Galilei, Galileo, Discourse concerning the natation of bodies, 1663

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb pagenum="409"/>
              though the Concluſions are true, yet are the Cauſes thus aſſigned
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              deficient, nor is it true, that the Solid in ſubmerging, raiſeth and
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              repulſeth Maſſes of Water, equall to the parts of it ſelf ſubmerged;
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              but the Water repulſed, is alwayes leſs than the parts of the Solid
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                <arrow.to.target n="marg1409"/>
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              ſubmerged: and ſo much the more by how much the Veſſell in
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              which the Water is contained is narrower: in ſuch manner that it
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              hinders not, but that a Solid may ſubmerge all under Water, with­
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              out raiſing ſo much Water in Maſs, as would equall the tenth or
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              twentieth part of its own Bulk: like as on the contrary, a very
                <lb/>
                <arrow.to.target n="marg1410"/>
                <lb/>
              ſmall quantity of Water, may raiſe a very great Solid Maſs, though
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              ſuch Solid ſhould weigh abſolutely a hundred times as much, or
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              more, than the ſaid Water, if ſo be that the Matter of that ſame
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              Solid be
                <emph type="italics"/>
              in ſpecie
                <emph.end type="italics"/>
              leſs grave than the Water. </s>
              <s>And thus a great
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              Beam, as ſuppoſe of a 1000 weight, may be raiſed and born afloat
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              by Water, which weighs not 50: and this happens when the Mo­
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              ment of the Water is compenſated by the Velocity of its Motion.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1404"/>
              How the ſub­
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              merſion of So­
                <lb/>
              lids in the Wa­
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              ter, is effected.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1405"/>
              What Solids
                <lb/>
              ſhall float on the
                <lb/>
              Water.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1406"/>
              What Solids
                <lb/>
              ſhall ſinke to the
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              botome.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1407"/>
              What Solids
                <lb/>
              ſhall reſt in all
                <lb/>
              places of the Wa­
                <lb/>
              ter.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1408"/>
              The Gravitie of
                <lb/>
              the Water and
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                <emph type="italics"/>
              S
                <emph.end type="italics"/>
              olid muſt be
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              compared in all
                <lb/>
              Problems, of Na­
                <lb/>
              tation of Bodies.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1409"/>
              The water re­
                <lb/>
              pulſed is ever leſs
                <lb/>
              than the parts of
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              the Sollid ſub­
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              merged.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1410"/>
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              ſmall quantity
                <lb/>
              of water, may
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              float a very
                <lb/>
              great Solid Maſs.</s>
            </p>
            <p type="main">
              <s>But becauſe ſuch things, propounded thus in abſtract, are ſome­
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              what difficult to be comprehended, it would be good to demonſtrate
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              them by particular examples; and for facility of demonſtration, we
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              will ſuppoſe the Veſſels in which we are to put the Water, and place
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              the Solids, to be inviron'd and included with ſides erected perpendi­
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              cular to the Plane of the Horizon, and the Solid that is to be put
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              into ſuch veſſell to be either a ſtreight Cylinder, or elſe an upright
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              Priſme</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The which propoſed and declared, I proceed to demonstrate the truth
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              of what hath been hinted, forming the enſuing Theoreme.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>
                <emph type="italics"/>
              THEOREME I.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>The Maſs of the Water whichaſcends in the ſub­
                <lb/>
                <arrow.to.target n="marg1411"/>
                <lb/>
              merging of a Solid, Priſme or Cylinder, or that
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              abaſeth in taking it out, is leſs than the Maſs of
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              the ſaid Solid, ſo depreſſed or advanced: and
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              hath to it the ſame proportion, that the Surface
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              of the Water circumfuſing the Solid, hath to the
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              ſame circumfuſed Surface, together with the Baſe
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              of the Solid.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1411"/>
                <emph type="italics"/>
              T
                <emph.end type="italics"/>
              he Proportion
                <lb/>
              of the water rai­
                <lb/>
              ſed to the
                <emph type="italics"/>
              S
                <emph.end type="italics"/>
              olid
                <lb/>
              ſubmerged.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Veſſell be A B C D, and in it the Water raiſed up to the
                <lb/>
              Levell E F G, before the Solid Priſme H I K be therein immerged;
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              but after that it is depreſſed under Water, let the Water be raiſed as
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              high as the Levell L M, the Solid H I K ſhall then be all under Water,
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              and the Maſs of the elevated Water ſhall be L G, which is leſs than the
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>