularly vpon the fides B.C,C.A,A D, and\C B,by the Corollary of the r2:0f the thirtenth, and chelafore they thal cut’ them: into two equal] partes ia the pointes K, L, M, bythe . 3.0f the third. Agayne letthelines which wer drawen from the folide angles to the oppofite bafes be diuided into two equal partes,name- ly, the line DG in the point T,the line C N id the point O, the line AF inthe point P,and the line BH in the oint R: and drawe the bites HT,ET,H O,and EO. Now forafmuch as the lines GK, and G L, which are drawen Demon fira- from the centre of one tron. and the felffame trian gle ABC to the fides, are equall,and the lines D KandD Lare equall, for they are thé perpen- diculars of equal & like Ad triangles : and the line Produce in D G iscommon to thé. the figure the Wherefore,by the 8.of line T Fro the firft,the angles K D- the print B. G & LDG,are equall. And forafmtich as the lines HD &DF are dra- wen from the centre of - equal circles which cé- eaine the equal triangles A DC & DBC, therfore they are equal, & the line D Tis comon to thé both, and they containeequal angles,as before hath bene proued, Wherfore the bafesH T and FT are equal by the 4.of the firft.And by thefame reafon.if we drawe thelines C Fand C H, may we proue that the other lines H.O.and F.O, are equalto the fame lines HT and. F:T,and alfo the one to the other. Wher- fore alfo.after the,fame maner may be proued that the reft of the lines, which couple the centres of the triangles and thefedtions.of the perpendiculars into two equal partes,as thelines NP,GR,G P,RN: NT,P H,GO,andRF,are equal, And forafmuche as from euery one of the centres of the bafes are drawne threright lines to the feCtions into two equal parts of the perpendiculers, and there are foure centres,it followeth,that thefe equal right lines fo drawne,aretweluein number,of which euéry three and three make.afolide angle in the foure.centres of the bafes, and in the foure{eétions into two equal partes of the perpendiculars: wherfore that folide hath 8.angles;contained vnder 12.equal fides;which make fixe. quadrangled figures, namely; HOF T,PGRN,PH OG,G OFR,FRN T,and TNPH. Now let vs proucthat.thofe quadrangled figures are rectangle, | Fora{muchasvpon D.C the common, bafe'of the triangles A D Cand BD C falleth the perpen- diculars AS and BS,which are drawne by:the centres H and F; either of thefe lines H S and $F fhalbe the third part ofeither of thefelines AS and § B:for the line AH is duple to the line H S,and deuideth the bafe D C into two equal partes by the corollary of the r2.0f che thirtenth. Wherefore in the trian- gle AB S the fides A S and B Sare cut proportionally in the poynts H and F:and therfore the line H F is a parallel to the fide AB, by the 2.of thefixth. Wherfore the triangles A S B and H S F are equiangle, by the 6.of the Gxth. Wherfore the bafe H F fhalbe thethird part of the bafe A B,by the 4.of the fixth. Weunay.alfo proue that theline T O isthe third part of the line D.C sfor the lies E C and ED, which gre drawhe from the centre of the {phere Which containeth the yramisare equal: and theline EN, (which is drawne from the centreto the bafe)is the third part of the line E C,foalfo is theline G E the third partof the line E D(by the corollary of the 13.0f the thirteath) for itis the ftxth parte of the dia~ meter of the {phere which containeth the pyramis: And the line O N,1s the halfof the whole line NC wherfore the refidue E O is the third part of the line E Ciand fo alfo is the line ET the third part of the fine E D. Wherfore the line T O in the triangle D E C isa parallel to theline D C,and is a third parte of the fame,by the former 2.and 4.0f the fixth,as the line H F was proued the third part of che line AB. But A B and DC being fides of the pyramis are equal. Wherfore the lines HF and TO, being the third partes of equal lines,are equalsby:the 1 s,ofthefiueth. Wherfore by the 8.of the firftthe angles H T F, and T F O are equal:and by the fame reafon,the angles oppofite ynto them, namely, the angles F O . YYy.¥. an The ffienth Booke ee and O'H Tare équalithe one to the other, and alfo areequa} to the faid anglesH T Rand TF'O SSS ee ee “ere The fiftenth Booke 7] Vppofe that the pyramisgeuen be AB GD ;eche of whofe fides let be cutteinto two €- Iquall partes:and draw the lines which couple the fections , which being diuided by an >| extreame and meane proportion , and right lines being drawne by the fections, fhall re- ‘ceaue 20.triangles making an Icofahedron , as in the former propofition it was manifeft. &@| Now then if we take the centres of thofe triangles, we fhall there finde the 20. angles of | : the dodecahedron inferibed init by the ¢.of thisbooke.And forafmuch‘as4.bafes of the forefayd Icofahedré are cécentricall with the bafes of the pyramis, as it was proued in the 2.corollary of the ¢.of this boke: there thal be placed 4.angles of the dodecahedr6,namely,the 4 angles E,F,H,D, in the 4.centres of the bafes:and of the other i1¢.angles, vader every one of the 6. fides of the pyramis are fubtended rwe:namely,vnder the fide A D,the angles C.K: vnder the fide B D theangles. LI : vn~ der the fide G DtheanglesM;.N: ynderghefide A B the angles T, S:: ynder the fideB G the angles P, O:and vnder the fide AG the angles RJQ:fo there reit 4. angles , whofe true place we willnow ap- bene uch.as.acubecontayned in one and the felfe fame {phere with a dodecahedron, is infcri- ed in thefame dodecahedronyas it was manifelt by-the 17.0f the thircenth,and 8.of this booke;it fol- loweth thata.cube anda dodecahedron circum{cribed aboutit., are contayned in one and the felfe fame bodies, for that their angles conourre in}one and the felfe fame poyntes.And it w2s proued in the 28.0f this booke,that 4.angles of the cube infcribed in the pyramisare fet in the middle fections of che A . « . t« - ~ ® 7 - e ; : ct . * y —_— ; ” FF 3 : ~ 4 : = . Fs - ~~ Pe? 2 os - > as s~ ’ ' ; B be t ¢ 4 . Y ; 7 < +t : : 4 , } . rom Do re oe% Fite S G <> j . 7 J _ : Fal > ' 4 : - a ‘ ~ ae “ wAhe ' -« erpendiculars which are drawne from the folide angles of the pyramisto the oppofite bafes : where- di the other 4.angles of the dodecahedron are alfo,as the angles of the cube, fetin thofe middle fee- tions of the perpendiculars.Namely,the angle V is fet in the middeft of the perpendicular A H: the ane gle Y in the middeft of the perpendicular B F; the angler in. the middeft of the perpendicular.G E:and Finty the angle D in the middetft of the perpendicular-D . which is drawne from the toppe aRERe PT” ramus to the oppofite bafe . Wherefore thofe4.angles of the dodecahedron may be fa i y vnder the folide angles of the pyramis,or they may be fayd to be fet at the fi sata Therefore the dodecahedron after this maner fet,is infcribed in the pyramis geuen ( by the firft diffinition of this _ booke ) for that vp6 euery one.of the bafes of the pyramis are fet an angle of theidodecahedré infcri- bed. Wherefore in.a erilater equilater pyramis is infcribeda dodecahedron. to > Thezt.Probleme. The 21-Propofition. °°" * In enery ene oF the Feo% + lolides toil ibea Sphere. 7 We Of the Veale folides to taliribe's Sphere. , of Euctides Elementes Fol.44.5. tN the 13. of thie thirtenth ahd the other 4. propofiti@tis followih¢ , it Was declared that =) the y.regular folidessarefo contaynedin afpherestharight lin¢s drawnefrom the cen- Thats propofies- 4p tre of the {phere or of the folide inferibed ,.to.eutry owt ofthe anglés of the folidein- 0” Campane jos {cribed,are equall . Which right lines theretore mike pyramids , whofetoppesarethe hath, & zs the ) centre of the {phere , or of thefolide, andthe bafes are culery one of the bafes of thofe laft alfoin ors @ {olides . And forafmuch as thofe bafes are in tuery folidé ‘equall and like the one to the soni : ; derof the x, other,and defcribed in equalheircles:thofé ingles fhall cutte the {phere : for the angles oaks ee ; 7 which touch the circumference of the circle,touch alfo the fuperficies of the fphere. Wherefore perpé- hiss ‘ diculars drawne from the centre of the fphere to the bafes , or to the playne fuperficieces of the equall circles,are equall,by the corollary of the aflumpt of the 16.0fthe twelfth . Wherefore making the cen- tre the centre ofthe fphere which contayneth the folide, and the {pace fome one of the equall perpen- diculars;defcribe afphere, abd it fhalltouch,etiery one of the bafes of that folide : neither thallthe fu- perficies of the {phere pafle beyond thofe bafes: when as.thofe perpéndiculars are the left lines which are drawne from the centre to the bafes,by the 3.corollary of the fame aflumpt. Wherefore we hauein every one of the regular bodies infcribed a {phere which regular bodies are in number onely fiue, by the corollary ofthe 18.of the thirtenth. A Corollary. |. T hevegular figures infcribed in-fpheres , and alfo the [pheres circum|crie bed about them or contayning them bane one'and the felfe fame centre. Namely; their pyramids’, the angles of whofe bafestouch the fuperficies of thefphere , doo from thofe angles caufe equall tight lines to be drawne to one and the felfe fame poynt's making the toppes of the pyta- mids in thefame poynt : and therefore they make the‘centresofthe {pheres in the felfe fame toppes: when as the right lines drawne from thofe angles to the crooked fuperficies, wherein are fer the an- gles ofthe bates of che pyramids,are equall. 3 hy : An aduertifment of Fluffas:: Of thefe folides,onely the OGohedron receaueth the other folides infcribed one within an other . Forthe O@ohedron contayneth the Icofahedron infcribed in it : and the fame Icofahedron contayneth the Dodecahedron in{cribed in the fame Icofahedron: and the famé dodecahedron contayneth thecube infcribed in the fame O@ohedron, and finally'the fame-cuibe eirchm{cribeth the Pyra- mis infcribed in the fayd O&ohedron . But this happe- neth notun the other folides. Pee GD Theendé of the fiuerenth Booke “of Euclides Elementes after’ Campane and F befvas. . = oS . i < = a ms — — . a . ' —— — a = — a = Aaa ae —— nm ci Se <3 sana RSinime : =. = = " 3 = SF ei ee ae So el a oe 3a —— so at Sap + 5. < pope, Pee eel a = SS —_— _ ~ cl — NE a > - Ses Sie a = — -- ke ons = eee = ~ Po iege Ps ase aS Re meatl iy j a ge. Hy ' —— + —— a ees The argn- ment of the 16. booke, the fame: lattly they are inf{cribe Lin one andt a fel fe 1 hefixtenthbooke of | ~~ theElementes of Geometrie 9 peehntah sibel toinferibe the fiue regular (olides one with in an other. : RR ~~ Nithe former fiuetenth booke hath bene taught how ‘ \ Now femeth to'reit,to copare thofe folids fo infcribed, prieties; which thing ,Fluffas confidering, in this fixcéth booke added by him, hath excellently well and moft conningly Rerorped - For which yvndoubtedly he hath of all then which‘haue aloueto the Mathematicals,de- UL . ferued much prayfe and commendacion:both for the Ae -Kteattrauailes and paynes (which itisnoftlikely ) he 1 hath taken in in yenting fuch dtraunge and wonderfull y : aS aes ey table, butalfo occafion of inuention of greater things pertayning tothe natures of the ° cular folides. PR SS 2 gg ap ae sae fiue resuiar Olle eSs. i 19030 SfF MISNESS9i NOPso/SO sii) Pre, OF O93 i : 4 = 1 . : : * » y - “Ostia 6 ae te t Se MGtie tt rar a °F i Fe ¢ 4 Sti As 2S ZAM SA wha had shay ml A 4 i ki34) P . t* € . oe” : a ~hrer - rt ore} ; : viee > . oun . 7. *ROWDsTE OD) 5: Sfi3 nt Sarin: : S$ Pes Jatt INA Giis bi, | v dee ; OS oki | t * “~ ’ : i P 1 ~ a m F - ay mae + <4 £3 7 “ +s . =» ~ - ” » i Srk Lag si 4b Ase uh EF } Bad wt be MV? Lise ViiboOLoay : 2 bs4 vodinsleniGfid 20x ET ODOMGOM e:: nos'- cc: Li dine ff 2€,. * L. \pofati ° ght? e f1C . ~— . - . . ” - . : ~ g : ro aer34 . > : i trek nTrfe ms : ao . . LPs 7 : Shawe “eggs eid 2M; GO" ater Seen Ab. i 2m | A Dodecahedron, anda cube’ infcribedin it, and a Pyramis infcribed in 1} Or the angles of the pyramisare fepimthe angles of thercube wherein it is infcri- | | béd (by ce firftof Wakalbdckhihad altchcdbplas bf tic cube are fer in the angles “¥ ¥! of the dodecahedron circumfcribed tit:(by the 8.of the fiuetenth) : Andall 3_S\\t the angles of rhe Dodecahedron, ate fet in the fuperficies of the fphere,by the t7. be = of the chirtenth. Werefope théte og: Bis .