Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/020.jpg" pagenum="349"/>
              ly. </s>
              <s>The like ſhall alſo hold true in the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion of the Sphære
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              leſs than an Hemiſphere that lieth with its whole Baſe above the
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              Liquid.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1145"/>
              * Or according
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              to the Perpendi­
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              cular.</s>
            </p>
            <p type="head">
              <s>COMMANDINE.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Demonſtration of this Propoſition is defaced by the Injury of Time, which we have re­
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              ſtored, ſo far as by the Figures that remain, one may collect the Meaning of
                <emph.end type="italics"/>
              Archimedes,
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                <emph type="italics"/>
              for we thought it not good to alter them: and what was wanting to their declaration and ex­
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              planation we have ſupplyed in our Commentaries, as we have alſo determined to do in the ſe­
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              cond Propoſition of the ſecond Book.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>If any Solid Magnitude lighter than the Liquid.]
                <emph type="italics"/>
              Theſe words, light-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1146"/>
                <lb/>
                <emph type="italics"/>
              er than the Liquid, are added by us, and are not to be found in the Tranſiation; for of theſe
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              kind of Magnitudes doth
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              Archimedes
                <emph type="italics"/>
              ſpeak in this Propoſition.
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              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1146"/>
              A</s>
            </p>
            <p type="main">
              <s>Shall be demitted into the Liquid in ſuch a manner as that the
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                <arrow.to.target n="marg1147"/>
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              Baſe of the Portion touch not the Liquid.]
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              That is, ſhall be ſo demitted into
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              the Liquid as that the Baſe ſhall be upwards, and the
                <emph.end type="italics"/>
              Vertex
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              downwards, which he oppoſeth
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              to that which he ſaith in the Propoſition following
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              ; Be demitted into the Liquid, ſo, as
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              that its Baſe be wholly within the Liquid;
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              For theſe words ſignifie the Portion demit­
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              ted the contrary way, as namely, with the
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              Vertex
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              upwards and the Baſe downwards. </s>
              <s>The
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              ſame manner of ſpeech is frequently uſed in the ſecond Book; which treateth of the Portions
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              of Rectangle Conoids.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1147"/>
              B</s>
            </p>
            <p type="main">
              <s>Now becauſe every Portion of a Sphære hath its Axis in the Line
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                <arrow.to.target n="marg1148"/>
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              that from the Center of the Sphære is drawn perpendicular to its
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              Baſe.]
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              For draw a Line from B to C, and let K L cut the Circumference A B C D in the
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              Point G, and the Right Line B C in M
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              :
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                <figure id="id.073.01.020.1.jpg" xlink:href="073/01/020/1.jpg" number="13"/>
                <lb/>
                <emph type="italics"/>
              and becauſe the two Circles A B C D, and
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              E F H do cut one another in the Points
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              B and C, the Right Line that conjoyneth
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              their Centers, namely, K L, doth cut the
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              Line B C in two equall parts, and at
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              Right Angles; as in our Commentaries
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              upon
                <emph.end type="italics"/>
              Prolomeys
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              Planiſphære we do
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              prove: But of the Portion of the Circle
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              B N C the Diameter is M N; and of the
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              Portion B G C the Diameter is M G;
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1149"/>
                <lb/>
                <emph type="italics"/>
              for the
                <emph.end type="italics"/>
              (a)
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              Right Lines which are drawn
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              on both ſides parallel to B C do make
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1150"/>
                <lb/>
                <emph type="italics"/>
              Right Angles with N G; and
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              (b)
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              for
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              that cauſe are thereby cut in two equall
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              parts: Therefore the Axis of the Portion
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              of the Sphære B N C is N M; and the
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              Axis of the Portion B G C is M G:
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              from whence it followeth that the Axis of
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              the Portion demerged in the Liquid is
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              in the Line K L, namely N G. </s>
              <s>And ſince the Center of Gravity of any Portion of a Sphære is
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              in the Axis, as we have demonstrated in our Book
                <emph.end type="italics"/>
              De Centro Gravitatis Solidorum,
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              the
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              Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is,
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              of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra­
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              vity of thoſe Portions of Sphæres. </s>
              <s>For ſuppoſe, if poſſible, that it be out of the Line N G, as
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              in Q, and let the Center of the Gravity of the Portion B N C, be V, and draw V
                <expan abbr="q.">que</expan>
              Becauſe
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              therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha­
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              ving the ſame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the
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              Portion B G C ſhall, by the 8 of the firſt Book of
                <emph.end type="italics"/>
              Archimedes, De Centro Gravitatis </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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