Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[61. ALEXANDRO FARNESIO CARDINALI AMPLISSIMO ET OPTIMO.]
[62. FEDERICI COMMANDINI VRBINATIS LIBER DE CENTRO GRAVITATIS SOLIDORVM. DIFFINITIONES.]
[63. PETITIONES.]
[64. THEOREMA I. PROPOSITIO I.]
[65. THEOREMA II. PROPOSITIO II.]
[66. THE OREMA III. PROPOSITIO III.]
[67. THE OREMA IIII. PROPOSITIO IIII.]
[68. ALITER.]
[69. THEOREMA V. PROPOSITIO V.]
[70. COROLLARIVM.]
[71. THEOREMA VI. PROPOSITIO VI.]
[72. THE OREMA VII. PROPOSITIO VII.]
[73. THE OREMA VIII. PROPOSITIO VIII.]
[74. THE OREMA IX. PROPOSITIO IX.]
[75. PROBLEMA I. PROPOSITIO X.]
[76. PROBLEMA II. PROPOSITIO XI.]
[77. PROBLEMA III. PROPOSITIO XII.]
[78. PROBLEMA IIII. PROPOSITIO XIII.]
[79. THEOREMA X. PROPOSITIO XIIII.]
[80. THE OREMA XI. PROPOSITIO XV.]
[81. THE OREMA XII. PROPOSITIO XVI.]
[82. THE OREMA XIII. PROPOSITIO XVII.]
[83. THEOREMA XIIII. PROPOSITIO XVIII.]
[84. THEOREMA XV. PROPOSITIO XIX.]
[85. THE OREMA XVI. PROPOSITIO XX.]
[86. THEOREMA XVII. PROPOSITIO XXI.]
[87. THE OREMA XVIII. PROPOSITIO XXII.]
[88. THEOREMA XIX. PROPOSITIO XXIII.]
[89. PROBLEMA V. PROPOSITIO XXIIII.]
[90. THEOREMA XX. PROPOSITIO XXV.]
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DE IIS QVAE VEH. IN AQVA.
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          <p>
            <s xml:space="preserve">
              <pb o="22" file="0067" n="67" rhead="DE IIS QVAE VEH. IN AQVA."/>
            dum eam, quæ per g ſurſum eleuabitur. </s>
            <s xml:space="preserve">non igitur manebit
              <lb/>
            portio ſic inclinata, nec conuertetur ita, ut axis ad ſuperfi-
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            ciem humidi ſit perpendicularis: </s>
            <s xml:space="preserve">quoniam quæ ex parte 1
              <lb/>
              <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/>
            deorſum; </s>
            <s xml:space="preserve">quæ uero ex parte a ſurſum ferentur, ut ex iam de
              <lb/>
            monſtratis apparere poteſt. </s>
            <s xml:space="preserve">Quòd ſi axis cum ſuperficie
              <lb/>
            humidi fecerit angulum minorem angulo b, ſimiliter de-
              <lb/>
              <anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/>
            monſtrabitur, nõ manere portionem, ſed inclinari, donec
              <lb/>
            utique axis cum ſuperficie humidi faciat angulum angulo
              <lb/>
            b æqualem.</s>
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          </p>
          <div type="float" level="2" n="1">
            <figure xlink:label="fig-0064-01" xlink:href="fig-0064-01a">
              <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0064-01"/>
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            <note position="left" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve">A</note>
            <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">B</note>
            <note position="right" xlink:label="note-0065-02" xlink:href="note-0065-02a" xml:space="preserve">C</note>
            <note position="right" xlink:label="note-0065-03" xlink:href="note-0065-03a" xml:space="preserve">D</note>
            <note position="right" xlink:label="note-0065-04" xlink:href="note-0065-04a" xml:space="preserve">E</note>
            <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a">
              <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0066-01"/>
            </figure>
            <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">F</note>
            <note position="right" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">G</note>
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        </div>
        <div type="section" level="1" n="39">
          <head xml:space="preserve">COMMENTARIVS.</head>
          <p>
            <s xml:space="preserve">QVARE quadratum b d magis excedit quadratum
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              <anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/>
