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

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb pagenum="450"/>
            <p type="margin">
              <s>
                <margin.target id="marg1513"/>
              Priſmes and
                <lb/>
              Cylinders
                <lb/>
              ving the ſame
                <lb/>
              Baſe, are to one
                <lb/>
              another as their
                <lb/>
              heights.</s>
            </p>
            <p type="head">
              <s>THEOREME.
                <lb/>
                <arrow.to.target n="marg1514"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1514"/>
              All Figures
                <lb/>
              of all Matters,
                <lb/>
              float by hep of
                <lb/>
              the Rampart
                <lb/>
              pleniſhed with
                <lb/>
              Air, and ſome
                <lb/>
              but only touch
                <lb/>
              the water.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              All ſorts of Figures of whatſoever Matter, albeit more
                <lb/>
              grave than the Water, do by Benefit of the ſaid
                <lb/>
              part, not only float, but ſome Figures, though of the
                <lb/>
              graveſt Matter, do ſtay wholly above Water, wetting
                <lb/>
              only the inferiour Surface that toucheth the Water.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>And theſe ſhall be all Figures, which from the inferiour Baſe up­
                <lb/>
              wards, grow leſſer and leſſer; the which we ſhall exemplifie for
                <lb/>
              this time in Piramides or Cones, of which Figures the paſſions sre
                <lb/>
              common. </s>
              <s>We will demonſtrate therefore, that,</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              It is poſſible to form a Piramide, of any whatſoever Matter propoſed,
                <lb/>
              which being put with its Baſe upon the Water, reſts not only
                <lb/>
              ſubmerging, but without wetting it more then its Baſe.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>For the explication of which it is requiſite, that we firſt
                <lb/>
              the ſubſequent Lemma, namely, that,</s>
            </p>
            <p type="head">
              <s>LEMMA II.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Solids whoſe Maſſes anſwer in proportion contrarily to
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1515"/>
                <lb/>
                <emph type="italics"/>
              their Specificall Gravities, are equall in Abſolute
                <lb/>
              Gravities.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1515"/>
              Solids whoſe
                <lb/>
              Maſſes are in
                <lb/>
              contrary
                <lb/>
              portion to their
                <lb/>
              Specifick
                <lb/>
              vities, are equall
                <lb/>
              in abſolute Gra
                <lb/>
              vity.</s>
            </p>
            <p type="main">
              <s>Let A C and B be two Solids, and let the Maſs A C be to the
                <lb/>
              Maſs B, as the Specificall Gravity of the Solid B, is to the Speci­
                <lb/>
              ficall Gravity of the Solid A C: I ſay, the Solids A C and B are
                <lb/>
              equall in abſolute weight, that is, equally grave. For
                <lb/>
                <figure id="fig273" number="12"/>
                <lb/>
              if the Maſs A C be equall to the Maſs B, then, by the
                <lb/>
              Aſſumption, the Specificall Gravity of B, ſhall be
                <lb/>
              quall to the Specificall Gravity of A C, and being
                <lb/>
              quall in Maſs, and of the ſame Specificall Gravity
                <lb/>
              ſhall abſolutely weigh one as much as another. </s>
              <s>But
                <lb/>
              if their Maſſes ſhall be unequall, let the Maſs A C be greater, and in it
                <lb/>
              take the part C, equall to the Maſs B. And, becauſe the Maſſes B
                <lb/>
              and C are equall; the Abſolute weight of B, ſhall have the ſame
                <lb/>
              portion to the Abſolute weight of C, that the Specificall Gravity of
                <lb/>
              B, hath to the Specificall Gravity of C; or of C A, which is the
                <lb/>
              ſame
                <emph type="italics"/>
              in ſpecie
                <emph.end type="italics"/>
              : But look what proportion the Specificall Gravity of
                <lb/>
              B, hath to the Specificall Gravity of C A, the like proportion, by the
                <lb/>
              Aſſumption, hath the Maſs C A, to the Maſs B; that is, to the Maſs C: </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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