Cardano, Girolamo, De subtilitate, 1663

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              <s id="s.010839">
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              lineis æqualibus DE, EF, FG, igitur propor­
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              tionis ABC: fiant igitur ſuper A, anguli
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              æquales M & E, per 25. erítque trigonus
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              ille ſimilis EMN: igitur ex 4. ſexti Elemen.
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              </s>
              <s id="s.010840">proportio A ad latera reliqua, vt E M ad
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              EN, & MN, ſed eadem erat, vt A ad B, &
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              C, igitur ex 11. quinti Element. </s>
              <s id="s.010841">& nona,
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              eiuſdem latera illa æqualia erunt B & C,
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              quòd eſt propoſitum. </s>
              <s id="s.010842">Demonſtrata hac 42.
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              demonſtrabimus 43. quæ erit 10. quarti Ele­
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              ment. </s>
              <s id="s.010843">Sit AB quam diuido (vt docet 11. ſe­
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              cundi Element.) in C, & per præcedentem
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              facio trigonum ſuper AB, cuius vnum latus
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              ſit æquale AB, & ſit AD aliud æquale AC,
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                <figure id="id.016.01.239.1.jpg" xlink:href="016/01/239/1.jpg" number="93"/>
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              & ſit BD: quia itaque AC eſt proportione
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              media ex 17. ſexti Elementorum, inter AB
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              & BC, erit & B D æqualis, ac proportione
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              media inter A B & B C: ducta igitur linea
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              DC: erit trigonus B A D & B D C angulo
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              communi B, & lateribus continentibus pro­
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              portionis eiuſdem laterum, ex 6. ſexti Ele­
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              mentor. </s>
              <s id="s.010844">quare BD æqualis CD, ideóque DC
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              æqualis CA, & ex ſecunda harum anguli
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              CDA, & A æquales, & ex 31. DCB, eſt
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              æqualis vtrique, îgitur duplus ad A. </s>
              <s id="s.010845">Sed
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              per 2. harum, DCB eſt æqualis B, & per
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              eandem B eſt æqualis A D B, igitur tam
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              ADB, quàm B duplus ad A, quod eſt propo­
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              ſitum. </s>
              <s id="s.010846">Ex his patet demonſtratas eſſe omnes
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              propoſitiones quarti, niſi quòd non licebit
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              circulum circumducere, aut inſcribere, ſed
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              ſolùm inuenire centrum: ipſáſque figuras
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              æquilateras & æquiangulas conſtruere, to­
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              tamque doctrina Euclidis,
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              ad finem 6. li­
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              bri, ſeu 9. iam peracta eſt. </s>
              <s id="s.010847">Vt verò ad finem
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              reliquorum librorum perueniamus, demon­
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              ſtrabimus 44. quæ eſt, ex quocunque puncto
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              in diametro propoſito perpendicularem eri­
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              gere, quæ ad contactum periferiæ circuli
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              perueniat. </s>
              <s id="s.010848">Nam per 35. inueniemus propor­
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              tione mediam, & per ſextam educemus il­
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              lam perpendicularem ad lineam propoſitam
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              ex puncto dato, pertingéntque ad periferiæ
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              locum circuli, cuius propoſita linea eſt dia­
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              meter ex demonſtratione decimætertiæ ſexti
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              Elementorum, qua Euclides vtitur. </s>
              <s id="s.010849">Cùm ve­
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              rò quadrageſimam quintam demonſtrare
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              voluerimus, quæ talis eſt: ſuper datam li­
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              neam triangulum rectum ſupremum ha­
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              bentem, reſpicientémque datam lineam,
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              cuius latus vnum aſſignatæ lineæ, quæ mi­
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              nor ſit prima, conſtituere, circulum deſcri­
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              bemus nobis iam conceſſum, ductáque dia­
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              metro ex duodecima ſexti, lineam ei ſubiun­
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              gemus, ad quam diameter ſe habeat, vt pri­
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              ma linea ad latus illud: erit igitur hæc li­
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              nea minor diametro circuli conceſſi, quare
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              ex quadrageſimaprima collocabimus eam
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              intra circulum, complebimúſque trigonum:
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              igitur ſuper primam lineam angulum in ex­
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              tremitate, angulo contento ex diametro, &
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              quarta linea æqualem per vigeſimamquin­
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              tam faciemus, & lineam hanc productam
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              ex decimatertia faciemus æqualem aſſigna­
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              tæ ſecundæ: quare ex ſexta, ſexti Element.
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              </s>
              <s id="s.010850">completo trigono, fiet ſimilis primo & or­
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              thogonius. </s>
              <s id="s.010851">Quo inuento, ad finem omnium
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              eorum, quæ ab Euclide ſcripta ſunt, tum ab
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              Hypſicle addita Alexandrino, abſque impe­
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              dimento perueniemus. </s>
              <s id="s.010852">Sed hæc (vt vbi) ac
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              ſimilia, ad oſtentationem ingenij, vtilitatem
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              verò penè nullam, inuenta ſunt. </s>
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            <p type="main">
              <s id="s.010853">
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              LIBER DECIMVSSEXTVS.
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            <p type="main">
              <s id="s.010854">
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              De Scientiis.
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            <p type="margin">
              <s id="s.010855">
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              Circuli pro­
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              prietates 12.
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              Recti & cir­
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              cularis crea­
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              tio.</s>
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            <p type="main">
              <s id="s.010856">AT non vocant vtilitate figu­
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              rarum proprietates illæ ſexa­
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              ginta, quas nunc hîc ſubiicere
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              propoſitum eſt. </s>
              <s id="s.010857">Fit enim cir­
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              culus motu non flexilis rei ex
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              altero termino fixæ: ſicut recta motu plani
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              eodem loco conſiſtentis, vtpote rotæ ſuper
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              rem fixam. </s>
              <s id="s.010858">Sic enim regulæ fiunt. </s>
              <s id="s.010859">Fiunt & re­
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              cta extentione. </s>
              <s id="s.010860">Manifeſtum eſt igitur ex hoc,
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              rectum arte prius eſſe circulari, circulum au­
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              tem natura. </s>
              <s id="s.010861">Extrema verò hæc, atque ideò
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              contraria vt periferiam: & quantò minoris
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              fuerit ambitus. </s>
              <s id="s.010862">Omnes igitur aliæ lineæ me­
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              diæ ſunt inter rectam, ac circularem, ac quaſi
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              ex his compoſitæ. </s>
              <s id="s.010863">Circuli igitur & rectarum
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              figurarum certa eſt ratio, aliarum in con­
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              ſtans, niſi vt altera ab altera generatione
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              pendet, vt coni ſuperficies à recta, à coni au­
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              tem ſuperficie paraboles. </s>
              <s id="s.010864">Ex ſuperficiebus
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              autem nulla præter circularem generari di­
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              citur, multò minus recti linearum corporum
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              vllum. </s>
              <s id="s.010865">Rectilinea enim fiunt, non generan­
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              tur. </s>
              <s id="s.010866">Simpliciſſimum enim quod rotundum
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              eſt, vt inter corpora ſphæra, & inter ſuper­
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              ficies circulus. </s>
              <s id="s.010867">Eius autem duodecim propria
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              ſunt. </s>
              <s id="s.010868">In hoc igitvr primo ſecantes ſe lineæ
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              ſub eadem proportione partes conſtituunt.
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              </s>
              <s id="s.010869">Angulúſque contentus ab illarum ſectio­
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              ne, angulis in circumferentia ſuper vtroſ­
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              que arcus conſtitutis pariter acceptis, eſt </s>
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          </chap>
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