Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/026.jpg" pagenum="355"/>
              remaining Figure I S L A. </s>
              <s>Becauſe now that N O is
                <emph type="italics"/>
              Seſquialter
                <emph.end type="italics"/>
                <lb/>
              of R O, but leſs than
                <emph type="italics"/>
              Seſquialter ejus quæ uſque ad Axem
                <emph.end type="italics"/>
              (or of its
                <lb/>
                <emph type="italics"/>
              Semi-parameter
                <emph.end type="italics"/>
              ;) ^{*} R O ſhall be leſſe than
                <emph type="italics"/>
              quæ uſque ad Axem
                <emph.end type="italics"/>
              (or
                <lb/>
                <arrow.to.target n="marg1170"/>
                <lb/>
              than the
                <emph type="italics"/>
              Semi-parameter
                <emph.end type="italics"/>
              ;) ^{*} whereupon the Angle R P
                <foreign lang="grc">ω</foreign>
              ſhall be
                <lb/>
                <arrow.to.target n="marg1171"/>
                <lb/>
              acute. </s>
              <s>For ſince the Line
                <emph type="italics"/>
              quæ uſque ad Axem
                <emph.end type="italics"/>
              (or
                <emph type="italics"/>
              Semi-parameter
                <emph.end type="italics"/>
              )
                <lb/>
              is greater than R O, that Line which is drawn from the Point R,
                <lb/>
              and perpendicular to K
                <foreign lang="grc">ω,</foreign>
              namely RT, meeteth with the line F P
                <lb/>
              without the Section, and for that cauſe muſt of neceſſity fall be­
                <lb/>
              tween the Points
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              and
                <foreign lang="grc">ω;</foreign>
              Therefore if
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ines be drawn through
                <lb/>
              B and G, parallel unto R T, they ſhall contain Right Angles with
                <lb/>
              the Surface of the Liquid: ^{*} and the part that is within the Li­
                <lb/>
                <arrow.to.target n="marg1172"/>
                <lb/>
              quid ſhall move upwards according to the Perpendicular that is
                <lb/>
              drawn thorow B, parallel to R T, and the part that is above the Li­
                <lb/>
              quid ſhall move downwards according to that which is drawn tho­
                <lb/>
              row G; and the Solid A P O L ſhall not abide in this Poſition; for
                <lb/>
              that the parts towards A will move upwards, and thoſe towards
                <lb/>
              B downwards; Wherefore N O ſhall be conſtituted according to
                <lb/>
              the Perpendicular.]</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1166"/>
              *
                <emph type="italics"/>
              Supplied by
                <emph.end type="italics"/>
              Fe­
                <lb/>
              derico Comman­
                <lb/>
              dino.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1167"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1168"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1169"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1170"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1171"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1172"/>
              G</s>
            </p>
            <p type="head">
              <s>COMMANDINE.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Demonſtration of this propoſition hath been much deſired; which we have (in like man­
                <lb/>
              ner as the 8 Prop. </s>
              <s>of the firſt Book) reſtored according to
                <emph.end type="italics"/>
              Archimedes
                <emph type="italics"/>
              his own Schemes, and
                <lb/>
              illustrated it with Commentaries.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>The Right Portion of a Rightangled Conoid, when it ſhall
                <lb/>
                <arrow.to.target n="marg1173"/>
                <lb/>
              have its Axis leſſe than
                <emph type="italics"/>
              Seſquialter ejus quæ uſque ad Axem
                <emph.end type="italics"/>
              (or of
                <lb/>
              its
                <emph type="italics"/>
              Semi-parameter] In the Tranſlation of
                <emph.end type="italics"/>
              Nicolo Tartaglia
                <emph type="italics"/>
              it is falſlyread
                <emph.end type="italics"/>
              great­
                <lb/>
              er then Seſquialter,
                <emph type="italics"/>
              and ſo its rendered in the following Propoſition; but it is the Right
                <lb/>
              Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we ſay
                <lb/>
              that Conoids are then conſtituted erect when the cutting Plane, that is to ſay, the Plane of the
                <lb/>
              Baſe, ſhall be parallel to the Surface of the Liquid.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1173"/>
              A</s>
            </p>
            <p type="main">
              <s>Which ſhall be the Diameter of the Section I P O S, and the
                <lb/>
                <arrow.to.target n="marg1174"/>
                <lb/>
              Axis of the Portion demerged in the Liquid.]
                <emph type="italics"/>
              By the 46 of the firſt of
                <lb/>
              the Conicks of
                <emph.end type="italics"/>
              Apollonious,
                <emph type="italics"/>
              or by the Corol­
                <lb/>
              lary of the 51 of the ſame.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1174"/>
              B</s>
            </p>
            <figure id="id.073.01.026.1.jpg" xlink:href="073/01/026/1.jpg" number="21"/>
            <p type="main">
              <s>And of the Solid Magnitude A P
                <lb/>
                <arrow.to.target n="marg1175"/>
                <lb/>
              O L, let the Centre of Gravity be R;
                <lb/>
              and of I P O S let the Centre be B.]
                <lb/>
                <emph type="italics"/>
              For the Centre of Gravity of the Portion of a Right­
                <lb/>
              angled Conoid is in its Axis, which it ſo divideth
                <lb/>
              as that the part thereof terminating in the vertex,
                <lb/>
              be double to the other part terminating in the Baſe; as
                <lb/>
              in our Book
                <emph.end type="italics"/>
              De Centro Gravitatis Solidorum Propo. </s>
              <s>29.
                <emph type="italics"/>
              we have demonſtrated. </s>
              <s>And
                <lb/>
              ſince the Centre of Gravity of the Portion A P O L is R, O R ſhall be double to RN and there­
                <lb/>
              fore N O ſhall be Seſquialter of O R. </s>
              <s>And for the ſame reaſon, B the Centre of Gravity of the Por­
                <lb/>
              tion I P O S is in the Axis P F, ſo dividing it as that P B is double to B F;
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1175"/>
              C</s>
            </p>
            <p type="main">
              <s>And draw a Line from B to R prolonged unto G; which let
                <lb/>
                <arrow.to.target n="marg1176"/>
                <lb/>
              be the Centre of Gravity of the remaining Eigure I S L A.] </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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