Cardano, Girolamo, De subtilitate, 1663

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              <s id="s.010937">
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              quod eò magis fiunt proximæ. </s>
              <s id="s.010938">Et ſufficiat
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              demonſtraſſe de vno, vtpote quòd G & M,
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              ſint propinquiores, quàm S & T: nam tunc
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              patebit quòd vbi magis procedent illæ duæ
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              lineæ, erunt eò proximiores. </s>
              <s id="s.010939">Capiatur igi­
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              tur gratia exempli circulus PSQ, & duca­
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              tur TSR, ita quòd perueniat ad oppoſitam
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              circumferentiæ partem: & ſimiliter duca­
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              tur M G N in ſuperficie H, ita quòd G N,
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              perueniat ad circumferentiam circuli VGX:
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              & ducantur rectè LT, & OM in ſuperficie
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              H, quæ contangent circulos QLP, & XOV,
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              quia ducuntur ex loco contactus: & quia
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              O & M, ſunt in ſuperficie circuli OXV,
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              nam M eſt terminus lineæ NM, quæ eſt in
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              ſuperficie circuli OXV, erit linea OM, in
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              ſuperficie eiuſdem circuli, & ita LT in
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              ſuperficie circuli P L
                <expan abbr="q.">que</expan>
              Sed tales ſuperfi­
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              cies æquidiſtant, quia ambæ à baſi circuli:
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              & ſunt lineæ OM & LT, in ſuperficie K,
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              ambæ: igitur æquidiſtantes. </s>
              <s id="s.010940">Et iam LO
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              & TM æquidiſtant, ſunt enim partes æqui­
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              diſtantium, igitur LT, & OM ſunt æqua­
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              les. </s>
              <s id="s.010941">Et cùm contangant circulos PLQ, &
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              VOX, igitur ex demonſtratis ab Euclide
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              in 3.Elementorum quadratum TL, eſt æqua­
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              le ei quod fit ex TR in TS, & quadratum
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              OM eſt æquale ei quod fit ex MN in MG
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              & quadratum TL, eſt æquale quadrato
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              OM, igitur quòd fit ex TR in TS, eſt æqua­
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              le ei quod fit ex MN in M G. </s>
              <s id="s.010942">Igitur ex de­
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              monſtratis 6.Elementorum ab Euclide pro­
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              portio ST ad GM, eſt vt MN ad TR. </s>
              <s id="s.010943">Sed
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              MN maior eſt TR, quia ſi duceretur per N,
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              ſuperficies æquidiſtans ipſum N, caderet
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              infra R, aliter occurreret K, quia diameter
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              QP, eſt minor XV, & ſuperficies circulorum
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              ſunt æquidiſtantes, igitur ST maior eſt GM.
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              </s>
              <s id="s.010944">Ducuntur igitur S Y & G Z, ad perpendi­
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              culum ſuper EF, & erunt anguli SYT, &
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              GZM æquales quia recti. </s>
              <s id="s.010945">Similiter anguli
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              STY, & GMZ æquales ſunt, quia ST, &
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              GM ſunt æquidiſtantes, ſunt enim ambæ in
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              ſuperficie eadem, quæ eſt H, & in duabus
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              ſuperficiebus æquidiſtantibus circulorum:
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              igitur ex 32. primi elementorum trigoni
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              STY, & GMZ, ſunt æqualium angulorum,
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              quare per quartam ſexti eiuſdem proportio
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              ST ad GM, vt SY ad GZ. </s>
              <s id="s.010946">Sed ST vt pro­
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              batum eſt, maior eſt GM, igitur S Y maior
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              GZ. </s>
              <s id="s.010947">Sed SY eſt minima quæ poſſit duci ex
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              puncto S, ad lineam EF, quia ad perpendi­
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              culum eò quòd omnis alia ducta ab eodem
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              puncto, ad lineam EF, ex quauis parte op­
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              ponitur maiori angulo quàm SY: quia op­
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              poneretur recto, igitur punctus G, eſt proxi­
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              mior lineæ EF, quàm punctus S,
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              quòd erat demonſtrandum. </s>
              <s id="s.010948">Ple­
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              rique deficiunt in hac vltima
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              parte, admittentes paralogiſ­
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              mum. </s>
              <s id="s.010949">Feci igitur conum ex ra­
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              pa, vt conſulit Rabbi Moyſes,
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              & feci ſuperficies K & H, ex pa­
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              pyro, & inſcriptis lineis A C,
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              EF, SG, viſæ ſunt non concur­
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              rentes, vt à latere vides. </s>
              <s id="s.010950">Sed eas
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              niſi ea arte inuentas difficile eſt
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              deſcribere. </s>
            </p>
            <figure id="id.016.01.243.1.jpg" xlink:href="016/01/243/1.jpg" number="101"/>
            <p type="caption">
              <s id="s.010951">
                <emph type="italics"/>
              Lineæ nunquam concurrentes.