* ~ & a y . ie “ [ear A at sae a T hefe three folides lth lelfe fameI cofabedron sor O Gohedron ,or yams | h ‘fame Icofahedron,by the,5.11.8 : fame Oftohedron, by the 4. ¢.and 16,0f nis, byithe firft,18.and 19.of the fame. vot the circumferibed Icofahedron, Nad 12.0f the fiuetenth: and they ae ‘ibed if * * — band é Ls = " For the angles ofall thefe folides or octohedron,or pyramis. a % * a, fr ff F: , | Ce 7 eS pole yy = incum/fcribed about a cube,toa Dodecas ) hedron ya. One toan other, and to fer forth their paffions and pro- Elementes of Géotmetrysadded by Flufas. Fol.446. ~ bédyainferibed in thefame cube is tripletolan extreme &meane propartid, j Orafmuch a5,in the.2.corollary of the 13.0F the fiuetenth, it was proued, that the fide Via of a Dodecahedron inforibed in a. cube,is the lefle fegment of rhe fide of that cube de- b) WY uided by an extreme and meane propertion:and the fide of the dodecahedron ernst ey | {eribed about the fame.cube,is A greater fegnyentof the fide of the fame cube(which eee thing alfo was taught in.the 13.0f the finetenth)the fide of the Dodecahedron circum- ») oi Fight line deuided;by an extreme.and meane proportion, is to the leffe fegment of the fameswhich preportion is called an extreme.and meane proportion, by the diffinition,and by the 30,0 fixthsBnethe preportign of like folide prolihedrons,is triple to the proportion of the fides of like pro- portion,by the corollary of the 17.0fthe twelueth. Wherefore the proportion of the Dodecahedron circumfcribed about the cubgsis to the dodecahedron infcribed in oF fame cube, in triple proportion of the fides ioyned together by.an extreme and meane proportion The proportion therefore of a Do- decahedron circumicribed abottga cube to a dodecahedron infcribed in the fame cube, is triple to an extreme and meane proportion. \ Pe | \, The 3.Propofition.” In enery equiangle and equilater Pen tagort, a perpendicular drarwne from one of the angles tothe bafe js deuided by anextreme and meane proporti> on.by aight line Jubtending the fame angle , | $ : a Vipofethat AB CDT bean eqitiangle ahd equilater » nee on A 4 Se pentagon : and fromon¢ of the angles iamely, from = ConStrattion. JCM yi A. let therebe drawne fo the bafe C D2 perpendicusaiss ai {hal rs! lar A G:and letthe line B F | fubtend the angleB ABgtiogg Demon? which line B Flet the line AD cutinthe poyotl. Then] fays |: tiow = wht@ that the line B-F,cutceth the line A G by an extreme and ngeane a [2 proportion. For forafmucheas the afiglesG A FandG A Bare: aj\qs> equal by the27.of the third, and theangles A BFandiAP Bfhre:) | \\e-; equal by the s.of theifirft-cherefore the anglés remaining at'the =)|:! \:» poynt E,ofthe triangles A EBancha’E Fareéqual:for'thatthéy:::\v -\. ate thé refidués of two right anglesby the corollary of the 3220610: \ the firit.: Buc the angle E G Cy is by conftruction.a right dnglec: yj! wherfore thelines B-F & C D are parallelsbyiche 28.0f the firlt) (oc. ‘Wherefore as:the line D I iis towhe line TD A,fo is theline-GiEta eo: 2) cic) _the line EA by the 2, of the fixth But the line DA, isin the pointe’: od. sot I deuided by an extreme and meaneproportidn, by the 8-oftlie thirteenth. Wherforetheline G Aisin the poytt E,deuided by an extreme and meane proportion(by'the.z.of the fourtenth). Wherfore in e- uery equiangle and'eyuilater pentagonja petpendicular drawne from one. of the angles to the bafe, is deuided:by an extreme and meane proportion by a right line fubtending the fame angle. eewites TOWNE gt > cTisgA ‘7 + ~ re *B 4 Pyra- sotaghtlines cutting the fayd fides by an extreme and meane proportion:they mis made fralb.containe the bx/eof the Icofahedron inferibed im the Pyramis, which ftand «Te Sr at Po é, | trabedron ; throughout al meane proportion. 13 this booke. *d> fctibed,fhalbe ro thefide ef the Dodecahedron infcribed, as the greater fesment of 4 eat A A AS ae oo ee oe : == — a ee Sita Ss —— See . The fixtenth Booke of the Z Vppofe that A B G be the bafe ofa prams, in which let be infcribed an equilatér triangle N CAAA E K H,which is done by deniding the fides into two equal partes . And in this triangle let Sif there be infcribed the bafe of the Icofahedré infcribed in the pyramis: which is defcribed by YANG deuiding the fides FK,KH,HF, by an extreme & meane proportié in the poyats C, D, E, by che 19.0f the finetéth. Againe let the fides of the pyramis,namely,A B,B G,and G A be deuided by an eXtreme and meane proportion in the poynts1,M,L,by the 30.of the fixth. And drawe thefe right bic Demon'ra- A M,BL,G1.Then I fay that chofelines defcribe the triangle CDE of the Icofahedren. For forafmuch pte as the lines B G and F H are parallels, by thez.of thefixth: by the point D let the line ODN be drawne : nS co either of the lines B G & F H. Wherfore the triangle HDN fhalbe like to the triangle HKG, y the corollary of the z.of the fixth. Wherfore either of thefe lines DN and N H fhall be equal to the line D H,the greater fegment of the line K H orF H. And forafmuchas the line F Oisa parallel to the line H K,and the line O Dto the line F H:theline O D hall be equal to the whole line F H in the pa- rallelogramme F O D H,by the 34.0f the A firft. Wherefore as the whole line F His to the greater fegment FE, fo fhall the lines equal to them be, namely, the line O D to the line DN, by the 7.0f the fifth. Wherfore the line O N is deuided by an extreme and meane proportion in the poynt D,by thez.of the fourtenth. Bue the triangles AO D,AF E,and ABM, are like the one to the other, and fo.alfo.are the triangies ADN, AEH, andAMG, by the corollary of the fecéd of the fixth- Wherefore as FEistoEH,fois O Dro D N,and BM to M G. Wherfore the line A M cutting the lines F H and ON, lyke vnto the line B G in the pointes E,D,M; defcribeth E Dthefide of the triangle of the Icofahedron EC D, which is defcri- bed in the feétions E,C,D, by fuppofitio. And by the famereafon the lines BL and G I thall defcribe the other fides EC and § R. i C D of the fame triangle.By the point E, fi G let there be drawne to G Ia parallel line P. E Q.Now fora{much as the lines B M and F Eare parallels, the line A M is in the poynt E, cut like to the line A B in the poynit F,by the 2.of the fixth. Wherefore theline A Eis equal to the line E M:and vnto theline EM alfo are equal either of the lines G Dand D I: which are cut like vnto the forfaid lines. Againe forafmuche asin the triangle ADI thelinesDI and EP are parallels,as the line DI is to the line E P, fois the line AD to theline AE: butas theline A Dis totheline AE, fo isthe line D Gro the line E Q by the2.of the fixth: wherefore as the lineD I is tothe line EP, fois the line D G to theline E Q:and alternately asthe lineD Listo the lineDG, fo isthe line E P to theline EQ :butthelinesD1I and lGare ¢qual: wherfore alfo the lines EP and EQ. are eaualAnd forafmuch as the line A H isequal to the line F H,whofe greater fegmét is the line HN: therfore the whole line A N,is deuided byan extreme and meane proportion in the poynt H, by the s.of thethirtenth, But as the line A N isto the line A H,fo is the line A D to theline A E,by thez.6f fixth(for the linés F Hand O Nare parallels: Jand againe as the line A Dis to theline A E, fo ( by the fame) is the line A G to theline A Q, and the line AT to theline AP « forthe lines P Q,andGI are parallels : Wherefore the lines A G and A Tare deuided by an extreme and meane proportion in the points Q & P: & theline A Q fhalbe the greater fegmét of the line A Gor A B.And forafmuch as the whole line A G is to the greater fegment A Q_, as the greater fegment AI is to the refidue A P : the line A P fhalbe the leffe fegment of the wholeline A B or A G.Wherfore the line PEQ (which by the poynt E paffeth Lead to the line GI) cutteth the lines A G and B A by an extreme and meane proportion in the poynts Q and P.And by the fame reafon the line P R (which by the poyntC, affeth parallelwife to the line A M)fhall fall vpon the fections P and R; fo alfo fhal the line RQ(which y the poynt D paffeth parallelwife to the line B L)fall vpon the feétions RQ. Whereforeeither of the lines P Eand E Q fhalbe equalto the line C D,in the parallelogrammes P D,and Qe. the 34.0f the firft. And forafmuch as thelines P Eand E Qare equal,thelines PC,C R,R DandDQ albe likewife equal. Wherfore the triangle P R Q is equilater,and cutteth the fides of the bale of the pyramis in the poyntes P,Q..R, by an extremeand meane proportion.And in itis infcribed the bafe E C D.of the Ico- fahedron contained in the forefayd pyramts. If therefore from the angles of the bafe ofa pyramis,be drawne to the oppofite fides;right lines cutting the fayde fides by.an extreme and meane proportion : they fhall conraine the bafe of the Icofahedron inferjbed in the ie which bafe fhall be infcribed in an equilater triangle;whofe anglescut the.fides of the bafe of the pyramis by an extreme & meane proportion. | SS ——— P Kot a La vat SE —"= =~ a = an as a oS es — a! Cenftrultion. y- a a eee > — =< ! H 4) i @A Elementes of Geometry,addedby Flufeas. Fol.aayy, q A Corollary. fier} Vhe fide of, an Icofahedron infcribed in an Offobedron; is the greater fegment of the line , which, being drawen from the angle of the bafe of the Otéohedron cutteth the oppofite fide by an extreameand meane proportion. For,by the 16,0f the fiuetenth,F KH is the bafe of the O&tohedron,which containeth the bafe of the I- cofahedron CDE: vnto which triangle F KH, the triangle H K G is'quall,as hath bene proued. By the point H draw vnto thé line M E 2 parallel line H T, cutting thelline D N in the point $. Wherefore ES, DT,and ET,are parallelogrammes : and therefore thelines EH and M T are equall : andthelines EM and HT are like cut in the pointes D afid S,by the 34.0f the firft . Wherefore the greater fegment of the line H T is the line H S,which is equall to ED the fide of the Icofahedron. But(by the 2.of the fixch )the line T K iscut like to the line H K by the parallel D M.And therefore(by the 2.of the fourtenth) it is di uided by an extreme and meane proportion.But the line T M is equall to the line EH. Wherefore alfo the line TK is equall to the line EF or DH . Wherefore the refidues EH and T Gare equal]. Forthe whole lines FH and.K G are equall. Wherefore KG the fide of the triangle H KG is in the point T diui- ded by an extreme and meane proportion in the point IT, by the right line HT ,and the greater fegment thereot isthe line ED the fide of the Icofahedron infcribed in the O@ohedron, whofe bafe is the trian- gle H K G (or the triangle FKH which is equall to the wiangle HKG) by the 16.0f the fiuetenth, . q The s. Propofition. T he fide of a Pyramis diuided by an extreme and meane proportion, mas keth the leffe fegment in power double to the fide of the Icofahedron ine feribed init. Ress ; . | Ne Ps oa i Vppofe thatAB G be the bafe of a pyramis:/ and lee the bafe of the Icofahedron infcri- Compu fion. Se EN ibed in it,be C D E,defcribed of three rightlines,which being drawen from the angles , ¥ LF) f oo, NOOK 2/2. lof the bafe A B G cut the oppofite fides by an extreme and meane proportion, by the for- be NY; ) -- | mer Propofition:namely,of thefe three lines A M, B I,and G L. Then I fay, that A lithe WS ealeffe fegment of the fide A G,is in power duple to C E the fide of the Icofahedron . For, ‘foraftfuch as by the former Propofition,it was proued thatthe triangle C DE is infcri- bed in an equilater triangle, whofe angles cut the fides of Demonfira- ABG the bafe of the pyramis by an extreme and*meane | £80Me proportion, let thar triangle be F HK, cutting the line AB A in the point F . Wherefore the leffe fegment FA is. equal | - tothe fegment Al, by the 2.of the fouretenth : (for the lines AB and AG are cutlike ). Moreouer thefide FH of the triangle FH Kis in the point D cutinto two equall partes,as in the former Propofition it was proued,and F C+ ED alfo by the (ane is a parallelogramme: Wherefore the ' lines C E and F D are equall,by the 33,0f the fitft.And for- afmuch as the line F H fubtendeth the angle B A G ofan €- quilater triangle, which angle is contained vnder the grea~ ter fegment A H and the leife fegment A F : therefore the Jine FH is in power double to theline AF or to theline Al the leffe fezment, by the Corollary ofthe 16. of the fue- tenth . But the fame line F H isin power quadrupleto the e oi G line CE, by the 4.of the fecond: (for the line F H is double & to the line CE ). Wherefoaé the line A 1 being the halfe of the fquare of the line F H is in ae duple to the line C E, to which the line F H was in power quadruple . Wherefore the fide A G of the pyramis being diuided by an.extrenfe and meane proportion,maketh the lefle fegment Alin powerdupletothe . fide CE of the Icofahedron infcribed in it. | ¢ A Corollary. The fide of an Icofahedron infcribed in a pyramis , is a refiduall line. « For the diameter of the Sphere which containeth the fiue aa ae rationall,is in power fef- quialtera to the fide of the pyramis,by the 13.of the thirtenth *+and therefote the fide of the pyramusis rationall by the definition : which fide being diuided by an extreme and — proportion, we 7 ij. ¢ 2 \oAk ‘The fixtenth Booke of the the lefle fegmenta refiduall line, by the ¢. of the thirtenth . Wherefore the fide of the Icofahedron be- ing comment{urable to the fame leffe fegment (for the fquare of the fide of the Icofahedron is the halfe of the fquare of the faidJefle fegment ) is a refiduall line, by that which was added after the 103. of the tenthbooke:* “* .