            f q, quàm b c quadratum: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">idcirco linea f q minor eſt,
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            quàm b c: </s>
            <s xml:space="preserve">itemq; </s>
            <s xml:space="preserve">f minor quam b r.</s>
            <s xml:space="preserve">] _Quoniani exceſſus, quo_
              <lb/>
            _quadratum b d excedit quadratum b c ad quadratum b d minorem_
              <lb/>
            _proportionem habet, quàm exceſſus, quo quadratum b d excedit qua_
              <lb/>
            _dratum f q, ad idem quadratum: </s>
            <s xml:space="preserve">erit ex octaua quinti exceſſus, quo_
              <lb/>
            _quadratum b d excedit quadratum b c, minor quàm exceſſus, quo ex_
              <lb/>
            _cedit quadratum f q. </s>
            <s xml:space="preserve">ergo quadratum f q minus est quadrato b c: </s>
            <s xml:space="preserve">&_</s>
            <s xml:space="preserve">
              <lb/>
            _propterea linea f q minor linea b c. </s>
            <s xml:space="preserve">Sed f q ad f eandem proportionẽ_
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            _habet, quam b c ad b r; </s>
            <s xml:space="preserve">utraque enim utriuſque ſeſquialtera est. </s>
            <s xml:space="preserve">cum_
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              <anchor type="note" xlink:label="note-0067-04a" xlink:href="note-0067-04"/>
            _igitur f q ſit minor b c, & </s>
            <s xml:space="preserve">f ipſa b r minor erit_.</s>
            <s xml:space="preserve"/>
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            <note position="right" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">A</note>
            <note position="right" xlink:label="note-0067-04" xlink:href="note-0067-04a" xml:space="preserve">14. quinti</note>
          </div>
          <p>
            <s xml:space="preserve">Et propterea k r ad ſ y maiorem habet, quàm dimidium
              <lb/>
              <anchor type="note" xlink:label="note-0067-05a" xlink:href="note-0067-05"/>
            ipſius k r ad ψ b.</s>
            <s xml:space="preserve">] _Est enim k r ad ſ y, ut quadratum p s ad qua_
              <lb/>
            _dratum ſ y
              <unsure/>
            : </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">dimidium lineæ K r ad lineam ψ b, ut quadratum e ψ_
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            _ad quadratum ψ b_.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="2">
            <note position="right" xlink:label="note-0067-05" xlink:href="note-0067-05a" xml:space="preserve">B</note>
          </div>
          <p>
            <s xml:space="preserve">Et s o minor quàm ψ b] _Est enim ſ y dupla ipſius ſ o._</s>
            <s xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">C</note>
          <p>
            <s xml:space="preserve">Et p h maior, quàm f.</s>
            <s xml:space="preserve">] _Nam p h eſt æqualis ſ ω, & </s>
            <s xml:space="preserve">r ψ_
              <lb/>
              <anchor type="note" xlink:label="note-0067-07a" xlink:href="note-0067-07"/>
            _ipſi f_.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="3">
            <note position="right" xlink:label="note-0067-07" xlink:href="note-0067-07a" xml:space="preserve">D</note>
          </div>
          <p>
            <s xml:space="preserve">Habebit ergo tota portio ad eam, quæ eſt extra humi-
              <lb/>
              <anchor type="note" xlink:label="note-0067-08a" xlink:href="note-0067-08"/>
            dum proportionem eandem, quam quadratum b d ad qua
              <lb/>
            dratum f q.</s>
            <s xml:space="preserve">] _Cum pars demerſa ad totam portionem ita ſit, ut_
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            _exceſſus, quo quadratum b d excedit quadratum f q ad b d quadratu
              <unsure/>
            :_</s>
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          </p>
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