Cardano, Girolamo, De subtilitate, 1663

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              <s id="s.010769">
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              Euclides etiam de non coniunctis demon­
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              ſtrat. </s>
              <s id="s.010770">Pro hac igitur demonſtranda, diuida­
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              tur angulus quem illæ continent indefinita
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              linea, poſtmodum facto
                <expan abbr="cẽtro">centro</expan>
              extremo lineæ
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              breuioris deſcribam circulum, qui in
                <expan abbr="termi-nũ">termi­
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                num</expan>
              minoris cadet, maiorem autem ſecabit ad
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              minoris æqualitatem. </s>
              <s id="s.010771">Tranſpoſitis enim tri­
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              gonis, quorum vertices ſunt in puncto con­
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              iunctionis propoſitarum linearum, fines
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              autem ſectiones circulorum cum lineis, ita
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              quòd media diuidens baſis ſit communis
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              vtriuſque ſecundum modum conceſſum ab
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              Euclide in ſua quarta primi elementorum,
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              ni datæ lineæ æquales fuerint, erit pars to­
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              ti æqualis, quod eſſe non poteſt. </s>
              <s id="s.010772">Si verò di­
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              cas circulum è minoris termino centrum
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              habentem ad mediam non peruenire, to­
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              ties per quartam anguli illi bifariàm ſecen­
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              tur, donec attingant: inde repetita demon­
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              ſtratione propoſitum habebitur, vt prius. </s>
            </p>
            <p type="main">
              <s id="s.010773">Quoniam verò trigonos tranſponimus,
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              id non ad conſtruendum quicquam licet. </s>
              <s id="s.010774">Par
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              enim fermè eſſet circuli æquilatationi, ſed
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              ſolùm in theorematibus, ad id quod ita ſit
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              demonſtrandum. </s>
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            <p type="main">
              <s id="s.010775">
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              PARS PRIMÆ VNDECIMÆ
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              </s>
            </p>
            <p type="main">
              <s id="s.010776">Vndecima erit, ſuper datam lineam
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              triangulum duum æqualium laterum deſcri­
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              bere: diuidemus eam bifariàm, erigemus
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              perpendicularem è ſectionis puncto per ſex­
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              tam, completóque trigono per primam pa­
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              tet propoſitum. </s>
            </p>
            <p type="main">
              <s id="s.010777">Ex hac & præcedente, abſque circulis,
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              per modum Euclidis, illius ſecundam de­
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              monſtrabimus, quæ erit duodecima noſtra.
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              </s>
              <s id="s.010778">At ex hac per modum Euclidis, tertia illius
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              demonſtrabitur generaliter, quæ erit
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              decimatertia harum.
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              </s>
              <s id="s.010779">Decimaſexta Euclidis,
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              & quinque ſequentes, vt
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              ab Euclide ponuntur,
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              demonſtrabuntur: habe­
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              búntque locum apud nos
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              decimæquartæ, & quin­
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              que ſequentium: quan­
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              doquidem nullis aliis
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              quàm demonſtratis iam
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              à nobis hucúſqne indi­
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              gent. </s>
              <s id="s.010780">Simili ratione vi­
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              geſimaſexta, & quatuor
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              ſequentes, vigeſimæ no­
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              ſtræ, & quatuor proxi­
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              mè ſequentium locum
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              obtinebunt. </s>
              <s id="s.010781">Vigeſima­
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              quinta noſtra erit apud
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              Euclidem vigeſimater­
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              tia, quæ ſic demonſtra­
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              pars primæ 12.
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              2 13
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              3 14
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              10 15
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              17 16
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              23 26
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              6 27
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              24 28
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              bitur: lineas continentes angulum æquales
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              inuicem ad datam circini latitudinem fa­
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              cies. </s>
              <s id="s.010782">Inde ſubtenſa recta, minor erit per
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              decimamoctauam ambobus lateribus trigo­
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              ni datum angulum continentibus. </s>
              <s id="s.010783">Huic ba­
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              ſi igitur per decimamtertiam hoc expuncto
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                <arrow.to.target n="marg1513"/>
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              dato in linea æqualem abſcindemus: inde
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              rurſus factis vtrinque terminis, lineæ iam
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              abſciſæ centris deſcribemus circulos, qui
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              ſe ſecabunt ex decimaoctaua, vt dixi: du­
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                <arrow.to.target n="marg1514"/>
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              ctis ergo linei ex communi circulorum ſe­
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              ctione ad extrema lineæ ſubiectæ, iam pa­
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              làm erit ex tertia angulum in dato puncto,
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              propoſito eſſe coæqualem. </s>
              <s id="s.010784">Sextam inde loco
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                <arrow.to.target n="marg1515"/>
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              demonſtrabimus facillimè ex decimatertia,
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              demonſtratione, quæ contradicentem dedu­
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              cat ad impoſſibile: ſed placet vera demon­
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              ſtratione oſtendere: alium igitur trigonum
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              ex præcedenti fabricabo baſim habentem ba­
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              ſi æqualem, & angulos qui ſunt ſupra baſim
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              angulis ſupra baſim propoſiti trigoni æqua­
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              les. </s>
              <s id="s.010785">Inde ſuperponendo baſim baſi ex prima
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                <expan abbr="harũ">harum</expan>
              conceſſa ab Euclide, fiet per communes
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              animi ſententias bis, ſuperponendo verſa vi­
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              ce, vt latera demonſtrentur æqualia. </s>
              <s id="s.010786">Quo
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              peracto vigeſimaoctaua ex præcedente de­
