Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/066.jpg" pagenum="400"/>
                <figure id="id.073.01.066.1.jpg" xlink:href="073/01/066/1.jpg" number="71"/>
                <lb/>
              ſtanding as before, we will again
                <lb/>
              demonſtrate, that N T is equall to
                <lb/>
              V I; and that the Portions A V Q
                <lb/>
              and A N Z are equall to each other.
                <lb/>
              </s>
              <s>Therefore, in regard, that in the
                <lb/>
              Equall and Like Portions A V Q L
                <lb/>
              and A N Z G, there are drawn
                <lb/>
              A Q and A Z cutting off equall Por­
                <lb/>
              tions, they ſhall with the Diameters
                <lb/>
              of the Portions, contain equall
                <lb/>
              Angles. </s>
              <s>Therefore, in the Triangles
                <lb/>
              N L S and V
                <foreign lang="grc">ω</foreign>
              C, the Angles at
                <lb/>
              the Points
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              and
                <foreign lang="grc">ω</foreign>
              are equall; and the Right Line B S equall to
                <lb/>
              B C; S R to C R; N X to V H; and X T to H I: And, ſince
                <lb/>
              V Y is double to Y I, N X ſhall be greater than double of X T.
                <lb/>
              </s>
              <s>Let therefore, N M be double to M T. </s>
              <s>It is hence again manifeſt,
                <lb/>
              that the Portion will not remain, but ſhall incline on the part
                <lb/>
              towards A: But it was ſuppoſed, that the ſaid Portion did
                <lb/>
              touch the Surface of the Liquid in one ſole Point: Therefore,
                <lb/>
              its Baſe muſt of neceſſity ſubmerge farther into the Liquid.</s>
            </p>
            <p type="head">
              <s>CONCLVSION V.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              If the Portion have leſſer proportion in Gravity to
                <lb/>
              the Liquid, than the Square F P to the Square
                <lb/>
              B D, being demitted into the Liquid, and in­
                <lb/>
              clined, ſo, as that its Baſe touch not the Liquid,
                <lb/>
              it ſhall ſtand ſo inclined, as that its Axis ſhall
                <lb/>
              make an Angle with the Surface of the Liquid,
                <lb/>
              leſſe than the Angle
                <emph.end type="italics"/>
                <foreign lang="grc">ψ;</foreign>
                <emph type="italics"/>
              And its Baſe ſhall
                <lb/>
              not in the leaſt touch the Liquids Surface.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Finally, let the Portion have leſſer proportion to the Liquid
                <lb/>
              in Gravity, than the Square F P hath to the Square B D; and
                <lb/>
              as the Portion is in Gravity to the Liquid, ſo let the
                <lb/>
              Square made of the Line
                <foreign lang="grc">ψ</foreign>
              be to the Square B D.
                <foreign lang="grc">ψ</foreign>
              ſhall be
                <lb/>
              leſſer than P F. Again, apply any Right Line as G I, falling
                <lb/>
              betwixt the Sections A G Q L and A X D, and parallel to B D;
                <lb/>
              and let it cut the Middle Conick Section in the Point H, and </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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