*" pM OBR Gi BSN Tie AT SON gO . a0 eee ve ‘\ - og The 6. Propofition. ——— as _ ba ——— The fide of a Cube containeth in power halfe the fide of an equilater trian- gulanP yvamis infcribed in the faidCube. — mx: 3 = = a x FES We Or fora{much as the fide of the pyramis infcribed in the cube'fubtédeth two fides of the cube js which containe a right angle, by the 1.0f the fiuetenth ;_ it is manifeft, by the 47. of the firft, Hiya westhat the fide-of the pyramis fubtéding the {aid fides, is in power duple to the fide of the cube: f e@XHZ Wherefore alfo the {quare of the fide of the.cubeis the halfe of the fquare of the fide of the pyramis. The fide therefore of a cube containeth in power halfe the fide ofan equilater triangular pyra- = _* \ if " / 44 : ; f i ‘ mis infcribed.in the faid cube. we * .* ~The 1-Propofition. The fide of a Pyramis is’ duple to the fide of an O&ohedron infcvi- T he fide of aCube is in power duple to the fide of an Octohedyon infcri« bed in tt. ie i ete i Slat SS ah eae _ ; q Tbe 9: Propofition, ia pin cae be ‘fie. of a Dodecabedron, is the greater feement of the line “which +s tontainethin'power halfe the fide of the Pyramis tnfcribed in the fayd Dodecahedron. | c 7 | : . ‘ 7 t “ BS 7 d eX 1 | Vppofe that ofthe Dodecahedron AB G Dthe fide be AB: and let the bafe of thecube OK eS vg | inicribed in the Dodecahedron be EC F H, by the #,of the fiuetenth . And letthe fide of p ies T he fide of an I cofabedron , is the meane proportional betwene the fide of Demonslra= $20Me the Cube circumfcribed about the Icofabedron, and the fide of the*Dodee » cabedron infcribed in the fame Cube. =) Vppofe that there bea cube A BF D, in which let there be inferibed an Icofahedron C L- 1G OR; by the 14.0f the finetemth. Let alfo the Dodecahedron infcribed in the fame be fe |}EDMNPS, by the 13.0fthe fame. Now forafmuch as CL the fide of the Icofahedron D SI is the greater fegmctof AB the fide of the cube circum{cribed about it,by the 3.Corolla- : Gai ry of the 14.0f the fiuetenthizand the fide E Dof the | : ~~. Dodecahedro infcribed in thefame cube is the leffe a tae = eel Sa fegmét of the fame.fide A B of the cube,by the 2,Corollary of the A 2 B r3.0f the fiieténth: it followeth that.A B the fide of rhe cube be? | ing diuided by an extreme and, meane proportion smaketh the greater {éemept C L' the fide ofthe Icofaliedron inicribed init,” and the leffe fegment E D thejfide of the Dodecahedton likewife * infcribed init. Wheréfore as th4 wholelineA B.the fide of the 7 cubejisto thegreater fegment C L thefide of the Icofahedron, fois the greater fegment CL the fide of the Icofahedron, t the lefle {egment BD the fide of the Dedecahedron, by the third «- definition of the fixth . Wherefore the fide of an Icofahedron,” is the meane prozortionall betwene the fide of the cube circum- F {cribed about the Icofahedron, and the fide of the Dodecahe- dron infcribed in the fame a : qT he rr Propofition, "The fide of a Pyramis, is in power + Offodecuple to the fide of the cube ine feribed init. ~ oe | | i mis is triple to the diameter of the bafe of the cube infcribedin it’s anid therefore it | is in power nonecuple to the fame diameter (by the 20,0f the fixth ) : But the dia- ' mer is in power double to the fide of the cube, by the 47. ofthe firit «And the dow- NSS ble of notrecuple maketh OGodecuple . Wherefore the fide of the pyramis isin UN power OGodecuplé to the fide of the cube infcribed in it. eires | Or,by that which was demonftrated in the.18.0f the fiuetenth the fide of the pyra- q The 12. Propofition. Se , T he fide of a Pyramis, isin power Olfodecuple to that right line, whofe | | AAAs. greater ~ — 4 Cont rublion. Demonitra- 20M T That is,as 18.20 Je Demonfra-. » 140i Po a eee ee . =< = nc al i ca ~~ ’ “ao — se . - eal a o . Seen nities . Senet ee ae S - ba RN OSS 2 ne - > / aT eof 4 “25 Cube infcibed mt, icone rf Yi scinie acre 07 1eiltwen ie) ahaohelmmwod o1 2aoghet fy mths ne hoe 3 he. Twas proved in the} corollary of the 13 of the fiueten th, thatthe fide ofa Dodecahe- ; Pa Wa reliPGileeet . rt kts: GE: % Sfit Of i Jiist wv ethi Ve pe | is 3U : wc >} drou,circumfcribed gbont.aC ube, is the gi ane deprp ent of théfide of the fame Cube.. af C3 Wherefore the, whote fide ‘of the Cube infertbee ig to the gretter, fegment > namely, - roa to M25 thefide of the dodecahedron circunifer ed.in anextreame znd meane proportion: ” \\ 3s tet cs i¢ Wis Sis 22 5 SETI 3 & MY SiO hide obsess: i ‘Te : : pan eka vy wherefore by conmneridnsthé ereater fegin ent, that is. the fide f the dodecahédron, is: gA to the whole, namely, to the fide of the Cube infcribed , in theconuérfe proportion of and meane proportion,by the .3.diffinition of the fiueth, a ie cilia : ‘+ y+ ac lw q Ihe 19.Propofition. ee mmhe ote ie. “ms eras! : , <1 The lidevf an Oohedronyis: fe/quialter-tothe fide of aPyraimis in ifCrie bed init. : . , > asda lool at OF (iby-theicordilaryof diér4i of che thirtenth.) the OGokedron is cutte into two Conftration. 4 REX 5), Quadrilacer pynamids,ondéf which let be A-B GD F.:-andlathecen tres of the cir Atel RS SEEM clesiwhich centayine therg:ibafes ofthe Odtohedron be K, §, 1, G : And draw thefe = Nal cightlines KEsEbLCiCKyaid EC Wherefore K EJ C isa{quare; and one of the Ti MRS: bates of theienbe in (cribed-in the OGtohedron ;by'the4.of the fuetenth. And foraf- the line EC is thefide of the-pyramis infcribedin % M fos ©) & capil end the OttohédronA BG DFE “ThairkBy thaeGDils sectorerw : ylssh | the fide of the O@ohiedron , is fefquigleer ted & visiloioD SMM BME’ che fide of phe pyramis infcribedin it. From the; ; ethos Qmmor\, poynt-A draw td the bafes BG and E erpendies |: o» | culars AN and AM: which(bythécor Lik, ehebe's basscui Demonffra- 12.0f the'thittenth) fhall paffe by the¢ehtres Band i: «|; %; ree C.Andidrawtheline NM. Now forafinuch sag Be » Vp: G D Fisa {quare , by the 14,/ofthe thittenth the lines N’'Gand MsD\fhall be parallels-and e Hallsccin: For thelines BG_and F D are bythe perpendi¢gu«.: i lars cutte into two equal partes in the poyntesiNn ++I and M(by the3.of the third), Wherefore thehifies'. N N Mand GD fhall be parallels and equall , bythe) =; » 33-0fthe-firlt, And forafmuch.as.thélinesa-N and. -- 4 M-whith; axe.theiperpendiculars of equall and':...|,. like trianglesarequt'alike in thepoyntesBand Cy 9 | | the lines EC and N Mofhalt be parallels.y by dieses. 6) sox badok Guetta 7: of the fixth : and therefore by the corollary of the fame, the triangles AEC, aid A NM fhall be like. Wherefore as the line A N is to the line A E,fo is the line NM.to the line E C ly the 4.of the fixth.But the line A N is fefquialter to the line AUB forthé line-A Eisd uple'to the line EN , by thecorollary of the 12,0f the caning : wherefore ae eo the line GD which is equal. vnto it, isfefquialter co the Hab BC hWherfore’G Dthe Gide of the O@ohedron|, is fefquialcer to E> che fide of the pyta- mis infcribed in it. Atoryoue cd ON OtH WEEMS ttt FS se04 q The a See ie agen Sat HS THA . 7 : O <> Sate t” * % Elementes of Geometry,added by Pluffas. Fol, 4.50. ; { The 20. Propofition. | If from the power of the diameter of an Icofahedron , be-taken away the power tripled of the fide of the cube inferibed in the Icofahedron: the power remayning hall be fefquitertiatothe power of the fide of the I+ cofabedron. rles of the cube, which are at the poyntes Fand Haré etat the centres of the bafes of the Icofahedron, by the rr.ofthe fiuetenth. Wherefore'the line F Hthall be perpendicular to both the bafes ofthe Icofahedré, by the corollary of the aflipr of the 15.0f the twelfth. Wherefore the line I B contayneth in power the two lines 1 Hand HB, by the 47. ofthe firft.But theline He - B, is drawne from the centre of the circle which con- tayneth the bafe of the Icofahedron namely,the angle Bis placed in the circumference, and: the poynt H is the centres Wherefore the whole lite B\G contayneth in power the witole lines F H and the diameter of the circle(namely,the double of the line B H)by the rs\of the fiueth.Bucthe diameter which is double to the line HB isin power fefquitertia,ta. the fideof the e- diss Oy * quilater triangle infcribed inthe fame cucle,by the corollary of theaz.of the thirtenthsForit isin pro- portion to the fide,as the fide is to. the perpendicular,by the‘orollary of the 8.ofthe-fixth.And FH the diameter of the.cube, is in. power,triple ro.E H the fide of the fame cube,by the rs;0f the thirtenth . If therefore from.the power of the diameter B.G.,be taken away the power tripled of EH the fide of the cube inforibedythatis,the power of the, line F Hi: the refidue Cnamely.ythe power. of the diameter of the circle whichis.dupleto the line HB )thall be fefquitertiato che fide ofthe triangle in{cribed in thac cixcleswhicirfelfe fide is A B the fide of the Icofahedron.If therfore from the power of thediameter of anIcofahedré,be.také away the power tripled of the fide of the cube inferibed in-the Icofahedron,the ey ~ powerremayning hall be {efguitertia to the power of the fide.of the Icofahedron.. 5/1 jorbo deloo AG typeiiienes? 227207 wes I he diameter of the Icofabedron, contayneth in power two lines, namely, the diameter of the cube infcrthed;soluch coupleth the centres of the oppofite ba- fes,and the diameter of the circle-which contayneth the bafe of the I cofahedron. For it was manifeft, that BG the diameter contayneth in'power the line F H which coupleth the centres,and the double of theline B.H, that is,the diameteraf the circle contayning the bafe wherein is the centre H. gq Uhez1.Propofition, The fide of 4 Dodecabedron is the le/Se fegment of that right line which decabedron. | ) Tao a | BEB.}. Lec 45 in power'duple tothe fide of the Offohedron infcribed im the fame Doe Dewonstra- ti0% Conftruttione T he fixtenth Booke of the Et ther be taken a Dodecahedron AB GDC T,one of whofe fides let be A B. And let the Odtohedron infcribed in the Dodecahedron be EFLKI: one of whofe q ,|| fideslecbe EF. ThenI fay that AB the fide of the Dodecahedron, is the leffe feg- i £e%s || ment of acertaynerightline(cut ‘At by .an extreame and meane bt8 A. EB | portion)which is in power duple * : i i BZ j h : & AN >) SS i TAN

