Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/028.jpg" pagenum="357"/>
                <figure id="id.073.01.028.1.jpg" xlink:href="073/01/028/1.jpg" number="24"/>
                <lb/>
                <emph type="italics"/>
              Angle K H M: Therefore
                <emph.end type="italics"/>
              (f) O G
                <emph type="italics"/>
              and H N are parallel,
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1184"/>
                <lb/>
                <emph type="italics"/>
              and the
                <emph.end type="italics"/>
              (g)
                <emph type="italics"/>
              Angle H N F equall to the Angle O G F; for
                <lb/>
              that G O being Perpendicular to E F, H N ſhall alſo be per-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1185"/>
                <lb/>
                <emph type="italics"/>
              pandicnlar to the ſame: Which was to be demon ſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1179"/>
              (a)
                <emph type="italics"/>
              By Cor. </s>
              <s>of 8. of
                <lb/>
              6. of
                <emph.end type="italics"/>
              Euclide.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1180"/>
              (b)
                <emph type="italics"/>
              By 17. of the
                <emph.end type="italics"/>
                <lb/>
              6.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1181"/>
              (c)
                <emph type="italics"/>
              By 14. of the
                <emph.end type="italics"/>
                <lb/>
              6.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1182"/>
              (d)
                <emph type="italics"/>
              By 33. of the
                <emph.end type="italics"/>
                <lb/>
              1.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1183"/>
              (e)
                <emph type="italics"/>
              By 4. of the
                <emph.end type="italics"/>
              1.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1184"/>
              (f)
                <emph type="italics"/>
              By 28. of the
                <emph.end type="italics"/>
                <lb/>
              1.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1185"/>
              (g)
                <emph type="italics"/>
              By 29. of th
                <emph.end type="italics"/>
                <lb/>
              1</s>
            </p>
            <p type="main">
              <s>And the part which is within the Liquid
                <lb/>
                <arrow.to.target n="marg1186"/>
                <lb/>
              doth move upwards according to the Per­
                <lb/>
              pendicular that is drawn thorow B parallel
                <lb/>
              to R T.]
                <emph type="italics"/>
              The reaſon why this moveth upwards, and that
                <lb/>
              other downwards, along the Perpendicular Line, hath been ſhewn above in the 8 of the firſt
                <lb/>
              Book of this; ſo that we have judged it needleſſe to repeat it either in this, or in the reſt
                <lb/>
              that follow.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1186"/>
              G</s>
            </p>
            <p type="head">
              <s>THE TRANSLATOR.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              In the
                <emph.end type="italics"/>
              Antient
                <emph type="italics"/>
              Parabola (namely that aſſumed in a Rightangled
                <lb/>
              Cone) the Line
                <emph.end type="italics"/>
              juxta quam Poſſunt quæ in Sectione ordinatim du­
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              cuntur
                <emph type="italics"/>
              (which I, following
                <emph.end type="italics"/>
              Mydorgius,
                <emph type="italics"/>
              do call the
                <emph.end type="italics"/>
              Parameter
                <emph type="italics"/>
              ) is
                <emph.end type="italics"/>
              (a)
                <lb/>
                <arrow.to.target n="marg1187"/>
                <lb/>
                <emph type="italics"/>
              double to that
                <emph.end type="italics"/>
              quæ ducta eſt à Vertice Sectionis uſque ad Axem,
                <emph type="italics"/>
              or in
                <emph.end type="italics"/>
                <lb/>
              Archimedes
                <emph type="italics"/>
              phraſe,
                <emph.end type="italics"/>
                <foreign lang="grc">τᾱς υσ́χρι τοῡ ἄξον<34>;</foreign>
                <emph type="italics"/>
              which I for that cauſe, and
                <lb/>
              for want of a better word, name the
                <emph.end type="italics"/>
              Semiparameter:
                <emph type="italics"/>
              but in
                <emph.end type="italics"/>
              Modern
                <lb/>
                <emph type="italics"/>
              Parabola's it is greater or leſſer then double. </s>
              <s>Now that throughout this
                <lb/>
              Book
                <emph.end type="italics"/>
              Archimedes
                <emph type="italics"/>
              ſpeaketh of the Parabola in a Rectangled Cone, is mani­
                <lb/>
              feſt both by the firſt words of each Propoſition, & by this that no Parabola
                <lb/>
              hath its Parameter double to the Line
                <emph.end type="italics"/>
              quæ eſt a Sectione ad Axem,
                <emph type="italics"/>
              ſave
                <lb/>
              that which is taken in a Rightangled Cone. </s>
              <s>And in any other Parabola, for
                <lb/>
              the Line
                <emph.end type="italics"/>
                <foreign lang="grc">τᾱς μσ́χριτοῡ ἄεον<34></foreign>
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              quæ uſque ad Axem
                <emph type="italics"/>
              to uſurpe the Word
                <emph.end type="italics"/>
              Se­
                <lb/>
              miparameter
                <emph type="italics"/>
              would be neither proper nor true: but in this caſe it may paſs
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1187"/>
              (a) Rîvalt.
                <emph type="italics"/>
              in
                <emph.end type="italics"/>
              Ar­
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              chimed.
                <emph type="italics"/>
              de Cunoid
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              & Sphæroid.
                <emph.end type="italics"/>
              Prop.
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              </s>
              <s>3. Lem. </s>
              <s>1.</s>
            </p>
            <p type="head">
              <s>PROP. III. THEOR. III.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Rightangled Conoid, when it
                <lb/>
              ſhall have its Axis leſſe than ſeſquialter of the Se­
                <lb/>
              mi-parameter, the Axis having any what ever pro­
                <lb/>
              portion to the Liquid in Gravity, being demitted into
                <lb/>
              the Liquid ſo as that its Baſe be wholly within the
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              ſaid Liquid, and being ſet inclining, it ſhall not re­
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              main inclined, but ſhall be ſo reſtored, as that its Ax­
                <lb/>
              is do ſtand upright, or according to the Perpendicular.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let any Portion be demitted into the Liquid, as was ſaid; and
                <lb/>
              let its Baſe be in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid;
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                <figure id="id.073.01.028.2.jpg" xlink:href="073/01/028/2.jpg" number="25"/>
                <lb/>
              and let it be cut thorow the
                <lb/>
              Axis, by a Plain erect upon the Sur­
                <lb/>
              face of the Liquid, and let the Se­
                <lb/>
              ction be A P O
                <emph type="italics"/>
              L,
                <emph.end type="italics"/>
              the Section of a
                <lb/>
              Right angled Cone: and let the Axis
                <lb/>
              of the Portion and Diameter of the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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