Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[21.] ARCHIMEDIS DE IIS QVAE VEHVNTVR IN AQVA LIBER SECVNDVS. CVM COMMENTARIIS FEDERICI COMMANDINI VRBINATIS. PROPOSITIO I.
[22.] PROPOSITIO II.
[23.] COMMENTARIVS.
[24.] PROPOSITIO III.
[25.] PROPOSITIO IIII.
[26.] COMMENTARIVS.
[27.] PROPOSITIO V.
[28.] COMMENTARIVS.
[29.] PROPOSITIO VI.
[30.] COMMENTARIVS.
[31.] LEMMAI.
[32.] LEMMA II.
[33.] LEMMA III.
[34.] LEMMA IIII.
[35.] PROPOSITIO VII.
[36.] PROPOSITIO VIII.
[37.] COMMENTARIVS.
[38.] PROPOSITIO IX.
[39.] COMMENTARIVS.
[40.] PROPOSITIO X.
[41.] COMMENTARIVS.
[42.] LEMMA I.
[43.] LEMMA II.
[44.] LEMMA III.
[45.] LEMMA IIII.
[46.] LEMMA V.
[47.] LEMMA VI.
[48.] II.
[49.] III.
[50.] IIII.
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            <s xml:id="echoid-s2582" xml:space="preserve">
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            ſuperficiem recto, ſit portionis ſectio anzg; </s>
            <s xml:id="echoid-s2583" xml:space="preserve">ſuperficiei
              <lb/>
            humidi ez: </s>
            <s xml:id="echoid-s2584" xml:space="preserve">a-
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              <figure xlink:label="fig-0098-01" xlink:href="fig-0098-01a" number="64">
                <image file="0098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0098-01"/>
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            xis portionis,
              <lb/>
            & </s>
            <s xml:id="echoid-s2585" xml:space="preserve">ſectionis dia-
              <lb/>
            meter b d: </s>
            <s xml:id="echoid-s2586" xml:space="preserve">ſece-
              <lb/>
            turq, b d in pũ-
              <lb/>
            ctis _K_r, ſicuti
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            prius; </s>
            <s xml:id="echoid-s2587" xml:space="preserve">& </s>
            <s xml:id="echoid-s2588" xml:space="preserve">duca-
              <lb/>
            tur n l quidem
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            ipſi e z æquidi-
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            ſtans, quæ con-
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            tingat ſectionẽ
              <lb/>
            a n z g in n; </s>
            <s xml:id="echoid-s2589" xml:space="preserve">& </s>
            <s xml:id="echoid-s2590" xml:space="preserve">
              <lb/>
            n t æquidiſtans
              <lb/>
            ipſi b d; </s>
            <s xml:id="echoid-s2591" xml:space="preserve">n s ue-
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            ro ad b d perpẽ
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            dicularis. </s>
            <s xml:id="echoid-s2592" xml:space="preserve">Itaq;
              <lb/>
            </s>
            <s xml:id="echoid-s2593" xml:space="preserve">quoniam portio ad humidum in grauitate eam proportio
              <lb/>
            nem habet, quam quadratum, quod fit à linea ψ ad quadra
              <lb/>
            tum b d: </s>
            <s xml:id="echoid-s2594" xml:space="preserve">erit ψ ipſi n t æqualis: </s>
            <s xml:id="echoid-s2595" xml:space="preserve">quod ſimiliter demonſtrabi
              <lb/>
            tur, ut ſuperius. </s>
            <s xml:id="echoid-s2596" xml:space="preserve">quare & </s>
            <s xml:id="echoid-s2597" xml:space="preserve">n t eſt æqualis ipſi u i. </s>
            <s xml:id="echoid-s2598" xml:space="preserve">portiones
              <lb/>
            igitur a u q, e n z inter ſe ſunt æquales. </s>
            <s xml:id="echoid-s2599" xml:space="preserve">Et cum in æquali-
              <lb/>
            bus, & </s>
            <s xml:id="echoid-s2600" xml:space="preserve">ſimilibus portionibus a u q l, a n z g ductæ ſint a q
              <lb/>
            e z, quæ æquales portiones auferunt; </s>
            <s xml:id="echoid-s2601" xml:space="preserve">illa quidem ab extre
              <lb/>
            mitate baſis; </s>
            <s xml:id="echoid-s2602" xml:space="preserve">hæc autem non ab extremitate: </s>
            <s xml:id="echoid-s2603" xml:space="preserve">minorem fa-
              <lb/>
            ciet acutum angulum cum portionis diametro, quæ ab ex-
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            tremitate baſis ducitur. </s>
            <s xml:id="echoid-s2604" xml:space="preserve">At triangulorum n l s, u ω c angu
              <lb/>
            lus ad l angulo ad ω maior eſt. </s>
            <s xml:id="echoid-s2605" xml:space="preserve">ergo b s minor erit, quam
              <lb/>
            b c: </s>
            <s xml:id="echoid-s2606" xml:space="preserve">& </s>
            <s xml:id="echoid-s2607" xml:space="preserve">ſ r maior, quàm c r: </s>
            <s xml:id="echoid-s2608" xml:space="preserve">ideoq; </s>
            <s xml:id="echoid-s2609" xml:space="preserve">n χ maior, quam u h; </s>
            <s xml:id="echoid-s2610" xml:space="preserve">& </s>
            <s xml:id="echoid-s2611" xml:space="preserve">
              <lb/>
            χ t minor, quàm h i. </s>
            <s xml:id="echoid-s2612" xml:space="preserve">Quoniam igitur u y dupla eſt ipſius
              <lb/>
            y i; </s>
            <s xml:id="echoid-s2613" xml:space="preserve">conſtat n χ maiorem eſſe, quàm duplã χ t. </s>
            <s xml:id="echoid-s2614" xml:space="preserve">Sit n m dupla
              <lb/>
            ipſius m t. </s>
            <s xml:id="echoid-s2615" xml:space="preserve">perſpicuũ eſt ex iis, quæ dicta ſunt, non manere
              <lb/>
            portionẽ; </s>
            <s xml:id="echoid-s2616" xml:space="preserve">ſed in clinari, donec eius baſis contingat ſuperfi-
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            ciem humidi: </s>
            <s xml:id="echoid-s2617" xml:space="preserve">contingat autem in puncto uno, ut patet in </s>
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