=r AA Vppofe that ABG DFE bea . ( aN yr FP Con Sirullion. (EN | pentagon, containing »fiue KZ fides of the Icofahedron by Abo | the 16.0f the thirtenth, and pe eet <4 Jet it be inferibed in a circle, whofe centre let be the point E.And vpon the fides of the pentagon, let there be rea- red vptriangles, makinga folide angle of the Icofahedron at the poynt I,by the 16, cS of the thirtenth: And in the circle AB D, * infcribe an equilater triangle A H K.From os / the centre Edraweto H K the fide of the H /\ Cc triangle, and G D the fide of the. penta- x K gon,a perpendicular line, which letbe E- CNM. And draw thefe rightlines EG, G : ba \ E D,I Gand 1 D.And denidetheline B G CN Dp into two equal parts in the poyntT, And | | sees BB... draw iM a drawe thefelinesIN,I] T,T N, ET. And forafmuche as in the perpendiculars1T & IN are the centresof the cifcles which containe the equilater triangles 1B G, &. I G D,by the corollaryé of the firtt of the thirde. Let thofe’centres be the points S and O.And draw theline S O. Detide ~~ the line T.B the hal€of BG the fide of the Icofahedron by.an-extreme and meane™ proportion in the poynt R, by the 30.0f the fixth, and Jet the lefle fegment therof be RB.And forafmuch as the line SO cou pleth the centres of the triangles I BG, 8 I GD, itis by.thes.of the fiuetenth, the fideof the Dodécahedré infcribed in the Icofahedron, whofe fide is the line B G. From the fide BG take away B R the defle fegment of the halfe fide. And from the refidue G R take away the third pare G V (by the 9.0f thefixth.)Then I fay that the refidue R V is equal toS O the fide of the Dodecahedron infcribed. For forafnuch as the perpendicular EN is in the poynt | , Demontira- € dettiderd by any extreme and sitinie sn,.O 2omlel proportion; by the corollary ofthefirit of the fourtenth,andthe greater fegmenttherofis thelineEC and vnto theline\E« the line G Mas equal; by the corollary of the 12.0f the thirtenth? wherefore the line-E C cis “to the ine © Ny as theliné C:M is to the fame line C N, by the 7. of the fiueth. But as the line-E.Ciis.to the line G N 5fo is the whole line EN’, to the greater fegment E- C, by the 3:ditinition of the fixth. ‘Wherefore ( by the 11 0f the: filueth) , as the whole line EN is co che grearerfegment EC fois tlic line CM to the line CN.’ Wherefore the line C M,is deuided by an extreme and meane proportion in the poynt N, namely, is deuided like vnto the line E N, by the z.0f the fourtenth:W herfore the line M excedeth the line EN-bythelefle fegment of his halfe, namely by MN. And forafnuchicas E G,Dis the triangleof an:equilater and equiangle pentagon A B G D F, and E T'Nislikewifethe triangle ofthe like pentagon infcribed in the pentagon AB GD F: Therefore by the 20.of the fixth, the triangleE T Nis like to the triangleEGD, Wherefore astheline E G is to the line E N,fo by the 4.of the fixth,is the line G D to the line N T. Wherefore the lineG D (orBG which is equal vnto it excedeth the line NT by theleffe fegment of the halfe of B G. For the line EG did in like fort excede theline E N.But thatleffe fegmentis the line B R.Wherefore the refidue R G is equal to the line T N.And forafmuch as 1 BG is an equilatersriangle: the perpendicular ST fhalbe the 7 halfe’ of the line SI'which is drawne from’the centre,by the corollary of the 12.0f the thirterith: wher- fore'the line IT excedeth thelineI S by his third part.And forafmuche.as the line S O-which coupleth the feétions,is a parallel to the line T N, by the 2.0f the ane, For ene equal perpendiculars L T,and IN are cut likefnithe sag S.8°O: therfore the triangles PT N'& 1S'O, are like by the corollaty of the {econd of the fixth, W herfore as the line LT isto the line1S, 0 by the.4,of the fixth is the line T N to the line SG. But the line I T excedetlithe line I'S by a third part: wherfore the line T N,excedeth the line $ O by a third part: but the line T N is proued equal to ie line R G.Wherfore the line R G exce- deth the line S O by athird part of himfelf,which is G V.Wherfore the refidue R V,is equal to theline S O,which is the fide ofthe dodecahedrominfcribed in the Icofahedron, whofe fide is the line B G. If therfore halfe of the fide of an Iopfahedr6,be deuided by an extreme & meane proportion:and if the 20%. ee Oe ee eee ee ew s © ; ALS : é ; yrs =< ‘a . = — , ro ve s) pl " wai Tag + SP Bs se a od . . = bs ed eee eo x ¢ = > es i germeatace —~. bs en : os 4s oa ~ te ae —2 ~~ —— - — Z - = - ge “ ; : = aggre — eerie age: Sas Ie ans ee x, _——— 2 — aps , ror . me Y at ae ee = “- a “ —s aques . - sp eanetl > ay = — —< = — = ——- a — = = _ 4 Bie = = ; amet ar % -_ Bg ee a i Se ea td a a EE ona = x _s a NOE OI LS = — ee — - : 7 rx : — — PO ee = = - ae — ——— = ; : new by ~~ 4 . F - ee pee, oot 7" ‘ . Ore ss ee ee ee “ . OO. gy Sp EM te Fon — eee Th aa : ae iefle fegment therof be taken away from the whole fide,andagaine from therefidue be také away the a CRG ae = : that which cemaineth fhall be equal to the fide of the dodecahedron infcribed in the fame | iy cofahedron, : | e ydno ee. \ Thezs.Propofitions: > To oe that a cube geuen ts to a trilater equilater pyramis infcribed init, triple. [SEK iit age © SaV pofe that the cube genen,be AB CH sand let the pyramis infcribed in it be AG I oN Yeo ‘Then I fay that the cube ABCHis triple to the ofbcaien AGDE, Ferforafnuche oe “ti Brice) f bate AF Dts commonto the pyramis AF DBandAF DG, the pyramis AF DB fhalbe fet Eee without the pyramis A FD G, Likewife the reit of the bafes of the inferibed pyramis are common tg the reft of the pyramids fette without : which are ‘thefe + the pyramis AG D C vppon the the bafe A G De the pyramis' A GPE ypon the bafeA GF: and the pyramis G D FH vpon the bafe.G. DB; Which py-. A ramids taken without,are foure inpumber, equal and ‘like boise Ar the oneto the other, by the 8.diffinition of the eleuéth. For | every one of themis contained vader thre halfe {quares of the cube, and one of the bafes of the pyramis-infcribed. Wherfore euery one of théis cotained vnderthe halfe bafe of the cube, & the altitudeof the cube.As thepyramis A E- G F,hath to his bdfe halfe of the {quare E H namely, the tri angle E G F,& hath to his altitude, the altitudeof the cube, namely,theline A E. Wherfore the fayd pyramisis the fixth part of the cube. For ifthe cube be deuided into two prif- mes,by the plaine C B F G,the prifme ACBGEF, fhalbe tri- ple to the pyramis AEGF,hauing one & the felfe fame bafe with it EGF,and one and the felfe fame altitude E A,by the firtt corollary of the 7.of the twelueth. Wherefore the fayd outward pyramis AEGF isthe fixth part of the whole cube. Wherfore alfo the fame pyramis together wyth the other thre outwarde pyramids A FD B,A G D C,and GDFH;, thal containe two third partes of the cube. Wherfore the refi- due,namely the pyramis infcribed A GDF, thal contain one third part of the cube. And therefore conuerfedly the cube fhall be triple to it: wherefore we haue proued thata cube geué triple to a trilater & equilater pyramis in{cribed init. . If Elementes of Geometry ,addedhy Flufias. Fola52. q The 26. Propofition. To prone that a trilater equilater Pyramis 1s duple toan O&ohedron ins [cribed in tt. yg Et there be takena trilater Pyramis ABCD: whofe fixe fides let be cut into two e- y WY quall partes, in the pointes E,K,F,L,G,and H : infcribing thereby an Oétohedron in Ag the pytamis, by the 2.ofthe figetenth . Wherefore the pyramids AEG H,BEFK, JCFGL, & DKHL, fall without the O&ohedron infcribed,by the fame fecond of the 4 fiuetenth. Butthe outward Pyramids ( namely, A E G H, and the three qther)are like 3X € vnto the whole pyramis, by the 7.definition of the eleuenth.For the bafes of the whole pyramis are by parallellines drawen in them cutinto like triangles,by the Corollary of the 2.0fthe fixth, of which the ferefayd pyra~ mids are made. Wherefore the whole pyramis | is to every one of them in treble proportion of that in which the fides of like proportionare, by the 8.of che twelfth But by conftrudtion,the proportion of the fide A B to the fide AE is duple . Wherefore the whole pyramisABCD is o€tuple to the pyramis AEG H,and fois itto euery one of the pyramids which are equall to AEG i. For duple proportion multiplyed into it felfe twife maketh o¢tuple. Wherefore it followeth that the 4.pyramids A E.GH,B EF K, CFEGL,and D KH L,taken together,make the halfe of the whole pyramis ABCD . Where- fore the refidne, namely, the OGohedron EG~ LKUF, isthe other half of the pyramis. W her- fore the pyramis is duple to the Oohedron. Wherefofe we haue proued that a trilater e- ujlater pyramis is duple to an Oétohedron in- cribed in it. oT he 27. Propofition. Fo prove that aC ubeis fextuple to an Ozohedron infcribed in it, BBB.iij. Let “ one 8 e = Eee ee “s = Sede — ' ‘ SSeS ee ae ee > Low aie #5 ~~ —— ne i dea = Lhe fixtenth Booke of the smne Etthete be takenacube ABCD, EFGH: whofe. 4.ftanding lines A EBFC H, &D G3 Y let be cut into two equall partes inthe pointes.1,K,M,L : and by thofe pointes let there be nk extended a plaine KL MI : which thall bea:fquare, and parallel to the {quares BC & FH, Kae by the 15.0f the eleventh . Wherefore in it fhall be the bafe which is common to the two “pyramids of the O@ohedron infcribed in the cube, by that which was demontftrated in the third of the fiuetenth . Let tharbafe be N P._R Q, coupling 5 A . the centres of the bafes of the cube : and vpon that bafelee oc) ss + be fertherwo pyramids of the Odtohedron,which Jet be NP QRS, andNP QRT.. Aid forafmuch as thofetwo pyramids taken together, haue their altitude equall with Demonflra- — the altitude of the whole cube, ech of themra part hath to £10» his alttinde halfe the altitude of the cube,namely, halfe of the fide of the cube,as theline KB. And forafmuchasthe: B {quare K*L M Lis double to the {quare N R Q P, by thei47. of the firit: the other{quares of the cube fhallalfo bedou- ble to the fquare NR QP. And forafmuchasthe cube, as it was manifelt by the lait ofthe fiuetenth, is refolued into: - fixe pyramids, whofe bafes are the bafes of the cube,8 the altitudes the lines drawen fré the centre to the bafes, which are equall.to halfe the fide of the cube : it followeth that e+ | uery one of the fixe pyramids of the cube, hauing his bafe Q: 7 double to the bafe of eche of the pyramids of the O¢tohe- dron, and the felfefame altitude that the faid pyramids of, F YY the Odtohedré haue,is double to either of the pyramids of i the oétohedré,by the 6.of the twelfth . And forafmuchas : euery one of the pyramids of the cube is equall to the rwo pyramids of the OG@ohedron,the fixe pyramids of the cube shall be fextuple to the whole O&ohedron .. Wherefore it is manifeft,thata cube is fextuple to an O@ohedron infcri- rc) bed in its | Fete a 51 - e q Ihe 28.Propofition. T'o proue that an Offohedron is quadruple fefquialter toa Cube infcria bed init. 1 FHS « | BZ SZ SSA V ppofe that the Odtohedron geuen be ABCDEF: andlet the cube in{cribedin it be ‘ Xx GH1IK,VQRS. Then] fay, that the O&ohedronis quadruple fefquialter to the cube r : RN y) infcribed init. Forafmuch as the lines drawen from the centre ofthe OGohedron,or of SK) x 0) the Sphere which containeth it, ynto the centres of the bafes of the O&ohedron,are pro- eee LX ued equall, by the 21.0f the fiuetenth : andtheangles of the cube are fetin the centres of tho{e bafes, by the 4. of the fiuetenth : it followeth, that the felfe fame right lines are drawen from one and the felfe famecentre of the cube and ofthe Oftohedron: for they haue eche one and the {elfe fame centre, by che Corollary of the 21. of the fiuetenth:, Let that centre be. the point T . Wherefore the bafe B DF C,which cutteth the O&ohedron intotwo equalland quadrilater pyra- mids, by the Corollary of the 14. of the thirtenth, fhall alfo cut the-cube into two equall partes , by the Corollarytof the 39.0fthe eleuenth . For it paffeth by the centre T,by that which was demonftra- ted in the 14.0f the thirtenth . And forafmuch as the bafe of the cube is inthe 4. centres G,H,1,K, of the bafes of the pyramis ABD F C, a plaine L N O M, extended by thofe pointes, fhall be parallel to the plaine B DF C, by that which was demonftrated in the 4.of the fiuetenth,'and fhall cuc the pyra- mis in the pointes-L,N,O,M : and thelines LN, BD, and NO,DE; fhall be parallels, fo alfofhall che | lines OM, EC, and LM,BC : andthe {quare G H1K ofthe cube thall be infcribed in the {quare LN OM, bythe fame. Wherefore the {quare LN O Meisiduple to. the