Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/021.jpg" pagenum="350"/>
              Planotum,
                <emph type="italics"/>
              be in the Line V Q prolonged: But that is impoſſible; for it is in the Axis
                <lb/>
              G: It followeth, therefore, that the Center of Gravity of the Portion demerged in
                <lb/>
              Liquid be in the Line N K: which we propounded to be proved.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1148"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1149"/>
                <emph type="italics"/>
              (a)
                <emph.end type="italics"/>
              By 29. of the
                <lb/>
              firſt of
                <emph type="italics"/>
              Encl.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1150"/>
                <emph type="italics"/>
              (b)
                <emph.end type="italics"/>
              By 3. of the
                <lb/>
              third.</s>
            </p>
            <p type="main">
              <s>But the Centre of Gravity of the whole Portion is in the Line
                <lb/>
                <arrow.to.target n="marg1151"/>
                <lb/>
              T, betwixt the Point R and the Point F; let us ſuppoſe it to be
                <lb/>
              the Point X.]
                <emph type="italics"/>
              Let the Sphære becompleated, ſo as that there be added of that Portion
                <lb/>
              the Axis T Y, and the Center of Gravity Z. </s>
              <s>And becauſe that from the whole Sphære,
                <lb/>
              whoſe Centre of Gravity is K, as we have alſo demonſtrated in the (c) Book before named, the
                <lb/>
              is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1152"/>
                <lb/>
                <emph type="italics"/>
              of the Portion E F H ſhall be in the Line Z K prolonged: And therefore it muſt of neceſſity
                <lb/>
              fall betwixt K and F.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1153"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1151"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1152"/>
              (c)
                <emph type="italics"/>
              By 8 of the
                <lb/>
              firſt
                <emph.end type="italics"/>
              of Archimedes.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1153"/>
              E</s>
            </p>
            <p type="main">
              <s>The remainder, therefore, of the Figure, elevated above the Sur­
                <lb/>
              face of the Liquid, hath its Center of Gravity in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine R X
                <lb/>
              prolonged.]
                <emph type="italics"/>
              By the ſame 8 of the firſt Book of
                <emph.end type="italics"/>
              Archimedes, de Centro Gravita­
                <lb/>
              tis Planorum.</s>
            </p>
            <p type="main">
              <s>Now the Gravity of the Figure that is above the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid ſhall
                <arrow.to.target n="marg1154"/>
                <lb/>
              preſs from above downwards according to S L; and the Gravit
                <lb/>
              of the Portion that is ſubmerged in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid ſhall preſs from be
                <lb/>
              low upwards, according to the Perpendicular R L.]
                <emph type="italics"/>
              By the ſecond Sup­
                <lb/>
              poſition of this. </s>
              <s>For the Magnitude that is demerged in the Liquid is moved upwards with as
                <lb/>
              much Force along R L, as that which is above the Liquid is moved downwards along S L; as
                <lb/>
              may be ſhewn by Propoſition 6. of this. </s>
              <s>And becauſe they are moved along ſeverall other Lines,
                <lb/>
              neither cauſeth the others being leſs moved; the which it continually doth when the Portion
                <lb/>
              is ſet according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes
                <lb/>
              do concur in one and the ſame Perpendicular, namely, in the Axis of the Portion: and look
                <lb/>
              with what force or
                <emph.end type="italics"/>
              Impetus
                <emph type="italics"/>
              that which is in the Lipuid tendeth upwards, and with the like
                <lb/>
              doth that which is above or without the Liquid tend downwards along the ſame Line: And
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1155"/>
                <lb/>
                <emph type="italics"/>
              therefore, in regard that the one doth not ^{*} exceed the other, the Portion ſhall no longer move
                <lb/>
              but ſhall ſtay and reſt allwayes in one and the ſame Poſition, unleſs ſome extrinſick Cauſe
                <lb/>
              chance to intervene.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1154"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1155"/>
              *
                <emph type="italics"/>
              Or overcome.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROP. IX. THEOR. IX.
                <lb/>
                <arrow.to.target n="marg1156"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1156"/>
              * In ſome Greek
                <lb/>
              Coppies this is no
                <lb/>
              diſtinct Propoſi­
                <lb/>
              tion, but all
                <lb/>
              Commentators,
                <lb/>
              do divide it
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              from the Prece­
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              dent, as having a
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              diſtinct demon­
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              ſtration in the
                <lb/>
              Originall.</s>
            </p>
            <p type="main">
              <s>^{*}
                <emph type="italics"/>
              But if the Figure, lighter than the Liquid, be demit­
                <lb/>
              ted into the Liquid, ſo, as that its Baſe be wholly
                <lb/>
              within the ſaid Liquid, it ſhall continue in ſuch
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              manner erect, as that its Axis ſhall ſtand according
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              to the Perpendicular.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>For ſuppoſe, ſuch a Magnitude as that aforenamed to be de
                <lb/>
              mitted into the Liquid; and imagine a Plane to be produced
                <lb/>
              thorow the Axis of the Portion, and thorow the Center of the
                <lb/>
              Earth: And let the
                <emph type="italics"/>
              S
                <emph.end type="italics"/>
              ection of the Surface of the Liquid, be the Cir­
                <lb/>
              cumference A B C D, and of the Figure the Circumference E F
                <emph type="italics"/>
              H
                <emph.end type="italics"/>
                <lb/>
              And let E H be a Right Line, and F T the Axis of the Portion. </s>
              <s>If
                <lb/>
              now it were poſſible, for ſatisfaction of the Adverſary, let it be
                <lb/>
              ſuppoſed that the ſaid Axis were not according to the Perpendicu­
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              lar: we are now to demonſtrate that the Figure will not ſo </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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