Cardano, Girolamo, De subtilitate, 1663

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              torum: ideóque ex decimaquarta eiuſdem,
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              ſi KC eſt maior AC, vel æqualis, vel minor,
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              ita CG maior, æqualis, aut minor CB. </s>
              <s id="s.010808">Cùm
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              igitur KC æqualis ſit AF, diuiſa per medium
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                <figure id="id.016.01.238.1.jpg" xlink:href="016/01/238/1.jpg" number="91"/>
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              in H, per quintam deſcribetur ſemicircu­
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              lus ſecundum propoſitam magnitudinem,
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              quia AF fuit dupla illi latitudini per ſextam:
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              erigo igitur CM perpendicularem, & ducta
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              GL ex ſectione circuli, & perpendicularis,
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              ducemus BM illi æquidiſtantem per trigeſi­
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              mam mediam: conſtat igitur CM, eſſe pro­
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              portione mediam inter AC & CB: nam (vt
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              demonſtratum eſt) vt KC ad CA, ita CG ad
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              CB, quare, vt KC ad CG, ita AC ad CB: at
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              ex quarta ſexti, vt CG ad CB, ita LC ad
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              CM: LC, autem ex octaua ſexti Elemento­
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              rum & trigeſimaquarta harum eſt in media
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              proportione K C & CG, igitur & CM, in
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              media proportione AC & CB. </s>
              <s id="s.010809">Ex hac ha­
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              betur vltima ſecundi Element. </s>
              <s id="s.010810">quæ eſt tri­
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              geſimaſexta. </s>
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            <p type="margin">
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              Quintus li­
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              ber Euclid.
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              totus.
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              </s>
              <s id="s.010812">Sexti libri
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              primæ duo­
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              decim propo­
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              ſitiones.</s>
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            <p type="margin">
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              13 35
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              </s>
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            <p type="margin">
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              Vltima ſe­
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              cundi
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                torum</expan>
              36.
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              Reliquum
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              ſexti Ele­
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              mentorum
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              præter vlti­
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              mam.
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              </s>
              <s id="s.010815">Tertij Eucl.
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              </s>
              <s id="s.010816">noſtræ,
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              17 37
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              </s>
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              <s id="s.010817">Eadem ratione perficiemus Euclidis de­
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              monſtrationibus omnes ſexti Elementorum
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              propoſitiones, vltima duntaxat excepta. </s>
              <s id="s.010818">Inde
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              decimamſeptimam tertij Elementorum ag­
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              grediemur. </s>
              <s id="s.010819">Hæc erit trigeſimaſeptima. </s>
              <s id="s.010820">Du­
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              cta igitur ex puncto præter circulum per
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              centrum recta, ſumam mediam per trigeſi­
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              mamquintam inter totam, quæ ex puncto
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              vſque ad circumferentiam interiorem, & il­
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              lam quæ ei exterius adiacet, inde ſuper ter­
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              minum inuentæ erecta perpendiculari ſe­
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              cundum quantitatem ſemidiametri circuli,
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              ad quem ex puncto propoſito contingens
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              ducenda eſt, concludo triangulum. </s>
              <s id="s.010821">Ergo
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              huic angulo contenta ex vltimo ducta &
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              perpendiculari, ſeu oppoſito conſimili ex
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              25. facio angulum in centro æqualem verſus
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              punctum propoſitum, ex quo ducta recta ad
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              extremum lineæ quæ angulum facit, vbi ſci­
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              licet circulum tangit, contingens erit: nam
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              ex ſexta ſecundi Element. </s>
              <s id="s.010822">& 47. primi eiuſ­
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              dem, linea ex puncto ad centrum æqualis
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              erit lineæ vltimò ductæ, quæ recto opponi­
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              tur: ex prima igitur harum, angulus ille in
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              baſi iuxta circumferentiam rectus, & ex 16.
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              tertij Element. </s>
              <s id="s.010823">producta contingens. </s>
              <s id="s.010824">Reli­
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              quæ omnes tertij libri, præter 24. & 33. de­
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              monſtrantur, ex iam demonſtratis. </s>
              <s id="s.010825">In 24.