fquare-GHIK, by the 47. of the firft”. From the folide angle A, let there be drawen tothe plaine fuperficies BD F.C, 2 perpen- ' dicular, which let fall vppon it in the point T,and let the fame perpendicular be A T, cutting the plaine LN OM inthepoint P. And itfhallalfo bea perpendicular to the plaine L N OM, by the Corollary of the 14. of the elenenth . Againe from the angle B AD of the triangle ADB, let there be drawen by thecentre H of the triangle, to the bafe aline AHX. Wherefore the line A X is fef- uialrer to the line AH, by the Corollaryof the 12. of the.thirtenth . Wherefore the line A His | duple to the line HX . But the other lines AB; A D, AF, AC/and the perpendicular AP T, are cutlike vnto thedine-A‘HX, by the 17. ofthe elesenth : Wherefore the line AP is double to the . line PT. Wherefore the line A P isthe altitude of the cube, forthe line PT is she chelfcuhereats t 7 m= fie an .e ‘2 - Fae i * \% ae ~ ae i ay Bin 4 —~ Elementes of Geometry,added by Fliafvas. Fol.4.53. And forafmuch as vpon the bafe ; A G HIK of the cube,and vnder the : alticude ‘A P of the fame cube, is fet the pyramis AGH IK: the faid pyramis is the third:part of the cube, by the Corollaty of the 7:0f the twelfth. But vnto the pyramis A GHIK the pyramisALNOM is duple, by the 6. of the twelfth, for the bafe of the one is double to the bafe of the other . Wherefore the pyramis ALN O Mistwo third : partes of the cube. And forafmuch —__ asthe pyramids ALN OM, and \ ee ABD FC,arelike, by the 7.'defi- nition of the eleuenth : therefore they are in triple proportion of thatin which the fides of like pro- portion AH toAX, or AL to AB, are, by the Corollary of the 8. of the twelfth . Bur the fide ABis proued to be fefquialter to the fide AL. Wherefore the pyramis A- BCD Fistothe pyramis ALN-~ O M, as 27. is to 8. ( that is, in fef- ee proportion tripled : for E the quantitie or denomination of | fefquialter* proportion , namely, pean > tae es I —- multiplied into it felfe once maketh 2——-,which againe multiplyed by s—- maketh 32 ; thatis,27.to 8.) . But of what partes the pyramis AL N O'Mcontaineth 8,0f the fame the ‘cube con- taineth 12 : namely, is {efquialter-to. the pyramis. Wherefore of what partes the cube containeth 12,0f the fame the whole O&ohedron (which is double to the pyramis ABD FC ) containeths4. Which 54. hath to r2.quadruple fefquialter proportion . Wherefore the whole Ottohedron is. co theicube ine {cribed in it, in quadruple felquialter and of the cube to ‘the pyramis, x. which is 3)as was raught in thediffinition of the fixth,there fhalbe producéd 13 — namely, the proportion of the octo hedron te the pyramis infcribed in it For4 — smultiplyed ve ; ; 4 y by 3.produce.:3 —, Wherefore the Octohedron isto the pyramis infcribed in itin tredecuplefefquialter proportion. Wherefore we hate proued thatan Ottohedron is to a tri- lacer equilater pyrantis infcribediin it, in tredecuple fefyut- alter proportion. ; e " The 30.Propofition. Lo prone that a trilater equilater Pyramis, 1s noncuple to a cube infcribed tbls: —t{ i boyiehiom-ofiees Bidw seater: aon zy eit oycht 6 ee em mr. se en ’ 3) ib* Sfcrthat thé pytaihis gelien Be A BOD, whole two bales lee be ABC; and DBC, Bey <: ’ F324, P ) —s : tcxs . . , : ; ; : 4 “ae fettheir centres be oe es ya G aid'P-Atid from the angle A, draw vnto the bafe B- as @ " . nt, o” ™ : ‘Oc oe pee Me on Le # = er ped = ees <- SS , is cr a . > = ’ o rs) td ©) w © is | nm ~? ; opel pe acmeaalhe line A D is the fide of the pyramis,the fame A D thal be the diameter of the bafe of the | cube which cétaineth the pyramis,by the wofthefiuetéth.p A : Demonfir4- Draw the line G I. And forafmuchas the line G1 coupleth’ $10 te the centres of the bafes of thepyramis: thefaidelineGI. |, thalbertticulianibeot ofthe baleotthecubein(aribed.in the)\11 909 of paris by the 18.0f the fiuetenth. . And forafmucheas the, —~ . . SAG is Udlible to théline GE; by the cOrollarye of thes ©" (°° 7’ ewelueth of the thirtenth: the whole line A E thal be triple: bse: inisrfodua tothe line GE: and foisaifo the line DEtothelineI E. Wherefore the lines A Dand GI are.paratlels, bythe 2,0f wry thefixth. And therefore the triangleés AE Diand GETare’ *7\' like, by the corollary ofthe fame.And forafmuch as the tri- ‘ktadt * anglesA ED,and GEL are like,the line AD fhalbettiple to, 44+ oe | the line GI, by the 4.ofthe fixth.But the line AD is the digcn BEN NE ‘© meter of the bafe osecube Siprecal bce chewpyRlind b oYrsiininy (9) shawaae ramis A BC D,and thelipéG1is the diameter of the bafe of the cube inftribed in the - | but the dianterer#of the bafes are equemultiplices to the fdesthnciy ase in power PT ike the fide ofthe cobéGituinferibed aban the pyramis ABCD, is triple to the fide of the cube, infcribed in the fame piramissby the 15-of the fiueth :burlikeeubesare in'triple proportion the orieto the other of that in which their fidesare,by the 33 .of thie eleuenth:and thefides are in triple proportion the one to the other: Wherfore triple taken thre tines bringeth forth twenty feuencuple,whiehis 27.to se | a the 4, ermes 27.9.3.1,being fet in triple propontion::the proportion of ‘the firft to th éfourth namely, of Re 27.to 1.fhalbe triple to the proportion of thefiritte the fecond, namelyjof27.t6 9,by thé ro.diffnition of the fiuethz which proportion of 27.9 1.is thé:proporti of thefidestripled;which proportis alfo is found in hike folides. Wherefore of what partesshe cube-circum feribedeontaineth 27.0f the fame,the cube infcribed'containeth one’s but of whatpartes thecube-circumferibed,containeth #926f the fame, the pyraniis infcribed-in ip,containeth 9.by the assofthis bookerwherfore of what partes the pyramis : oe RL - a ie om - = a. 2s = av? o a se cal as SST = Se ee — —— -. Pim SS —Mers PA a wre k ae TES i ge ae = a a te <> all i eo — ins ~ pr Pe . = S-. : . ON ne _ sd < SE ee = oe r- % te Pars oe ss d —_ : eins ee ee ee Ee ee he ee ae = 5 . as - Oe =~ a ~~ ~ ——— ~ ‘ny eae = — = — 7 Se a gS ee — — f — 2 ~~ - fein owt ms ~~ ~~ - - 2 ; = ns = SS = we <= =— = an wenn — = ABCD contameth 9.ofthe fame, the cube inferibed in che: pyramis, containeth one. Wherefore we’ haue proucd that a trilater and equilater pytamais,is nonecuple to'acube in{cribed in it.°°™ ee moe ~ . : - ¥ * jee ie ee Be - - vert ry ’ met as a * 5 ~ ae yom a A CLL ALA > : = Ekementes of Geometry added by Flufvas. Fol.45h0 : toetied owns q Ehe 3 1.Propofition. An Oétobedron hath to an Icofohedron infcribed init, that proportion, which two bafes of the Ofobedron bane.ta five bafes of the Icofahedron. i Vppofe thatthe octohedron geuen'be A B C D,and let the Iced fahedron inferibéd in it,be F- Care GHMKL1YO. Then I fay thar the o&chedronis tothe Icofahedron, as two bafes of the San ye, octohedron,aré to fiue bafes of the Icofahedron.For forafmuche as the folide of the o@tohe- ume dron confitterh of cight pyramids, fet vpon the bafes ofthe oftohedron,and Hating to theyr 7 altitude a perpendicular line drawne from.the centre to the bafe:let that perpendicular beER,orES, “ emon|trde being drawne fromthe centre E (which centre is common to either of the folides, by the corollary of $60%e the 21.of the fiuetenth)to the centres of the hafes;namely, to the poyntes Rand $. Wherefore for that thre pyramids are equal and like, they fhalbe equal to a prifiie fet vpon the felfe fame bafe, and ynder the felfe fame altitude, by the corollary of the feuenthof the rwelueth. But vnto this prifme is double that prifme which is fet vpon the felf {ame bafe,and hath his altitude duple, namely,the whole line RS by the corollary of the 2s.of the eleuenth:forit is equal to the two equaland like prifines whefeof it is compofed. Wherfore the priftne fet vpé the bafe of the o€tohedronand hauing to his altitude theline R S is equal to fix pyramids,fet vpon fix bafes of the OGohedron, and haning to their altirude rhe line ER. So thereremaine two pyramids (forin the o@ohedron are 8.bafes) which fhall be equal to-the prifme which is fet vpon the third part of the bafe of the o€tohedron,and vnder thé altitude RS. For prifmes vider one amd the felfe fame altitude; are in proportion the one to the other,as are their bafes, by the corollary of the 7 of the : twelucth . Wherefore the two prifmes which are fet vppon the bafe of the oftohedron, and vp- on 2 third part therof, and vnder the altitude R S, are equal tothe 8. pyramids of the Odtohedron, orto the whole folide of the oc- tohedron. And forafinuch as the Icofahedron infcribed in the oc- tohedron, hathe his bafes fetin the bafes of the OGohedron, by the 17.ofthe fiuetenth: it follow- eth thatthe pyramids fet vppon the bafes of the Icofahedron, & hauing to their toppes one and the felfe fame centre E, are con- tained ynder the felfe fame alti- tude, that the pyramids-of the oftohedron are cotained vnder. namely ,vnder the line ER,or ES. And therefore a prifme, fet vpon the bafe of the Icofahedron, and hauing his altitude double to the alritude of the pyramis, namely, | the whole line R S, is equal to | . fixe pyramids fet ypon the bafe of the Ieofahedron, and ynder the altitude E RorES, as we haue pro- ued inthe oficktdain, Wheckors the 20.pyramids,fet vpon the 20.bafes of the Icofahedron, are equal to thre prifmes fet vpon the bafe ofthe Icofahedron,and vnder the altitude R S$; and moreouer to an'o- ther prifine fet vppon athirde part of the bafe of the Icofahedron and vnder the fame altitude RS; which prifmets a thirde part,of the former prifme,by the corollarye of the 7.0f; the twelueth = for chéir proportion is as the proportion of the bafes. Wherfore two prifmes fet vpon the bafe of the o€to~ Hedi onjand'a thixd part therofand ynder the aleiaide R S,is to 4.prifmes fet vpon thrée’bafes of the = dofahiedron,ahd a third part thercof,and vader the famealtitude'R'$, in the fame proportion that the hafesare,that is,as 4.third partes of the bafeofthe O&odron (which are equal to one bafe, and —) to ten third parces of che bafe of the Icofahedron' (which are equal to'thre bafes & or as two third partes of the bafe éf the OGohedron, areto fiue thirde partes of the bale of the Icofahedron. But two thirde partes of the bafe of the Ottohedron, are to fiue' thirde partes of the bafe of the’ Teof- hedron, as two bafes are to fiue bafes(by the 15.of the fifth, for they are partes of equemultiplices:) And two prifmes of the Odtohedron are;to:4yprifmes of the Ieofahedron , as the folide of the Ofgtohedron is to the folide of the Icofahedrot, when as eche ‘are equal to eche of the folides : Wherefore (by. che.11.0fthedinerh), the folide of che Oftohédron, isto the folide ofthe lcofahedron infcribed in it, as two bafés of the O&ohedron, are to fiue bafes of the I¢ofahedron. An Oc- CCC jj. tokedron, “— oF = -* css a= SS ie — > t 1 i ; EE ~ - Oe. = = ~ a = SS eee —s = . : ~4 ee i ge == ——- naa ‘ a ee ee ees ne ee Ee = yA — Se es nS Or ae ni Se ee, Sa si a pate. wae ae Se . as ane ot a7 > -—- = ae a ae Sy ee ee - ee ee ~ rs peasant . _—_—— — — : . a a 7 . ae 9 —————S —s ‘ = . ———— —— = te —— = ie os : x. et ” —— —— 3 : — ~ , a . = ‘ ——- a = - : —sS- ee: en Ee en = = iar ——- SSS eS eS ee as Se ee SS : —— - ——_- olin ease ae Ie 5 See eee - — roe Sh : ’ : rr * : SSS Baie ree ~ : nee aut —-— = Sudiee SS —*= —- = —— ey tte ~- _—, ConSruion. Demon fra. tion 6 The fixtenth Booke of the tohedron therforeis to an Icofahedron infcribed in it; in chat proportion, that two bafes of the O&- hedron,are to finebafes ofthe Icofahedron. ows q The 32. Propofition. | T he proportis of the folide of an Icofahedron to the folide of a Dodecahes © dron inforsbed init , confifteth of the proportion of the fide of the Icofahee dron to the fide of the Cube contaynedin the fame [phere , and of the pros portion tripled of the diameter to the line which coupleth the centers of the oppofite bafes of the Icofabedron. LAS) Vppole that there be 2 Dodecahedron,whofe diameter let be H I, and let the Icofahe- S vs dro1containedin the fame {phere be AB G C, whofe dimetient let be A C.And let the SX). right line which coupleth the centres of the oppofite bafes be BG . And let the