Archimedes, Archimedis De insidentibvs aqvae

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            <s xml:id="echoid-s641" xml:space="preserve">
              <pb o="12" file="0039" n="39" rhead="_LIBER II._"/>
            faciat angulum, æqualem angulo x. </s>
            <s xml:id="echoid-s642" xml:space="preserve">Si autem portio ad humidum ingra- uitate habeat proportionem minorem proportione, quam habet tetra- gonum, quod ab f, p, ad tetragonum, quod a, b, d, dimiſſa in humidum, &</s>
            <s xml:id="echoid-s643" xml:space="preserve"> poſita inclinata ita, ut baſis ipſius non tangat humidum conſiſtet inclina ta ita, ut axis quidem ipſius ad ſuperficiem humidi, faciat angulum mi- norem angulo x, baſis autem ipſius, nec ſecundum vnum tangat ſuperfi- ciem humidi. </s>
            <s xml:id="echoid-s644" xml:space="preserve">Demonſtrabitur itaque hæc deinceps.</s>
            <s xml:id="echoid-s645" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s646" xml:space="preserve">Habeat itaque primo portio ad humidum in grauitate proportionem
              <lb/>
            quidem maiorem ea, quam habet tetragonum, quod ab x, o, ad id, quod a,
              <lb/>
            b, d, minore autem ea, quàm habet tetragonũ, quod ab exceſſu, quo axis
              <lb/>
            eſt maior, quàm hemiolius eius, quæ uſque ad axẽ ad tetragonum, quod
              <lb/>
            a, b, d, & </s>
            <s xml:id="echoid-s647" xml:space="preserve">ſupponatur prius diſpoſita figura. </s>
            <s xml:id="echoid-s648" xml:space="preserve">Quam autem proportionem
              <lb/>
            habet portio ad humidum in grauitate, hanc tetragonum, quod a, x, ad
              <lb/>
            id, qnod a, b, d, eſt autẽ quæ x, maior qui
              <lb/>
              <figure xlink:label="fig-0039-01" xlink:href="fig-0039-01a" number="31">
                <image file="0039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0039-01"/>
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            dem quàm x, p, minor autem exceſſu,
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            quo axis eſt maior, quàm hemiolius eius
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            quæ uſque ad axem. </s>
            <s xml:id="echoid-s649" xml:space="preserve">Inaptetur autem
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            quædam inter media conicarum ſectio-
              <lb/>
            num apol. </s>
            <s xml:id="echoid-s650" xml:space="preserve">a, z, d, quæ u, o, æqualis ipſi x,
              <lb/>
            & </s>
            <s xml:id="echoid-s651" xml:space="preserve">ſecet ipſa reliquam coni ſectionem pe
              <lb/>
            nes ipſa autem r, s, rectam
              <lb/>
            penes b, demonſtr abitur autẽ quæ
              <lb/>
            o, u, ipſius a, n, ſicut demonstratum est,
              <lb/>
            quæ p, s, ipſius s, x, dupla ab o, autem du
              <lb/>
            catur, quæ o, s, cõtingens ſectionem apol
              <lb/>
            quæ autem o, c, perpendicularis ſuper
              <lb/>
            b, d, & </s>
            <s xml:id="echoid-s652" xml:space="preserve">ab a, ad n, copuletur, erunt autẽ
              <lb/>
            quę a, n, q, u, æquales inuicem. </s>
            <s xml:id="echoid-s653" xml:space="preserve">Quoniam
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            enim in ſimilibus portionibus apol. </s>
            <s xml:id="echoid-s654" xml:space="preserve">a, x,
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            d, producto ſunt ab axibus ad portiones, quæ a, n, a, q, æquales angulos fa
              <lb/>
            cientes ad baſes eandem proportionem habebunt quæ q, a. </s>
            <s xml:id="echoid-s655" xml:space="preserve">a, n, cum ipſis
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            l, a. </s>
            <s xml:id="echoid-s656" xml:space="preserve">a, d, propter ſecundam figuram præſcriptarum æqualis, ergo quæ a,
              <lb/>
            n ipſi q, n, & </s>
            <s xml:id="echoid-s657" xml:space="preserve">æquediſtans ipſi o, s, demonstrandum, quòd demiſſa in hu-
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            midum ita, ut baſis ipſius, non ſecundum vnum tangit axis ad
              <lb/>
            ſuperficiem humidi angulum acutum faciat maiorem exceſſu Di
              <lb/>
            mittatur enim, & </s>
            <s xml:id="echoid-s658" xml:space="preserve">conſiſtat ita, ut baſis ipſius tangat, ſecundum vnum ſi
              <lb/>
            gnum ſuperficiem humidi. </s>
            <s xml:id="echoid-s659" xml:space="preserve">Secta autem portione per axem plano recta
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            ad ſuperficiem humidi, ſuperficiei quidem portionis ſectio ſitq́ apol. </s>
            <s xml:id="echoid-s660" xml:space="preserve">re-
              <lb/>
            ctanguli coni ſectio, ſuperficiei autem humidi, quæ o, a, axis autem ſectio
              <lb/>
            nis, & </s>
            <s xml:id="echoid-s661" xml:space="preserve">dyameter, quæ b, d, & </s>
            <s xml:id="echoid-s662" xml:space="preserve">ſecetq́ b, d, penes K, r, ut dictum eſt </s>
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