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              demonſtrabimus locum centri, vt Euclides:
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              perficere circulum non eſt poſſibile, cum re­
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              pugnet iam promiſſis, illa
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              vtemur, quia
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              non eſt opus, niſi in circumſcribendis, aut in
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              ſcribendis circulis niſi centri inuentione, vt
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              demonſtrabimus. </s>
              <s id="s.010826">In 33. etiam tertij abſol­
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              uemus quotquot voluerimus angulos ſupra
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              datam lineam: quia omnes, ſi circulus ſupra
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              illam deſcriberetur, in circumferentia illius
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              eſſent. </s>
              <s id="s.010827">Id prius auxilio trigeſimæquartæ ter­
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              tij, quæ abſque 33. demonſtratur in propo­
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              ſito tuo circulo, abſoluemus, inde per 25. ſu­
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              pra conſtitutam lineam: ergo erunt hæ no­
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              bis loco trigeſimęoctauæ, & trigeſimęnonæ,
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              ſicut vltima ſexti Elementorum pro 40. Pòſt
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              demonſtrabimus primam quarti Elemento­
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              rum, hæc erit nobis quadrageſimaprima: per
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              duodecimam ſexti Elementorum conſtitu­
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              tam, vt ſit latitudinis circini propoſiti ad A,
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              lineam, vt ſemidiametri circuli, in quo eſt
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              linea inſcribenda ad lineam inſcribendam,
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              inde collocata A in circulo mihi permiſſo,
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              expleo trigonum duûm æqualium laterum,
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              & angulo in centro circuli permiſſi, quem
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              ſubtendit linea A æqualem, ex vigeſima­
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              quinta facio in propoſiti circuli centro.
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              </s>
              <s id="s.010828">Conſtabit itaque ex trigeſimaprima harum
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              trigonos eſſe ſimiles, ſubducta ſemidiame­
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              tris recta, quare ſemidiametri conceſſi ad A,
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              vt ſemidiametri propoſiti ad ſubductam ex
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              quartaſexti Elementorum, talis verò fuit
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              ſemidiametri propoſiti circuli ad lineam
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              propoſitam, igitur ſubducta eſt æqualis pro­
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              poſitæ. </s>
            </p>
            <p type="margin">
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              Reliquum
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              tertij libri
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              excepta 24.
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              & 33.
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              24 37
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              </s>
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            <p type="margin">
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              33 39
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              </s>
            </p>
            <p type="margin">
              <s id="s.010831">
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              Eucl. </s>
              <s id="s.010832">noſtræ
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              Vltim. </s>
              <s id="s.010833">ſexti
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              40.1. quar­
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              ti 41.</s>
            </p>
            <p type="main">
              <s id="s.010834">Poſt hæc demonſtranda eſt 22. primi Ele­
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              mentorum: quamuis ad Euclidis finem non
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              ſit neceſſaria, ſed propter 22. ab eodem ad­
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              iecta fuerit ſolum, quæ iam ſuperius eſt de­
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              monſtrata. </s>
              <s id="s.010835">Sint igitur propoſitæ tres lineæ
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              ABC, ſub conditione ibidem adiecta, & aſ­
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              ſumo circulum mihi conceſſum, cuius dime­
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              tiens DH, & medium eius DE, & ſit A ma­
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              ior B, & B maior C, & ex 12. ſexti Elemen­
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              torum iam demonſtrata, fiat DE ad EF, vt
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              A ad B, & EF ad FG, vt B ad C: & quia B
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              & C, ſupponuntur longiores A, erit tota EG
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              longior E D, igitur G punctus cadet extra
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              circulum: fiat ex eadem 12. ſexti Elemento­
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              rum, vt GF ad FH, ita DF ad K, cui K adii­
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              ciatur L, æqualis GF: igitur vt DF ad K, ita
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              L ad F H. </s>
              <s id="s.010836">Quia verò H F eſt minima 4.
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              quantitatum proportionis vnius, erit D F
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                <figure id="id.016.01.238.2.jpg" xlink:href="016/01/238/2.jpg" number="92"/>
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              maxima, totáque DH maior tota KL, ex 25.
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              quinti Elementorum: igitur per 41. præce­
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              dentem collocabimus KL, quomodolibet in
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              circulo, vt ſit M O, & ex 13. faciam M N
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              æqualem L, erítque NO æqualis K: & du­
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              cam ex centro ENP producendo ex aduerſo
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              in Q, & iterum ex eodem centro EM. </s>
              <s id="s.010837">Ex 16.
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              ſexti Element. </s>
              <s id="s.010838">productum KL, id eſt MN,
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              in NO, æquale eſt producto DF in FH, quia
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              K & L fuerunt proportione mediæ inter
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              DF & FH: & ex 35. tertij Element. </s>
              <s id="s.010839">produ­
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              ctum P N in NQ, æquale eſt producto MN
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              in NO, igitur ex PN in NQ, fit quantam ex
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              DF in FH: igitur cum PQ, ſit æqualis DH,
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              erit NP, æqualis FH, & EN æqualis EF:EM
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              autem eſt æqualis E D, & FG æqualis L, &
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              L æqualis MN, igitur F G æqualis M N,
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              igitur trigonus E M N, conſtat ex tribus </s>
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