dodeca- (D hedron infcribed in the Icofahedron be that which is fet vpon the diameter B G, by the AA ZS| ¢.0f the fiuetenth. And let the fide of the cube be D E,and let the fide of the Icofahe- droa be D F,both the fayd folides being defcribed in one and the felfe fame fphere. Thé I fay that the proportion of the folide of the Icotshedran ABCG tothe folide of the EE by ymin fet vpon the diameter B G, infcribed in it, confifteth of the proportion of the line D F tothe line DE, and of the proportion tripled of the line A C to theline B G . For forafmuch as the folide,of the Ico hedron A BG C isto the folide of the dodecahedron H 1, being contayned, 12 one and the felfe fame {phere ,asDFistoDE, by the 8.of the fourcenth Bue the dodecahedron whofediameteris H I,is to the dodecahedron whofe diamer is B G, in treble proportio of that in which the diameter H Lis to the diameter B- G, by the corollary of < the 17.0f theewelfth: &: the lines HT and AC are equall by fuppoition (namely , the diameters of one and the felfefame {phere).Whereforeas H._. Lis toBG, foisACto B G, Wherefore the pro- portion of the extremes, namely , of the Iccfahe- dron AB G C tothe Do- decahedron fet vopon the diameter B G which coupe h the centres,cO- fuiteth, (by the 5 .difiniti- on of the fixr)of the pro- BRON D ak meanes, . | . | | 4 =f amely,of the proportid EN Ore me obits pies bu ofthe isthe f am BC G to the dodecahedron HI (which is one and the fame with the proportion ofD Eto DE jand of the proportion of the fame H I to the other dodecahedron fet vpon the diameter B G,infcribed in the fame Icofahedron AB G C,by.the fame 5. of rhe fiuetenth :_ which proportion is riple to the proportié of the line HI (or the line A C)to G B which coupleth the centres of the oppo; fice bafes of the Icofahedron . The proportion therefore of the folide of the Icofahedron ‘to the folide ofa Dodecahedron infcribed in it,confifteth of the. proportion of the fide of the Icofahedron.to the, fide of the Cube contayned.in the fame {phere , and of the proportion tripled of the diameter tothe line which coupleth the centres of the oppofite bafes of the Icofahedron. | | | jiaieme” nn T he folice of a Dodecahedron excedeth the folide of a Cube inferibed in st a > root : | fe 2 — we - - « Shce _ — © test / 3 > el > =< pe. 4 : 4fi ¥ . E Elementes of Geometry,added by Flufias. Fol.4so, it;bya parallelipipedon whofe bafe wanteth of the bafe of the (ube by a third part of the leffe feoment and whofe altitude wantethof the altitude ofthe (ube , by the lefse.fegment of the leffe-fegment of helfe the fide of the Cube. . aie : Orafmuch as by the 17, of the thirtenth,and 8.oF the fiuerenth,it was manifeft, that the bafe mad Of a cube infcribed in a dodecahedron ,doth with his fides fubtend the ancles of 4 pentagons : : baz wiacOcurring at one and the felfe fame fide of the dodecahedron: let that bal ofthe cubebe A- Consfrnufions Wi Cee? BOC D:and let the fide wherat 4.bafes of the dodécahedtoncircumferibedconcurre,be E G: which fhaltcontayne a folide A E B DGC fet vpon the bafe AB C D.Diuide the fides A B and D C in- to two equall partes in the poyntes L and N.And draw the line L N, which isa paralel to the fide E G, as itwas manifeft by the 17.0f the thirtenth. The perpendiculars alfo E R and G O ‘vhich couple thofe parallels,are eche equall,to halfe of the fide E G,and eche is the greater fegment of falfe the fide of the cube , and therefore the whole line EG is the greater fegment of the whole line 1N the fide of the cube(by the-forefayd 17.0f the thirtenth) . By the poyntes Rand O , draw vnto thefides AB andC D parallel lines F H and { K.And draw thefe right lines E F,EH,G Land GK. Now foafmuch as the two lines F H & ER touching the one the otherare parallels to the two lines I KandG 0 touchingalfo the DemonSfra~ one the othér, & not being in the felfe fame playne with the two firft lines: therforethe playnefuperfi- é#om. cieces EF H and G I K patting by thofe lines are parallels , by the rs.of the eleuenth: which playnes dao cutte the folide AEBD GC. Whererefore there are made fower quadrangled pynmids fet vpon the rectangle parallelogrames L H,L F,N K, and NI, and hauing their toppes the poymesE andG . And forafmuch as thé trianglesG O Kand E R H are equall and like, by the 4.0f the firit namely, they con- A L B , Sa —— in the ure a line F fe | Ea E oa on the point E tothe point G st é fe OE Gi Sate ee K D I. ce tayne eqliall ancles comprehéded vnder equall fides,and they are parallels by conftriCtion, being fet in the playnes GI K and EF H: the figuresG KH E,OKH R,and G O RE fhall be panllelogrammes, by the diffinition of a parallelogramme,and therefore the folide G OK E R His a prifme, by the rr .dffini- tion of the eleuenth.And by the fame reafon may the folide GO IER F be proued :o be a prifme.And forafmuch as vpon equall bafes NOK C,andRLB H,and vider i ie altitudes OG andRE are fee pytamids: thote pyramids fhall be equall to that pyramis which is fet ypon the bafe C KID ( whichis double to cither of the bafes N O K C,and RLB H)and vnder the fame altitude OG , by the. of the twelfth. And forafmuch as the fide G Eis the greater fegment of the line C B, the ine KH , which by the 33.0f the firft,is equall to the line G E, fhall Be the greater fegment of the fame Ine CB, by the 3.0f the fourtenth, Wherefore the refidues CK and H B fhall make the leffe fegment f the whole lineC- B.But as the greater fegment KH is to the twolines CK and HB the leffe fegment, fois the rectangle parallelogramme O H to the two reftangle parallelogrammes O C and H L,by the 1.of the fixt. Wher- fore the leffe fegment of the parallelogramme NB , fhall be the two parallelogrammes OC and HL. Put the line K M double to the line K C and draw theline MS parallel to the line CN. Wherefore the parallelogramme O K M S is equall to the parallelogrammes O CandH L,by the ofthe fixth. Wher- fore the pyramis fet vpon the bafe O K M S contayneth two third partes of the prifne fet vpon the felfe fame bafe, by the 4.corollary of the 7.0f the twelfth .Wherfore the prifme which isfet vpon two third CCC.u. partes ee Extendin the figure 2 line from the potnt E tothe point 2 The fixtenth Booke ofthe \e © partes of the bal@O.K MS is equallto the twopyramids NO KC Gand RLB HE For the fedtions of 2 prilme aré one to the orier,as the fections of the bafe are , by the firft corollary of the 25. of. the ele- neuth.But the fections of the bafé até as the fettions of the line'C B or K M, by thet.of the fixe. Wher- fore thetwopynmids NOKC:GandRLB H E,adde ynto the prifme.G OK ER H,two third. partes of the prifme fet vpon the bafe O KM'S.And forafniuch as the'line KM is the leffe fegmét of the whole line B C(for it is equall tc to the two lines CK and HB ) , and the prifme fet vpon the bafe O KH R is cutte like vnto the line K M , namely , in eche are taken two thirdes , as hath before bene proved : the prifne equall to the two pyramids, {hall adde ynto the prifme G;O K E RH, which is fet vpon the grea- ter feement K H, two thirds of the lefle fegment.. Whereforein the line BC there fhall remayne one third part of the leffe fegment:and therefore in the reCtangle parallelogramme N B which is halfe the bafe of the. cube,there {hall remayne the fame third part of theleflefegment . And by the famereafon may we prove thatin theother pyramids ONDIG,andRLAFE, and inthe prifme GOIERFis, left the felfe fame excefle of the bafe L A.N D,namely, the third part of the leffe fegment . Wherefore che whole prifme contayned betwene the triangles 1 G K and F EH, and vader the length of the greater be A_ wf ssensien Reet METS d BR is ee Ss : fegment and two third purtes of the leffe fegment of B C the fide of the cube , is equall to the folide compofed of 4.bafes oo dodecahedron and fet vpon the bafe ofthe cube. Wherfore the bafe of thae prifme wanteth of the whole bafe of the cube onely.a third part’of the lefle fegment : and the altitude of that prifme was the line G O, which is the greater fegment_of halfe the fide of the cube . And foraf- much as ynto the triangle G K , is doublethe r€tangle parallelogramme fetvpon the fame bafe IK, (the fide of the cube) anc vnder the altitude G O, by the 41. of the firlt:itfolloweth that three reCtan- gle parallelogrammes fetvpon the fame bafe LK,the fide of the cube’, and vnder the alutude O G the greater fegment of halfe the fide of the cube,are fextuple to the triangle IG K . Wherefore thofe three rectangle parallelogrammes doo make one rectangle parallelogramme fet vpon the bafe 1K and ynder the altitude of the ine G O tripled. But by the 7. diffinition of the eleuenth,there are fixe prifmes equal and like vnto the forefayd prifme, being fet vpon euery one of the fixe bafes of the cube: which prifmes are in proportion the one to the other as their bafes.are by the 3 corollary of the 7.oftwelfth. Where- fore the folide compofed of thefe fixe prifmés, fhall want of the bafe AB C D the third part of che lefle fegment,and taking his adtitude of the forefayd reftangle parallelogramme,the fayd altitude fhiall be e- quall to three greater fepmentes(one of which is G O)of halfe the fide of the cube. Now refteth to prone chat thefe three fegmentes want of the fide of the cube by the leflefegment of the leffe fegment of halfe the fide of the cube . Suppofe that AB the fide of the cube be diuided into the greater fegment A C,and into the leffe fepment C B(by the 30.0fthe fixt) . Anddiuide into twoe-. quall partes the line A Cin the poynt G., and the line CB in the poyntE . And ynto the line CG put the line C L equall . Now forafmuch as the lines AG and GC are the greater fegmentes of halfe the line A B,for eche of them is ee of the greater fegment RSE epee ee ome mre Sep of ee: whole ie A- S G c E L & B: the lines EB and | EC hall be the leffe : AS fegmentes of halfe the line A B. Wherefore the whole lineC L is the greater fegment , and the ne C- md + ~~ sive laed = "+ aaa Elementes of Geometrysadded by Flufeas. Fol.456. Eis the leffe fegment.Butas the line C L isto thelineC E,fo is the line C I to therefidue BL . Wher- fore the line E Lis the greater fegment of the line C E,or of the line EB whch is equall vnto ic. Wher- for e therefidue L B is the leffe fegment of the fame E B ( which is the lefle egment of halfe the fide of the cube)...Bur the lines A G,GC,andCL are three greater fegmentes of the halfe of the whole line A.B: which thre greater-fegmentes make the altitude of the foretaydfolide : wherefore the altitude of the fayd folide wanteth of AB the fide of the cube by the line LB, which is neleffe fegment of the line B E.Which line B Eagayne is the leffe fegment of haife the fide AB of the ctbe . Wherefore. the fore- fayd folideconfi fting of the fixe folides,wheéreby the dodecahedron exceedeth the cube inftribed in it, is‘fet vpon a bafe which wariteth ‘of the bafeof the cube by a third part of he lefle feamenty and. is ynder an altitude wanting of the fide of the cube by the leffe fegment of th+leffe Segment of halfe the fideof the cube. The folide therefore ofa dodecahedron exceedeth the folid: ofa cube inicribed in it, bya parallelipipedon, whofe bafe wanteth of the bae of the cube by athirdpart of the lefle fegment; and whofe altitude wanteth of the altitude of the cube , by the leffefegmeie of the leffe fegmentof halte the fide of the¢ube. | | ¢ A Corollary. | A Dodecahedron is double to a Cube infcribed in it, tating away the third art of the lefJe feoment of the cube,and moreouer the leffe eoment of the Ie e Pp IS > a's fegment of. half e of | that exceffe. For if there be geuen a cube, from {which is cut ofa folide fet vpon a third’part of the leffe fegment of the bafe,and vnder one and ‘the famealtitude with the cube: that folide taken away hath tothe whole folide the proportion of the fection of the bafe to the bafe, by the 32.of the cleuenth. Wherefore from the cube is taken away a third part ofthe leffe fegment . Far- ther,forafmuch as the refidue wanteth of the altitude of the cube,by the lefle egment of the leffe feg- ment of halfe the altitude or fide,and that refidue is a parallelipipedon, if it be «ut by a plaine fuperficies parallel to the oppofite plaine fi uperficieces cutting the altitude of the cube bya point,it fhall take away from that parallelipipedon a folide,hauing to the whole the proportion of the [ection to the altitude, by the 3-Corollary of the 25. of the eleuenth , Wherefore the excefle wanteth ¢f the fame cube by the third part of the lefle fezment ,and moreouer by the leffe fegment of the kffe fegment of halfe of that exceffe. q The 34. Propofition. I'he. proportion of the folide of a Dodecahedron to thi folide of an Icofas hedron infcribed in it, confifteth of the proportion triped of the diameter to that line which coupleth the oppofite bafes of the Dolecahedron, and of the proportion of the fide of the Cube to the fide of the I cofahedron infcrie bed in one and the feife fame Sphere. 