                <emph.end type="italics"/>
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              </s>
            </p>
            <p type="margin">
              <s id="s.010952">
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              Defectionis
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              priuilegia. </s>
              <s id="s.010953">2.</s>
            </p>
            <p type="main">
              <s id="s.010954">Defectionis duo ſunt priuilegia: primum
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              quòd proportio eius ad circuli ſuperficiem
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              eſt, velut rectanguli diametrorum defectio­
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              nis ad rectangulum diametrorum circuli,
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              quod eſt quadratum. </s>
              <s id="s.010955">Secundum ex hoc du­
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              cit originem, quòd proportio defectionis ad
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              defectionem eſt, velut rectangulorum ſub
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              diametris earum propriis contentorum. </s>
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            <p type="main">
              <s id="s.010956">Paraboles autem priuilegia ſex propria
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              ſunt. </s>
              <s id="s.010957">Primùm, ratio axis partium, in ea eſt,
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              vt deductarum ex ipſis punctis perpendicu­
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              larium ad paraboles circumferentiam dupli­
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              cata. </s>
              <s id="s.010958">Secundum, cùm fuerit ipſa perpendi­
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              cularis æqualis axis parti, quæ ad verticem
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              ab extremo eiuſdem perpendicularis termi­
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              nabitur, vocabitur ipſa perpendicularis la­
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              tus rectum Paraboles, erítque hæc ſemper
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              talem habens proportionem ex axe ad cir­
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              cumferentiam, qualis eſt perpendicularis ip­
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              ſius ad partem axis, quæ ipſam perpendicu­
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              larem, & verticem ſectionis interiacet: vo­
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              cantur verò hæ lineæ perpendiculares ordi­
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              natæ. </s>
              <s id="s.010959">Manifeſtum eſt igitur, quòd cuilibet
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              parti axis Paraboles, ac ſuæ perpendiculari
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              ſemper eadem linea in continua proportio­
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              ne ſubtenditur. </s>
              <s id="s.010960">Tertium, quòd ſi in ea pun­
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              ctus præter axem ſignetur, ab hoc contin­
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              gens ducatur, huic verò æquidiſtantes plu­
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              rimæ à circumferentia ad circumferentiam
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              ducta ex eodem puncto contactus æquidi­
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              ſtans axi, omnes lineas æquidiſtantes à
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              contingenti ductas per æqualia ſecabit. </s>
              <s id="s.010961">Por­
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              tiones quoque quomodolibet ſumptæ, æqua­
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              les habentes diametros, etiam æquales ſunt.
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              </s>
              <s id="s.010962">Ipſa verò ſuperficies æqualis eſt rectangu­
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              lo ex tota baſi in duas è tribus axis partes.
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              </s>
              <s id="s.010963">Sextum, cùm tres contingentes periferiam
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              Paraboles concidunt, duas quidem extremas,
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              media ſecante, erit proportio partium trium
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              linearum vna, ſcilicet partis inferioris ad
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              ſuperiorem, & ſuperioris alterius ad infe­
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              riorem, & mediæ illarum, quæ ad perife­
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              riam Paraboles terminantur. </s>
            </p>
            <p type="margin">
              <s id="s.010964">
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              Paraboles
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              priuilegia 6</s>
            </p>
            <p type="main">
              <s id="s.010965">Spiralis autem lineæ priuilegia ſex etiam
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              ſunt. </s>
              <s id="s.010966">Primum quidem, quòd ducta contin­
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                <arrow.to.target n="marg1534"/>
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              gens ex fine illius occurrit perpendiculari ex
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              initio, ſemper tantum abſcindens ex conti­n
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              genti, vt proportionem habeat ad circuli ſub
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              eodem ordine periferiam, ſecundum ordinem
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              ſeriei numerorum. </s>
              <s id="s.010967">Vnde patet, quòd portio
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              primæ ſpiralis ex perpendiculari, erit æqua­
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              lis periferiæ primi circuli, & portio perpen­
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              dicularis ex ſecunda ſpirali dupla circuli ſe­
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              cundi periferiæ, & portio ex tertia ſpirali
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              tripla circuli tertij periferiæ, atque ita dein­
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              ceps. </s>
              <s id="s.010968">Secundum, ex quocunque puncto pri­
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              mæ ſpiralis educta contingens, occurrit
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              perpendiculari ex initio eiuſdem dimetien­
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              tis ductæ, tantam ex illa abſcindens par­
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              tem, quanta eſt portio circumferentiæ cir­
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              culi, cuius ſemidiameter eſt linea ex initio
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              lineæ ſpiralis, vſque ad punctum contin­
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              gentis, clauſa inter primam lineam rectam
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              ſpiralis, quæ moueri intelligitur, & locum
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              ad quem per motum peruenerit ipſa ſpi­
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              ralis è directo loci contingentis. </s>
              <s id="s.010969">Eſtque
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              tertium priuilegium, quòd ſpatia ſpira­
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              lium ita ſe habent: primùm quidem vni­
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              tatis, ſecundum ſenarij, tertium duodena­
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              rij, quartum decemocto, atque ita deinceps
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              additione perpetua per ſenarium facta.
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              </s>
              <s id="s.010970">Quartum proportio cuiuſlibet circuli, ad </s>
            </p>
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