Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[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.]
[91. THEOREMA XXI. PROPOSITIO XXVI.]
[92. THEOREMA XXII. PROPOSITIO XXVII.]
[93. PROBLEMA VI. PROPOSITIO XX VIII.]
[94. THE OREMA XXIII. PROPOSITIO XXIX.]
[95. THEOREMA XXIIII. PROPOSITIO XXX.]
[96. THEOREMA XXV. PROPOSITIO XXXI.]
[97. FINIS LIBRI DE CENTRO GRAVITATIS SOLIDORVM.]
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DE IIS QVAE VEH. IN AQVA.
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          <p>
            <s xml:space="preserve">
              <pb o="38" file="0087" n="87" rhead="DE IIS QVAE VEH. IN AQVA."/>
            æqualibus, & </s>
            <s xml:space="preserve">ſimilibus, a p o l, a m q l ab extremitati-
              <lb/>
            bus baſium ductæ ſint a o, a q ita, ut portiones ablatæ
              <lb/>
            faciant cum diametris angulos æquales; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">anguli, qui
              <lb/>
            ad y g: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">lineæ y b, g b, & </s>
            <s xml:space="preserve">b c, b s inter ſe æquales erunt.
              <lb/>
            </s>
            <s xml:space="preserve">quare & </s>
            <s xml:space="preserve">ipſæ c r, s r: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">m u, p z: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">u n, z t. </s>
            <s xml:space="preserve">Quo-
              <lb/>
              <anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/>
            niam igitur m u minor eſt, quàm dupla u n; </s>
            <s xml:space="preserve">conſtat p z ip-
              <lb/>
            ſius z t minorem eſſe, quàm duplam. </s>
            <s xml:space="preserve">Sit p α dupla ipſius
              <lb/>
            ω t: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">iuncta α k ad e producatur. </s>
            <s xml:space="preserve">ergo totius quidem por
              <lb/>
            tionis centrum grauitatis erit puntum κ; </s>
            <s xml:space="preserve">partis eius, quæ
              <lb/>
            in humido eſt, centrum ω; </s>
            <s xml:space="preserve">eius uero, quæ extra humidum
              <lb/>
            in linea k e, quod ſit e. </s>
            <s xml:space="preserve">Sed linea k z perpendicularis erit
              <lb/>
            ad ſuperficiem humidi. </s>
            <s xml:space="preserve">quare & </s>
            <s xml:space="preserve">lineæ quæ per puncta e,
              <lb/>
            ω, æ quidiſtantes ipſi κ z ducuntur. </s>
            <s xml:space="preserve">non ergo manebit por-
              <lb/>
              <anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/>
            tio, ſed reuoluetur ita, ut baſis ipſius ſuperficiem humidi
              <lb/>
            nullo modo contingat: </s>
            <s xml:space="preserve">quoniã nuncin uno puncto contin
              <lb/>
            gens, ſurſum fertur ex parte a. </s>
            <s xml:space="preserve">perſpicuum eſt igitur por-
              <lb/>
              <anchor type="note" xlink:label="note-0087-03a" xlink:href="note-0087-03"/>
            tionem conſiſt ere ita, ut axis cum ſuperſicie humidi faciat
              <lb/>
            angulum maiorem angulo χ.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="1">
            <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">C</note>
            <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a">
              <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0085-01"/>
            </figure>
            <note position="left" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">D</note>
            <note position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">E</note>
            <note position="right" xlink:label="note-0085-04" xlink:href="note-0085-04a" xml:space="preserve">F</note>
            <note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">G</note>
            <note position="left" xlink:label="note-0086-02" xlink:href="note-0086-02a" xml:space="preserve">H</note>
            <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a">
              <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0086-01"/>
            </figure>
            <note position="left" xlink:label="note-0086-03" xlink:href="note-0086-03a" xml:space="preserve">K</note>
            <note position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">L</note>
            <note position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">M</note>
            <note position="right" xlink:label="note-0087-03" xlink:href="note-0087-03a" xml:space="preserve">N</note>
          </div>
        </div>
        <div type="section" level="1" n="53">
          <head xml:space="preserve">COMMENTARIVS.</head>
          <p>
            <s xml:space="preserve">Siportio ad humidum in grauitate minorẽ proportio-
              <lb/>
              <anchor type="note" xlink:label="note-0087-04a" xlink:href="note-0087-04"/>
            nem habeat; </s>
            <s xml:space="preserve">quàm quadratum s b ad quadratum b d; </s>
            <s xml:space="preserve">ma-
              <lb/>
            iorem uero, quàm quadratum x o ad b d quadratum.</s>
            <s xml:space="preserve">] _Hæc_
              <lb/>
            _eſt ſecunda pars propoſitionis, quam aliæ deinceps, postea ipſarum_
              <lb/>
            _demonstrationes eodem ordine ſequuntur_.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="1">
            <note position="right" xlink:label="note-0087-04" xlink:href="note-0087-04a" xml:space="preserve">A</note>
          </div>
          <p>
            <s xml:space="preserve">SI portio ad humidum in grauitate maiorem quidem
              <lb/>
              <anchor type="note" xlink:label="note-0087-05a" xlink:href="note-0087-05"/>
            proportionẽ habeat, quàm quadratũ f p ad quadratũb d.</s>
            <s xml:space="preserve">]
              <lb/>
            _Hãc quartá parté nos reſtituimus, quæ ĩ trãſlatione deſiderabatur_.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="2">
            <note position="right" xlink:label="note-0087-05" xlink:href="note-0087-05a" xml:space="preserve">B</note>
          </div>
          <p>
            <s xml:space="preserve">Erit ψ maior quidem, quàm x o, minor uero, quàm ex-
              <lb/>
              <anchor type="note" xlink:label="note-0087-06a" xlink:href="note-0087-06"/>
            ceſſus, quo axis eſt maior, quam ſeſquialter eius, quæ uſque
              <lb/>
            ad axem,] _Sequitur illud ex decima quinti libri elementornm_.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="3">
            <note position="right" xlink:label="note-0087-06" xlink:href="note-0087-06a" xml:space="preserve">C</note>
          </div>
          <p>
            <s xml:space="preserve">Demonſtrabitur m h duplaipſius h n, ſicuti demonſtra
              <lb/>
              <anchor type="note" xlink:label="note-0087-07a" xlink:href="note-0087-07"/>
            tũ eſt o gipſius g x duplam eſſe.</s>
            <s xml:space="preserve">] _Vt in prima parte huius, &_</s>
            <s xml:space="preserve">
              <lb/>
            _exijs, quæ nos proxime in ipſam conſcripſimus_.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="4">
            <note position="right" xlink:label="note-0087-07" xlink:href="note-0087-07a" xml:space="preserve">D</note>
          </div>
          <p>
            <s xml:space="preserve">Quoniam enim in ſimilibus portionibus a p o l, a x d,
              <lb/>
              <anchor type="note" xlink:label="note-0087-08a" xlink:href="note-0087-08"/>
            </s>
          </p>
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