2 Cahedron, whofe diameter let be AB: and Jet theline which coupleth the cé- , tres of the oppofite ba- fes be KH: and Jetthe Icofahedron infcribed in the Dodecahedron ABC, be DEF : whofe diameter let be DE. Now forafinuch as one andthe felfe fame circle oO coraineth thepentagon C ofa Dodecahedron, & the triangle of an Ico- fahedron defcribed in one and the felfe fame Sphere, bythes4.of the fourtenth: Let thatcir- ri CCC. ij. cle T he fixtenth Booke of the cle bel G O.Wherfore ¥O is the fide of the cube, and 1G the fide ofthe Icofahedron, by A y the fame. Thé 1 fay,that the proportion of the , fim Dodecahedron AH B- | C K to the Icofahedron DEF infcribed im it,co- filteth of the proporti6 tripled of the line AB to the line KH, and of. the proportion of the lineI O to the line1G. Demon? For forafmuch as the I- PemOnsra-— — eofshedron DE F is in- 560% {cribedin the Dodeca~ ay ) % 7 hedré ABC,by fuppo- oO fiid, the diameter DE. . . Fa fShalbe equal to the line KH, by the 7..of the fiue- tenth . Wherefore the Dodecahedron fer vpo the diameter KH ‘fhail beinfcribed in thefame a i Sphere, wherein the I- cofahedron D E Fis in- fribed : butthe Dodecahedron AH BCX is to the-Dodecahedron vpon the diameter KH in tri- ple proportion of that in which the diameter AB is to the diameter K H,by the Corollary of the 17. 0f the twelfth : andthefame Dodecahedron whichis fet vpon the diameter K H; hath to the Icofahedron DEF (which is fet vpon the fame diameter, or vpon a diameter equall vnto it, namely,D E ) that pro- portion which IO the fide of the cube hath to'l G the fide of the Icofahedron, infcribed in one & the {elfe fame Sphere, by the 8 of the fouretenth. Wherefore the proportion of the Dodecahedron A He BC X to the Icofahedron D E Finlcribed in it,confifteth ofthe proportion tripled of the diameter AB to the line K H, which coupleth'the centres of the oppofire bales of the Dodecahedron (which propor- tion is that which the Dodecahedron A H BC k hath to the Dodecahedron fer vpon the diameter K H) and of the proportion of 1O the fide of the cube to I G'the fide ofthe Icofahedron ( which is the pro- portion of the Dedecahedron fet vpon the diameter K Heo the Icofahedron DE F defcribed in one and the {elfe fame Sphere ) by thes. definition of the fxth.The proportion therefore of the folide of a Do- decahedron to the folide of an Icofahedronin{cribed init, confifteth of the proportion tripled of the diameter to that line which coupleth the oppofite bafes of the Dodecahedron,and of the propestion of the fide of the cube to the fide of the Icofshedral in{cribed in one and the felfe fame Sphere. T he 35. Propofition. T he folide ofa Dodecahedron containeth of a Pyramis circumfcribed ae bout it two ninth partes taking away a third part of one ninth part of the leffe feement ( of aline diuided by an extreme and meane proportion ) and moreouer the leffe fegment of the lefSe fegment of halfe therefidue. ee | i | : : : : i 4 ' | io | i : f t b \ \ \ y - t y f : } 5 | | : t ' 4 ; - J ’ Al 1 : it | i f} h : : t : * : oP ft ' , ' 4 i ne} k : is i é oH) MA ! : * : A { - : yj mf r) : " ; | + ; ‘ a)? ' 1 ’ ' ¥ i ‘ aff : ” ’ ie a 4 | a " ohn : J 4 Bi i « "AE nf = : ’ “ iat r $ { | 4 ‘ | ‘ , : My : | if ‘ - 1S. - / ‘ y 4 Rt, amet 4 ,* iP ius P Lt ‘ - ye! ; i | ne te e 4 ' i Hi HY ‘ J by & (2 ; : , ‘ ; Y : pi 7 y t % Hh at 4) $4 } i MW ul at Mit ¢ iD “AY a m Pa ia ; } ’ i ! ‘ ‘ ah : 4 > | - re aaf lhe ) We ‘ oe A. . wa ; is ‘& ned in one and the felfe fame pyramis, by the Corollary of the firft of this booke . And by the © ¥ Corollary of the 33.0f this booke, it is manifeft, that the Dodecahedron is double tothe = — “a = = + 3s ae : A | : T hath bene proued that the Dodecahedron together with the cube infcribed in it is contai- a fame cube, taking away the third part of the lefle fegment ,and moreoner the lefle fegment of the leffe fegment of halfe the Fefidue,or of this excefle. But a pyramis is to the fame cube infcribed in it nonecuple,by the 30.0f this booke. Wherefore the Dodecahedron infcribed in the pyramis,and con- taining the fame cube rwife,taking away the felfe fame third of the leffe fegment , and moreouer the ; lefle fegment of the leffe fegnrent of halfe the refidue, fhall containe two ninth “partes of the folide of 7 the pyramis ( of which ninth partes eche isequall vnto the cube ) taking away this felfe fame exceffe. The lide therefore ofa Dodecahedron containeth ofa Pyramis circum{cribed about it two ninth 3 partes, taking awaya third part of one ninth part of the leffe fegment ( of aline dinided by an extmere_ 9 and meane proportion ) and moreouer the lefle fegment of the leffe fegment of halfe the refidue. . ql he Elementes of Geometry,added by Flufas. Folge q Uhe 36. Propofition. An Offohedron exceedeth an I. cofahbedron infcribed init, by a parallelipie pedon Jet vpon the [quare of the fide of the Icofahedron , and hauing to his altitude the line which is the greater fegment of halfe the femidiamec ter of the Oohedron. : | Vppofe that there bean O@ohedron AtB CEPL : in which let there be inftribed an : Wg Icofahedron HKEGMXNVDS QT , by the 16. of the fivetenth And draw the dia- \ Ze ag AZRC.,BROIF, and the perpendicular K O paralleltotheline AZR. Then VY ay I fay that the O&ohedron AB C FP Lis greater then the Icofahedron infcribed in it,by SSS) 2 parallelipipedon fet vpon thefquare of the fide H K or G E, and hauing to his altitude the line K O or RZ : whichis the greater fegment of the femidiameter A R. Forafmuch asinthe fame 1:6. ithath bene proned,that the triangles KD G and KE Q_are defcribed in the bafes AP Fand AL F ofthe Oétohedron :therefore about the folide angle there remaine vppon the bafe FE Gthree triangles K E G,K F E,and KF G,which containea pytamis KEFG . Vnto which pyramis shall be equall and like the oppofite pyramis MEFG fet vpon the fame bafe F E G, by the 8.definition of che eleuenth., And by the famereafon fhall there at euery folide angle of the Odtohedron remayne pd ee equal and like : namely, two vpon the iy H K, two vpon the bafe B NV,two vpon moreouer two vp- on the bafe QL T. Now thé there ihal be made twelue pyramids, fet vpon a bafe contained of H = the fide of the Ico-~ fahedron, and yh- der two ‘leffe' ‘feg- mentes of the fide of the O&ohedron ve a right angle, as for exam- ple the bafe GEF, And forafmuch as the fide G E fubré- diag 2 right angle, is, by the 47. ofthe firft, in power du- leto-cither of the Fities EP-and FG, and fo the fide KH is in power duple to’ either of the fides AH andA K: and either of the lines AH, AK,or EF, FG,is in pow- er duple to eyther ofthe lines AZ or C ZK. which cétayne : oe 3 a right angle, made in the triangle or bafe A H K by the perpendicular A Z. Wherfore it followeth thas the fide G E or H K,is inpower quadruple to the triangleE FG orAHK. But the pyramis K E F G,ha- uing his bafe EF G in the plaine F LB P of the O&ohedron, fhall haue to hisaltitude the perpendicu- lar K O (bythe 4. definition of the fixth ) which is the greater feement of the femidiameter of the Octohedron, by the 16.of the fiuetenth. Wherfore three pyramids fer vnder the fame altitude and vpon equall bafes,fhall be equalt to one prifime fet vpon the fame bafe,and vnder the fame altitude, by the r, Corollary of the7.of the rwelfth. Wherefore 4.prifmes fet vpon the bafe GE F quadrupled ( which is equall to the {quare of the fide G E ) and ynder the altitude KO (orR Z the greater fegment which is equallto.KO ) fhall containe a folide equall to the twelue pyramids, which twelue pyramids make the exceffe of the O@ohedren aboue the Icofahedron infcribed in ic. An O&ohedron therefore excedeth an Icofahedron infcribed in it, by a parallelipipedon fer vpon the fquare of the fide of the Icofahedron, and hauing to his altitude the line which is the greater fegment of halfe the femidiameter of the Ofto- hedron. @ A Corollary. i KG/ ConStruditon. Demon ffra- b0%e . ty The jixtenth Booke of the )\) 96. « @ACorollary. . | | A Pyramis exceedeth the double of an Icofahedron infcribedinit sby a foe lide fet wpon the {quare of the fide of the Icofahedron infcribed in it,and haning +o bis altitude that whole line of which the fide of the Icofahedron 1s the greater comet. Forit is manifelt by the r9.ofthe fiuetéth that an oftohedré & an Icofahedré infcribed in ie “are in{cribed in one & the felffame pyramis.It hath moreouer bene proved in the 26.0f this boke,that a pyramisis double to an oftohedro infcribed init. Wherforethe two excefles of the two.octohedrons (vnto which thepyramis is equal)aboue thetwo Icofahedrons(in{cribed in the faid two odtohedrons) being brought into an folide, the faid folide fhalbe fet vpon the felfe fame fquare of the fide of the Ico- fahedrossand Mall have to his aleieude the hee beter K O doubled: whofe double coupling the oppofite fides H'K and XM maketh'the greater fegment the fame'fide of the Icofahedron, by the firlt and fecond coréllaty ofthe 24.6f the fiuecenth. s, | _ The 39.Propofiteon. Tf in a triangle haning to his bafe arational lmefet, the Jides be commene. furablein power to the bafe, and from the toppe-be drawn to the bafe a pers pendicular line cutting the hafe:T he Jethions of the bafe hall be commens farable in length to the whole bafe, and the perpendicular {hall be commen’ jfurable in power to the fard whole bafe. t ) ~~ T-t Vppofe chat there be acriangle AB.G, whofe bafe B G let bea rational line fet of purpolg. | 2\ Steg. And jet the fides Band A G be vnto the fame B Gcommenfurable at che leaft in power. *4,9.