Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/034.jpg" pagenum="364"/>
              ſaid K
                <foreign lang="grc">ω</foreign>
              in H, and A S is parallel unto the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine that toucheth in
                <lb/>
              P; It is neceſſary that P I hath unto P H either the ſame proportion
                <lb/>
              that
                <emph type="italics"/>
              N
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
              hath to
                <foreign lang="grc">ω</foreign>
              O, or greater; for this hath already been de­
                <lb/>
              monſtrated: But
                <emph type="italics"/>
              N
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
              is ſeſquialter of
                <foreign lang="grc">ω</foreign>
              O; and P I, therefore, is
                <lb/>
              either Seſquialter of H P, or more than ſeſquialter: Wherefore
                <lb/>
                <arrow.to.target n="marg1218"/>
                <lb/>
              P H is to H I either double, or leſſe than double.
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              et P T be
                <lb/>
              double to T I: the Centre of Gravity of the part which is within
                <lb/>
              the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid ſhall be the Point T. </s>
              <s>Therefore draw a
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine from T
                <lb/>
              to F prolonging it; and let the Centre of
                <lb/>
                <figure id="id.073.01.034.1.jpg" xlink:href="073/01/034/1.jpg" number="30"/>
                <lb/>
              Gravity of the part which is above the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid
                <lb/>
              be G: and from the Point B at Right Angles
                <lb/>
              unto
                <emph type="italics"/>
              N O
                <emph.end type="italics"/>
              draw B R. </s>
              <s>And ſeeing that P I is
                <lb/>
              parallel unto the Diameter
                <emph type="italics"/>
              N O,
                <emph.end type="italics"/>
              and B R
                <lb/>
              perpendicular unto the ſaid Diameter, and F
                <lb/>
              B equall to the Semi-parameter; It is mani­
                <lb/>
              feſt that the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine drawn thorow the Points
                <lb/>
              F and R being prolonged, maketh equall
                <lb/>
              Angles with that which toucheth the Section
                <lb/>
              A P O L in the Point P: and therefore doth alſo make Right An­
                <lb/>
              gles with A S, and with the Surface of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid: and the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ines
                <lb/>
              drawn thorow T and G parallel unto F R ſhall be alſo perpendicu­
                <lb/>
              lar to the Surface of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid: and of the Solid Magnitude A P
                <lb/>
              O L, the part which is within the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid moveth upwards according
                <lb/>
              to the Perpendicular drawn thorow T; and the part which is above
                <lb/>
              the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid moveth downwards according to that drawn thorow G:
                <lb/>
                <arrow.to.target n="marg1219"/>
                <lb/>
              The Solid A
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              O L, therefore, ſhall turn about, and its Baſe ſhall
                <lb/>
              not in the leaſt touch the Surface of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid, And if
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              I do not
                <lb/>
              cut the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine K
                <foreign lang="grc">ω,</foreign>
              as in the ſecond Figure, it is manifeſt that the
                <lb/>
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              oint T, which is the Centre of Gravity of the ſubmerged
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion,
                <lb/>
              falleth betwixt
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              and I: And for the other particulars remaining,
                <lb/>
              they are demonſtrated like as before.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1215"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1216"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1217"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1218"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1219"/>
              E</s>
            </p>
            <p type="head">
              <s>COMMANDINE.
                <lb/>
                <arrow.to.target n="marg1220"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1220"/>
              A</s>
            </p>
            <p type="main">
              <s>It is to be demonſtrated that the ſaid
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion ſhall not continue
                <lb/>
              ſo, but ſhall turn about in ſuch manner as that its Baſe do in no wiſe
                <lb/>
              touch the Surface of the Liquid.]
                <emph type="italics"/>
              Theſe words are added by us, as having been
                <lb/>
              omitted by
                <emph.end type="italics"/>
              Tartaglia.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              N
                <emph.end type="italics"/>
              ow becauſe N O hath greater proportion to F
                <foreign lang="grc">ω</foreign>
              than unto </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1221"/>
                <lb/>
              the Semi parameter.]
                <emph type="italics"/>
              For the Diameter of the Portion N O hath unto F
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              the
                <lb/>
              ſame proportion as fifteen to fower: But it was ſuppoſed to have leſſe proportion unto the
                <lb/>
              Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <lb/>
                <emph type="italics"/>
              than unto the Semi-parameter: And therefore
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              the Semi-parameter ſhall be greater
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1222"/>
                <lb/>
                <emph type="italics"/>
              than the ſaid F
                <emph.end type="italics"/>
                <foreign lang="grc">ω.</foreign>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1221"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1222"/>
              (a)
                <emph type="italics"/>
              By 10. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Foraſmuch, therefore, as in the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion
                <emph type="italics"/>
              A P O L,
                <emph.end type="italics"/>
              contained, be­
                <lb/>
                <arrow.to.target n="marg1223"/>
                <lb/>
              twixt the Right
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine and the Section of the Rightangled Cone K
                <lb/>
                <foreign lang="grc">ω</foreign>
              is parallel to A L, and
                <emph type="italics"/>
              P I
                <emph.end type="italics"/>
              parallel unto the Diameter, and cut by </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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