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-s1562" xml:space="preserve">
              <pb o="27" file="0065" n="65" rhead="DE IIS QVAE VEH. IN AQVA."/>
            æqualis r ψ: </s>
            <s xml:id="echoid-s1563" xml:space="preserve">& </s>
            <s xml:id="echoid-s1564" xml:space="preserve">ducatur ψ r perpendicularis ad b d, quæ
              <lb/>
            posſit dimidium eius, quod ipſis k r, ψ b, continetur. </s>
            <s xml:id="echoid-s1565" xml:space="preserve">Dico
              <lb/>
            portionem in humidum demiſſam adeo, ut baſis ipſius to-
              <lb/>
            ta ſit in humido, ita conſiſtere, ut axis cum ſuperficie humi
              <lb/>
            di faciat angulum angulo b æqualem. </s>
            <s xml:id="echoid-s1566" xml:space="preserve">Demittatur enim
              <lb/>
            portio in humidum, ſicuti dictum eſt; </s>
            <s xml:id="echoid-s1567" xml:space="preserve">& </s>
            <s xml:id="echoid-s1568" xml:space="preserve">axis cum humidi
              <lb/>
            ſuperficie non faciat angulum æqualẽ ipſi b, ſed primo ma
              <lb/>
            iorem: </s>
            <s xml:id="echoid-s1569" xml:space="preserve">ſecta autem ipſa plano per axem, recto ad ſuperfi-
              <lb/>
            ciem humidi, ſectio portionis ſit a p o l rectanguli coni ſe-
              <lb/>
            ctio; </s>
            <s xml:id="echoid-s1570" xml:space="preserve">ſuperficiei humidi ſectio c i; </s>
            <s xml:id="echoid-s1571" xml:space="preserve">ſitq, axis portionis, & </s>
            <s xml:id="echoid-s1572" xml:space="preserve">ſe
              <lb/>
            ctionis diameter n o, quæ fecetur in punctis ω t, ut prius. </s>
            <s xml:id="echoid-s1573" xml:space="preserve">& </s>
            <s xml:id="echoid-s1574" xml:space="preserve">
              <lb/>
            ducantur y p quidem ipſi ci æquidiſtans, contingensq; </s>
            <s xml:id="echoid-s1575" xml:space="preserve">ſe
              <lb/>
            ctionem in p; </s>
            <s xml:id="echoid-s1576" xml:space="preserve">m p uero æquidiſtans n o: </s>
            <s xml:id="echoid-s1577" xml:space="preserve">& </s>
            <s xml:id="echoid-s1578" xml:space="preserve">p s ad axem
              <lb/>
            perpendicularis. </s>
            <s xml:id="echoid-s1579" xml:space="preserve">Quoniam igitur axis portionis cum ſu-
              <lb/>
            perficie humidi facit angulum maiorem angulo b; </s>
            <s xml:id="echoid-s1580" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s1581" xml:space="preserve">
              <lb/>
            angulus s y p angulo b maior. </s>
            <s xml:id="echoid-s1582" xml:space="preserve">quare quadratum p s ad
              <lb/>
            quadratum s y maiorem habet proportionem, quàm qua
              <lb/>
            dratum ψ e ad quadratum ψ b: </s>
            <s xml:id="echoid-s1583" xml:space="preserve">& </s>
            <s xml:id="echoid-s1584" xml:space="preserve">propterea _K_ r ad s y ma
              <lb/>
              <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">B</note>
            iorem habet, quàm dimidium ipſius κ r ad ψ b. </s>
            <s xml:id="echoid-s1585" xml:space="preserve">ergo s y
              <lb/>
            minor eſt, quam dupla ψ b; </s>
            <s xml:id="echoid-s1586" xml:space="preserve">& </s>
            <s xml:id="echoid-s1587" xml:space="preserve">s o minor, quam ψ b. </s>
            <s xml:id="echoid-s1588" xml:space="preserve">quare
              <lb/>
              <note position="right" xlink:label="note-0065-02" xlink:href="note-0065-02a" xml:space="preserve">C</note>
            s ω maior, quàm r ψ; </s>
            <s xml:id="echoid-s1589" xml:space="preserve">& </s>
            <s xml:id="echoid-s1590" xml:space="preserve">p h maior, quàm f. </s>
            <s xml:id="echoid-s1591" xml:space="preserve">Itaque quoniã
              <lb/>
            portio ad humidum in grauitate eam habet proportionẽ,
              <lb/>
              <note position="right" xlink:label="note-0065-03" xlink:href="note-0065-03a" xml:space="preserve">D</note>
            quam exceſſus, quo quadratum b d excedit quadratum f q
              <lb/>
            ad quadratum b d: </s>
            <s xml:id="echoid-s1592" xml:space="preserve">quam uero proportionem habet por-
              <lb/>
            tio ad humidum in grauitate, eandem pars ipſius demerſa
              <lb/>
            habet ad totam portionẽ: </s>
            <s xml:id="echoid-s1593" xml:space="preserve">ſequitur partẽ demerſam ad to
              <lb/>
            tam portionem, eam proportionem habere, quã exceſſus,
              <lb/>
            quo quadratum b d excedit quadratũ f q, ad quadratū b d.
              <lb/>
            </s>
            <s xml:id="echoid-s1594" xml:space="preserve">habebit ergo tota portio ad eam, quæ eſt extra humidum
              <lb/>
              <note position="right" xlink:label="note-0065-04" xlink:href="note-0065-04a" xml:space="preserve">E</note>
            proportionem eandem, quam quadratum b d ad quadra-
              <lb/>
            tum f q. </s>
            <s xml:id="echoid-s1595" xml:space="preserve">Sed quam proportionem habet tota portio ad eã,
              <lb/>
            quæ eſt extra humidum, eandem habet quadratum n o ad
              <lb/>
            quadratum p m. </s>
            <s xml:id="echoid-s1596" xml:space="preserve">ergo p m ipſi f q æ qualis etit. </s>
            <s xml:id="echoid-s1597" xml:space="preserve">demonſtra
              <lb/>
            ta eſt autem p h maior, quàm f: </s>
            <s xml:id="echoid-s1598" xml:space="preserve">quare m h minor </s>
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