¥ || And from the toppe A, draw vnto the bafe B G,a perpendicular cutting the bafe in the point, fe P Then I fay thar the fections of the bafe, are commenfurablein lengthe to the whole line: B G,and that the perpendicular A P,is vnto the fame dafe B G comenfurable at theleatt in power,Pro- duce on either fide the lime'B Gto the poyntes C and E. Andyntotheline A G put the line GE equal, and vato theline A B'put the line B C equal.And ypon the lines C B, B Gand GE defcribe {quares B- KB D,and G.L.And from the greater of the fquares:‘of the linesA B or AG, which let be GL cutof a. ‘paraliclogrdmme EM equal to the leflefquare B Kk (by the 45.0f the firft:)) And (by thé fame) yato the. Fir part of refidue G M lec therebe applied vpon the line G D an equall rectangle paralielogramme OD-Now fore, ideale aftfiuch as the angles A P B and AP Gare tight angles, erfore(by the 47.0f the firft)}the line AG con pipegas taineth in power the two lines A P and P Gard the line AB thetwolines AP and P B. Wherfore how ftration. much the line A G containeth in power more then the line AB, fo much alfo doth theline PG contain in power tore then the line B P: namely, taking away the common {quare of AP, there isleft the ex-. ceffe of the fquare of P G aboue the fquaré of B P. But the {quare of A G (which is GL) exceedeth the fouare of A B (namely,the fquare B K) by the retangle parallelogramme G M or O D, by conftruction*. Wherfore the (quare of P G exceedeth the {quareof B P, by the retangle parallelogramme O D.And fora{much as ynto the {quares of A Band A G,are equal the {quares of A P and P B,and of AP and P= G:and their excefle is taken away ,namely,the reGlangle paralelogramme O D : there fhallbe left the ' fauares of A P and P O equal tothe fyuares of A P and P B.And taking away the fquare of A P which Cen ftruttion. is cOmon,therefidues | namely, the fquares of | A. BPandP O thalbee- qual : and therefore their fides(namely,the lines B P and P O) are equal. And forafmuch asthe fquares.G Laad . BK are(by fuppofitio) rational, and therefore coméfurable their ex- céffe OD, fhalbe'com- méfurable ynto thé by the 15.ofthe téth, And therfore it is rationall by the 9. diffattion of theréth. Wherfore the « rational parallelogram © D,being appliedvp- on the rational line G- D (or BG)maketh the Elementes of Geometry added byFlufjas. Fol 458, bredth O G rational andcomen{utablée id légei to tht. whole liné B,G, by. thé 20, of thetenth Pitt if the whole lIméB G-be comnienfurable to‘one of the partes OG the lines BOO Gjand BG {halPbe com- menfurable,by the fame 15.0f the tench, Wherfore alfo thelinerOG ihalbe commen furable to the half of the line B O,namely,to the line P O,or P B, by the 12.0f the tenth. And forafmuch as the two lines P Oand O Gare commenturable,the whole line P G fhalbe commen{urable to the line P O,or tothe line P B,by the fame 15.0f the tenth. Wherfore either of the lines P G and P B fhall ‘be cémenfurable vnto the Wholedline P B,by the fame. Wherefore the linesB G,P Band P G haue the one to.the other that proportion which numbers hau, by thes.of thirrenth. Wherfore the feétions P Band PG of the bafe B G are commenfurableth lénzth to thé fame bafe, by the 6.of the tenth. ° ¢ | And now that the perpendicular A P is commenfurable i power to the bale B G, is ‘thus pro- ied . Forafmuch as the fquare of A Bis by fuppofition, commenfurable to the f{quare of B G : and ynto the rational {quare of A Bis commmenfurable the rational {quare of B P( by the 12.0f the-eleuenth) Wherfore the refidue namely, the fquare of P Ais commenfurable to the famefquate of BP, by the 2.part of the 15.0f the eleuenths Wherefore by the 12.0f the tenthsthe fquare of PA is commenfurable to the wholefquare of BG. Wherefore the perpendicular A Piscommenfurablein powerto the bale B G,by the 3.diffinition of the tenth: which was required to.be proned. In demonitrating of this, we madeno mention at all of the length of the fides A Band A G,bue only‘of the length of the bafe B G+ for that the line B G-is the rational line firft {et : and the other lines ABandAG arefuppofed to be commenfurable in powcrionly tothe line BG. Wherefore if that be plainely demonftrated,when the fides are commenfurable in power. only to. the bafe,much more eafily wil it tollow,if the fame fides be fuppofed to be commenfurable both in length and in power to the bafe:thatis,if their lengthes be'expreffed by the rootes of fquare nombers. @y A’ Corollary. iii By the former things demonftrated >it is manife/t that if from the powers of the bafe and of one of the fides be taken away the power of the other frde,and if the halfe of the power.remaining ,be appled ppon.the whole bafc,it [hall make the bredth that fection of. the bafe-which ss coupled.tacthe fir[t-fide. For from the po- wers of the bafe B G,and ofione of the fides. A G, that is,from the fquares BD and GL,the power of the other fide"A B,namiely the {quare B i¢(or the parallelogramme E M)is,taken away. And the refidue, (namely, of the fquare B D,and of the paralldlogrammé' © D,or'D R; which by fappofition is ¢qual vn- to. O'D)the halfe (namely,of the whole'B R,which is P._D)for the’ lines G Rand P Bareequalto the lines,G O.and.P. O)is applied ro.the whole line B G ar G Ds\and maketh the bredthe the'line P G the feétion ofthe bafe B G,which feétionis coupled ro the firft fide A G. And by thefame reafon in the o- ther fidéjiffrom thefquaresB Dand BK be taken’ away the fquate GL; there fhall'remiaine the reftan- gle paralielogramme FO: Forthe:parallalelogramme.E-M isiequabeoithe {quare'B K,/and the parallelo~ ramme G M to the parallelogramme O D. Wherefore F B.théhalfe of the refiduc.b.O, maketh the Bendel BP,whichiscoupledtothefiritfidetakenAB. = oP beh A 4 A Corollary... 2. Ifa perpendicular. dratone from an angle of a triangle do cut the bafe: UPD Bions are to the other fides in power proportional by an Ae rithmetical proportion. For it was prowed that the éxceffe of the powers > of the lines A Gand A Bis one and the fame with the exceffe of the powers ofthe lines P Gand P B.If therfore the powers do equally excede theone thie otherythey fhall by an Arithmetical proportion, be proporuonall, 5 Sioa: | tiHasos . Bia" * 5 WO . wh rh MODS 7 e ry. . . 7 - ” © = . . . . ™% “Ey hi 5, ' qj ' ‘es 4) 3 ie g CWA DANS A Jas wD + $)\ oe os Theende of thefixtenth Booke *™: of the Elementes of Geometrie added rep Toafric by Flafare 9 21ND STtToonv i9 “-* . 4%Q @ Ae ; — as} ot . fis : : aL ESE SAPs GC . : Second pare of the De- | MONIT AbIOMe Flufas,of mixtand >: i Sap Abriefe treatife, added by Fluflas, of mixt and compofed regular folides, Egularfolides are fayd to be compofed and mixt,when ech of them is tranfformed into other folides, keeping ftill the forme, number, and inclination of the bafes, which they : [before had one to the other : fome of which yet are tranf ¥g 1, formed into mixtfolides, and other fome into fimple. Into mixt, as a Dodecahedron and an. Icofahedron : which are tranfformed or altered, if ye diuide their fides into two e- quall partes, and take away the folide. angles fubtended of plaine fuperficiall figures made by the lines coupling thofe middle fections : for the folide remayning after the taking away of thofe folide an- gles, is called an Icofidodecahedron. Ifye diuide the fides of a cube and of an Octohedron into two equall partes, and couple the fections,the folide angles {ub- tended of the plaine fuperficieces made by the coupling lines, being taken away, Exofobedro. there fhall be left a folide,which is called an Exoétohedron. So that both ofa Do- decahedron and alfo ofan Icofahedron, the folide which is made, {hall be called an Icofidédecahedron: and likewile the folide made'ofa Cube & alfo of an O&o- hedron,thall\be called an Exo@tohedron . Butthe other folide, namely,a Pyramis (or Tetrahedron ) is tran{formed into a fimple folide: for if ye diuide into two equall partes enery one of the fides ofthe pyramis, triangles defcribed of the lines which couple the fections, and fabtending, and ‘taking ‘away folide angles of the yramis, arc equall and like vnto the equilater triangles left in euery one of the safes ;, ofall which triangles 4s produced an Octohedron, namely, a fimple and not acompofed folide . Forthe Octohedron hath fower bafes, like in number, forme, atid mutuall inclination with the bafes of the pyramis : and hath the other | fower bafes with like fituation oppofite.and parallel to the former, Wherefore the application of the pyramis taken twife, maketh a fimple Oaohedron, as the other 4 i Teofrdodecahee row € folidesniake'amiixt compoundfolide. q Firft Definition. An E-xoffehedron is a flide igure cantained of fixe equall fquares, and eight equilater and equall triangles. oo. soaroaone os ayue lar Lid oy Second Definition, An Icofidodecahedron is a , sean figure , contained bnder twelue eqn s 4 later equall and equiangle Pentagons , and twentie equall and equilater j ho <. Oeee Tee « 1 COE TE ae ae i Tat cL. o a — - . iy For the better vnderftanding of the two former definitions, and alfo of the two Propofitions following, I haue here fettwo figures, whofe formes, ifye firk defcribe compofed regular folides. Fol.459. deferibe vpon pafted paper or fisch like matter; and then cutthem and folde them accordingly, they will reprefent ynto you the perfect formes ofan Exoct and ofan Icofidodecahedron. : ore oConedron q I he first Probleme. T 0 defcribe an equilater and equiangle exottohedron , and to contayne it ina [phere geuen:and to prone that the diameter of the {phere is double to the fide of the fayd exoltobedron. Vppote that there bea fphere geuen, whofe diameter letbe AB. And about the diameter A B let there be defcribed a {quare by the fixth of Con Struttion of the exotio- bedron. S*I the fourth:and vpon the {quare let there be defcribed a cube by the 15. stile! of the thirtenth: which letbe C DEF QT VR: and let the diameter thereofbe Q R,and the centre S.And diuide the fides of the cube into two equall artes,in the poyntes G,H,1,K,L,M,N,O,P.é&c.And couple the middle fections Be the right lines I N,N O,O P,P I and fuch like,which fubtend the angles of the fquares or bafes'of the cube:and they are equall by the 4. firft, and contalne right angles,as the angle N I P. For the angle NID which is at the bafe of the Ifofceles triangle N D IJ, is the halfe ofa right angle, and fo likewife is the oppofiteangle R- IP. Wherefore the refidue N I P isaright angle,and fo the reft. Wherefore N I P- O is afquare.And by thefame reafon fhallthe reft NMLK,KGHI &c.infcribed DDD.4j. in That the ixote tobedron ts cantayned in 4 Sphere. 7 bat the exoc~ tohedvron ts contayne din the fpbere Eten ¢ That the dia- meter of the foberetsdou- ble to the fide of the exotto- kedyou * & they are the limmits or borders Flufsas,of mixt and in the bafes of the cube,be fquares:and they fhall be fixein number, according to ae | thentiberofthe bafes ofthecube. C VE ; . Agayne forafmuch as the triangle /\ i KIN fubtendeth the folide angle . i D of the cube, and likewife the tr1- N 08 : me angle K G L the folide angle C, & i/ 3 a fo the reft, which fubtend the eight folide angles of the cubezand thefe triangles are equall and equilater, namely, being made of equall fides of the fquares , and the {quares the, , \ - iE 7 iA Jimmits or -borders of thé , as hath © . }/ / “a ye = before bene proued: wherefore L- Va a Ne : = ae RK MNOPHGK is an exoctohedro, ig by the diflinition , and is equilater, “p for itis contayned of equall {ub- tendent lines:it isalfo equiangle, <*'—— ‘B for eucry folide angle thereof, is contayned vnder two fuperficiall angles of two fquares , and two fuperficiall an- gles of two equilater triangles. | And now forafinuchas the oppofite fides and diameters of the bafes of the cube are parallels, the playne extended by therightlines QT, VR, fhall bea pa- rallelogramme. And for thatalfo in that playnelyeth QR the diameter of the cube,and in the fame playne alfo is the line MH, which diuideth the fayd playne into two equall parts ,and alfo coupleth the oppofite angles of the exoctohedron this line M H therefore diuideth the diameter into two equall partes , by the co- rollary of the 34.0f the firft , and alfo diuideth it felfe in the fame poynt, which let ——— ee be S,into two equall partes, by the 4 of the firft. And by the fame reafon may we proue that the reft of the lines, which couple the oppofite angles of the exoétohe- dron ,dooinS thecentre of the cube diuide the one the otherinto two ¢ uall parts.For euery one of the angles of the exoétohedron are fet in cuery one o the bafes of the cube. Wherefore making the centre the poynt S,and the {pace SH or S Mj defcribe a {phere, andit thall touch edery one of the angles equediftant from the poyntS. " Be POU! © PE ES : And forafmuch as A B the diameter of the {phere geuen , 1s put teat tothe diameter of the bafe of the cube,namely,to the line R'T , and the fame line R Tis equiall to'theline M Hyby the 33. of the firft: which line M Hco upling the oppo- fitc angles of the exoCtohedron , is drawne by the centre: wherefore itis the dia- meter of the {phere geuen which contayneth the exoctohedron. | Finally forafinuch asin the triangle R FT ,the line P O doth cutte the fides into two equall partes, it (hall cutte them proportionally with the bafes,namely,as F Risto FP, fo thall RT beto PO, by thez. of thefixth. But FR is double to F- P,by fuppofition: wherefore RT, or the diameter H M , 1salfo double to theline P.O the fide of the exo¢tohedron. Wherefore we haue defcribed an equilater & e- quiangle exo@tohedron,and com prehended itin a {phere geuen,and haue proued that the diameter of the fphere is double to the fide of the exoctohedron. The compofed regular folides. Fol.460. q Ube z. Probleme. To defcribe an equilater