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              monſtretur. </s>
              <s id="s.010787">Huic ſuccedunt vigeſimaquar­
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              ta, & trigeſimaprima Euclidis: prima autem
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              eiuſdem, trigeſimoſecundo loco ſic demon­
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              ſtrabitur, facto trigono æquilatero iuxta cir­
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              cini latitudinem eodem modo quo facit Eu­
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              clides, in terminis verò datæ lineæ duobus
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              angulis æqualibus illis trigoni ex vigeſima­
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              quinta harum, quare ex trigeſimaprima erit
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              tertius tertio, ac prioris trianguli ex ſecunda
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              harum anguli ſunt æquales, igitur & ſecundi
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              trigoni: igitur ex vigeſimaſexta erit trigonus
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              ſecundus ſuper
                <expan abbr="datã">datam</expan>
              lineam
                <expan abbr="cõſtitutus">conſtitutus</expan>
              æqui­
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              laterus. </s>
              <s id="s.010788">Trigeſimatertia erit duodecima pri­
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                <arrow.to.target n="marg1516"/>
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              mi Elemen. </s>
              <s id="s.010789">ex dato puncto per trigeſimam
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              harum datæ lineæ duco æquidiſtantem, inde
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              per ſextam ſuper deductam ex eodem puncto
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              duco perpendicularem, donec ex eadem par­
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              te occurrat datæ lineæ, cui cùm occurrerit
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              perpendicularis inſiſtet ex vigeſimatertia,
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              cùm prior iam ſit rectus. </s>
              <s id="s.010790">Poſt hæc quando­
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                <arrow.to.target n="marg1517"/>
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              quidem nil aliud ſupponitur præter demon­
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              ſtrata, liberum erit vſque ad vltimam primi,
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              relicta ſola vigeſimaſecunda, procedere. </s>
            </p>
            <p type="margin">
              <s id="s.010791">
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              Euclid. </s>
              <s id="s.010792">No­
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              ſtræ.</s>
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            <p type="margin">
              <s id="s.010793">
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              6 26
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              </s>
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            <p type="margin">
              <s id="s.010794">
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              Demonſtra­
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              tio ſextæ pri­
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              mi elemento­
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              rum per ar­
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              gumenti con­
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              cluſionem,
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              ſeu ducens
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              ad neceſſa­
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              rium.
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              </s>
              <s id="s.010795">7 37
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              24 28
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              31 30
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              32 31
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              Prima 32
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              </s>
            </p>
            <p type="margin">
              <s id="s.010796">
                <margin.target id="marg1516"/>
                <lb/>
              12 33
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              </s>
            </p>
            <p type="margin">
              <s id="s.010797">
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              Reſiduum
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              primi lib.
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              propter 23.
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              Totus ſecun­
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              dus lib. præ­
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              ter
                <expan abbr="vltimã">vltimam</expan>
              .
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              </s>
              <s id="s.010798">Tertij libri
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              primæ ſexde­
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              cim propoſi­
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              tiones.
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              </s>
              <s id="s.010799">31 34
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              </s>
            </p>
            <p type="main">
              <s id="s.010800">Eadémque ratione totum ſecundum
                <expan abbr="librũ">librum</expan>
              ,
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              vltima dumtaxat propoſitione excepta. </s>
              <s id="s.010801">Primas
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                <expan abbr="quoq;">quoque</expan>
              ſexdecim tertij libri, & partem
                <expan abbr="primã">primam</expan>
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              trigeſimę primæ proportionis eiuſdem, quam
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              trigeſimamquartam huius dicemus: nam du­
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              cta linea ex centro, per ſecundam harum,
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              conſtat angulum ſupremum æqualem eſſe
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              duobus, qui ſunt ſupra baſim pariter acce­
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              ptis: cùm verò tres ipſi æquales ſint duobus
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              rectis ex trigeſimaprima, neceſſe erit fateri
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              ſupremum, qui in circuli dimidio conſiſtit
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              eſſe rectum. </s>
              <s id="s.010802">Imò eodem modo quo ibi de­
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              monſtrantur, reliquæ huius propoſitionis
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              partes patêre poſſunt. </s>
              <s id="s.010803">Totus etiam quintus
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                <arrow.to.target n="marg1518"/>
                <lb/>
              liber, cùm ex aliis non pendeat, demonſtra­
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              bitur liberè ea ratione quæ ab Euclide: tum
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              verò & duodecim primi ſexti Elementorum
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              propoſitiones, cùm demonſtratis iam tantùm
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              indigeant. </s>
              <s id="s.010804">Iam verò decimamtertiam ſexti
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              demonſtrare pro trigeſimaquinta oportet:
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              iunctis igitur ligneis ad punctum ſecundum
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              rectitudinem per decimamtertiam, quæ ſint
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              AC, & CB, ducam per ſextam AF, quàm per
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              decimamtertiam faciam duplam circini la­
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              titudini, ductáque BF, ducam per trigeſimam
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                <arrow.to.target n="marg1519"/>
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              CB, æquidiſtantem BF, & faciam CG, æqua­
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              lem EF, & CK, æqualem EA, per decimam­
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              tertiam. </s>
              <s id="s.010805">Cùm igitur ſit proportio ex quarta
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              ſexti Element. </s>
              <s id="s.010806">AF, ad B, vt AE, ad AC, erit
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              ex decimanona quinti Element. </s>
              <s id="s.010807">AF, ad AC,
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              vt EF, ad CE, quare KC, ad AC, vt CG, ad
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              CB, per ſeptimum eiuſdem quinti elemen-</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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