Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/061.jpg" pagenum="391"/>
              and touching the Section in P, and T P parallel to B D; and P S perpen­
                <lb/>
              dicular unto B D. </s>
              <s>It is to be demonſtrated that the Portion ſhall
                <lb/>
                <figure id="id.073.01.061.1.jpg" xlink:href="073/01/061/1.jpg" number="61"/>
                <lb/>
              not ſtand ſo, but ſhall encline until
                <lb/>
              that the Baſe touch the Surface of
                <lb/>
              the Liquid, in one Point only, for let
                <lb/>
              the ſuperior figure ſtand as it was,
                <lb/>
              and draw O C, Perpendicular to B D;
                <lb/>
              and drawing a
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine from A to
                <emph type="italics"/>
              X,
                <emph.end type="italics"/>
                <lb/>
              prolong it to Q: A X ſhalbe equall
                <lb/>
              to
                <emph type="italics"/>
              X
                <emph.end type="italics"/>
                <expan abbr="q.">que</expan>
              Then draw O X parallel
                <lb/>
              to A
                <expan abbr="q.">que</expan>
              And becauſe the Portion
                <lb/>
              is ſuppoſed to have the ſame pro­
                <lb/>
              portion in Gravity to the Liquid
                <lb/>
              that the ſquare X O hath to the
                <lb/>
              Square B D; the part thereof ſubmerged ſhall alſo have the ſame
                <lb/>
              proportion to the whole; that is, the Square T P to the Square
                <lb/>
                <arrow.to.target n="marg1374"/>
                <lb/>
              B D; and ſo T P ſhall be equal to
                <emph type="italics"/>
              X
                <emph.end type="italics"/>
              O: And ſince that of the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortions
                <lb/>
              I P M and A O Q the Diameters are equall, the portions ſhall alſo be
                <lb/>
                <arrow.to.target n="marg1375"/>
                <lb/>
              equall.
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              gain, becauſe that in the Equall and
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ike
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortions A O Q L
                <lb/>
                <arrow.to.target n="marg1376"/>
                <lb/>
              and AP ML the Lines A Q and I M, which cut off equall
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              or­
                <lb/>
              tions, are drawn, that, from the Extremity of the
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              aſe, and this
                <lb/>
              not from the Extremity; it appeareth that that which is drawn from
                <lb/>
              the end or Extremity of the
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              aſe, ſhall make the Acute Angle with
                <lb/>
              the Diameter of the whole
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion leſset.
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              nd the Angle at
                <emph type="italics"/>
              X
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1377"/>
                <lb/>
              being leſſe than the Angle at N, B C ſhall be greater than B S; and
                <lb/>
              C R leſſer than S R:
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              nd, therfore O G ſhall be leſſer than P Z;
                <lb/>
              and G
                <emph type="italics"/>
              X
                <emph.end type="italics"/>
              greater than Z T: Therfore P Z is greater than double of
                <lb/>
              Z T; being that O G is double of G X. </s>
              <s>Let P H be double to H T;
                <lb/>
              and drawing a Line from H to K, prolong it to
                <foreign lang="grc">ω.</foreign>
              The Center of
                <lb/>
              Gravity of the whole Portion ſhall be K; the Center of the part
                <lb/>
              which is within the Liquid H, and that of the part which is above
                <lb/>
              the Liquid in the Line K
                <foreign lang="grc">ω</foreign>
              ; which ſuppoſed to be
                <foreign lang="grc">ω.</foreign>
              Therefore it
                <lb/>
              ſhall be demonſtrated, both, that K H is perpendicular to the Surface
                <lb/>
              of the Liquid, and thoſe Lines alſo that are drawn thorow the Points
                <lb/>
              Hand
                <foreign lang="grc">ω</foreign>
              parallel to K H: And therfore the Portion ſhall not reſt, but
                <lb/>
              ſhall encline untill that its Baſe do touch the Surface of the Liquid
                <lb/>
              in one Point; and ſo it ſhall continue. </s>
              <s>For in the Equall Portions
                <lb/>
              A O Q L and A P M L, the
                <lb/>
                <figure id="id.073.01.061.2.jpg" xlink:href="073/01/061/2.jpg" number="62"/>
                <lb/>
              Lines A Q and A M, that cut off
                <lb/>
              equall Portions, ſhall be dawn
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              from the Ends or Terms of the Baſes;
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              and A O Q and A P M ſhall be
                <lb/>
              demonſtrated, as in the former, to
                <lb/>
                <arrow.to.target n="marg1378"/>
                <lb/>
              be equall: Therfore A Q and A M,
                <lb/>
              do make equall Acute Angles with
                <lb/>
              the Diameters of the Portions; and </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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