Archimedes, Archimedis De insidentibvs aqvae

Table of handwritten notes

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            autem axis ad humidum faciat angulum minorem angulo b, conſimili-
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            ter prioribus demonstrabitur, quod non manebit portio ſed inclinabitur
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            donec utique axis ad ſuperficiem humidi faciat angulum æqualem, an-
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            gulo b.</s>
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        <div xml:id="echoid-div40" type="section" level="1" n="28">
          <head xml:id="echoid-head38" xml:space="preserve">DECIMVS.</head>
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            <s xml:id="echoid-s578" xml:space="preserve">Recta portio rectanguli conoydalis, quando leuior exi-
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            ſtens humido habuerit axem maiorem, quàm ut habeat pro-
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            portionem ad eam, quàm uſque ad axem, quam habent quin
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            decim ad quattuor demiſſa in humidum ita, ut baſis ipſius nõ
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            tangat humidum, quandoque quidem recta conſiſtet, quin-
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            que autem inclinata: </s>
            <s xml:id="echoid-s579" xml:space="preserve">& </s>
            <s xml:id="echoid-s580" xml:space="preserve">quandoque quidem ita inclinata, ut
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            baſis ipſius, ſecundum vnum ſignum tangat ſuperficiem hu-
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            midi: </s>
            <s xml:id="echoid-s581" xml:space="preserve">& </s>
            <s xml:id="echoid-s582" xml:space="preserve">hoc in duabus diſpoſitionibus faciet: </s>
            <s xml:id="echoid-s583" xml:space="preserve">& </s>
            <s xml:id="echoid-s584" xml:space="preserve">quandoq;
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            </s>
            <s xml:id="echoid-s585" xml:space="preserve">ita inclinata conſiſtet, ut baſis ipſius ſecundum ampliorem
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            locum humefiat: </s>
            <s xml:id="echoid-s586" xml:space="preserve">quandoque autem ita, ut baſis@ipſius, nec ſe
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            cundum vnum tangat ſuperficiem humidi. </s>
            <s xml:id="echoid-s587" xml:space="preserve">Quam autẽ pro-
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            portionem habeant ad humidum in grauitate fingula ho-
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            rum demonſtrabuntur.</s>
            <s xml:id="echoid-s588" xml:space="preserve"/>
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            <s xml:id="echoid-s589" xml:space="preserve">SIt portio qualis dicta est, & </s>
            <s xml:id="echoid-s590" xml:space="preserve">ſecta ipſa plano recto ad ſuperficiem hu-
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            midi ſectio in ſuperficie ſit quæ apol. </s>
            <s xml:id="echoid-s591" xml:space="preserve">rectanguli coni ſectio axis autẽ
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            & </s>
            <s xml:id="echoid-s592" xml:space="preserve">dyameter ſectionis ſit quæ b, d. </s>
            <s xml:id="echoid-s593" xml:space="preserve">Secetur autem quæ b, d, ſecundũ
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            K, ita ut dupla ſit quæ b, d, ipſi K, d, ſecundum c, autem, ut quæ b, d, ad K,
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            c, habeat proportionem, quam habent quindecim ad quattuor. </s>
            <s xml:id="echoid-s594" xml:space="preserve">Palam
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            igitur quòd quæ K, c, eſt maior ea, quæ uſque ad axem ipſius autem K, r,
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            ſit hemiolia, quæ eſt autem, & </s>
            <s xml:id="echoid-s595" xml:space="preserve">quæ s, b, hemiolia ipſius b. </s>
            <s xml:id="echoid-s596" xml:space="preserve">r. </s>
            <s xml:id="echoid-s597" xml:space="preserve">Co-
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            puletur autem ipſa a, b, & </s>
            <s xml:id="echoid-s598" xml:space="preserve">ipſa c, e, recta producta ducaturq́ue e, Z, æque
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            diſtanter, ipſi b, d, & </s>
            <s xml:id="echoid-s599" xml:space="preserve">rurſum ipſa a,
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            b, ſecta in duo æqualia penes t, duca-
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            tur æquediſtanter ipſi b, d, quæ t, h, & </s>
            <s xml:id="echoid-s600" xml:space="preserve">accipiatur rectanguli coni ſectio,
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            quæ a, e, circa dyametrum e, Z, & </s>
            <s xml:id="echoid-s601" xml:space="preserve">quæ a, t, circa dyametrum t h, ita, ut ſi
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            milis ſit, quæ a, e, i, a, t, h, portioni a, b, l, deſcribetur autem quæ a, e, i, co-
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            ni ſectio per K. </s>
            <s xml:id="echoid-s602" xml:space="preserve">Quæ autem a, b, r, recta producta ipſi b, d, ſecat ipſam a,
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            e, i, ſecet, ſecũdum y g, cum per y, g, ducantur æquediſtanter ipſi b, d, quæ
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            p, y, q. </s>
            <s xml:id="echoid-s603" xml:space="preserve">Secet antem ipſe ſectionem a, o, d, penes x, f, ducantur autem, & </s>
            <s xml:id="echoid-s604" xml:space="preserve">
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            quæ p, x, o. </s>
            <s xml:id="echoid-s605" xml:space="preserve">contingentes ſectionem apol. </s>
            <s xml:id="echoid-s606" xml:space="preserve">ſecundum o, p. </s>
            <s xml:id="echoid-s607" xml:space="preserve">Sunt
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            tres quædam portiones quæ apol. </s>
            <s xml:id="echoid-s608" xml:space="preserve">a, e, i, a, t, d, contentę arectis, & </s>
            <s xml:id="echoid-s609" xml:space="preserve">a ſe-
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            ctionibus rectangulorum conorum rectæ, & </s>
            <s xml:id="echoid-s610" xml:space="preserve">ſimiles, & </s>
            <s xml:id="echoid-s611" xml:space="preserve">inæquales, & </s>
            <s xml:id="echoid-s612" xml:space="preserve">
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            @angentes ſuper pnamquanque baſem a, b, n, autem ſurſum
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            ducta eſt, quæ n, x, p, n, o. </s>
            <s xml:id="echoid-s613" xml:space="preserve">o, g, ergo ad g, x, habet </s>
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