Archimedes, Natation of bodies, 1662

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    <archimedes>
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          <chap>
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              <s>
                <pb xlink:href="073/01/049.jpg" pagenum="379"/>
              parameter be equall to K R: and
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                <figure id="id.073.01.049.1.jpg" xlink:href="073/01/049/1.jpg" number="47"/>
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                <arrow.to.target n="marg1303"/>
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              let D S be Seſquialter of K R: but
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              S B is alſo Seſquialter of B R:
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              Therefore, draw a Line from A to
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              B; and thorow C draw C E Per­
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              pendicular to B D, cutting the Line
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              A B in the Point E; and thorow E
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              draw E Z parallel unto B D. Again,
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              A B being divided into two equall
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              parts in T, draw T H parallel to the
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              ſame B D: and let Sections of
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              Rightangled Cones be deſcribed, A E I about the Diameter E Z;
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              and A T D about the Diameter T H; and let them be like to the
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              Portion A B L: Now the Section of the Cone A E I, ſhall paſs
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                <arrow.to.target n="marg1305"/>
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              thorow K; and the Line drawn from R perpendicular unto B D,
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              ſhall cut the ſaid A E I; let it cut it in the Points Y G: and
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              thorow Y and G draw P Y Q and O G N parallels unto B D, and
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              cutting A T D in the Points F and X: laſtly, draw P
                <foreign lang="grc">Φ</foreign>
              and O X
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              touching the Section A P O L in the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              oints P and O. </s>
              <s>In regard,
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                <arrow.to.target n="marg1306"/>
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              therefore, that the three
                <emph type="italics"/>
              P
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              ortions A P O L, A E I, and A T D are
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              contained betwixt Right Lines, and the Sections of Rightangled
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              Cones, and are right alike and unequall, touching one another, upon
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              one and the ſame Baſe; and N X G O being drawn from the
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                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              oint N upwards, and Q F Y P from Q: O G ſhall have to G X
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              a proportion compounded of the proportion, that I L hath to L A,
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              and of the proportion that A D hath to DI: But I L is to L A,
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              as two to five: And C B is to B D, as ſix to fifteen; that is, as two
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                <arrow.to.target n="marg1307"/>
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              to five: And as C B is to B D, ſo is
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              E B to B A
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              ; and D Z to
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                <arrow.to.target n="marg1308"/>
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              D A: And of D Z and D A, L I and L A are double: and A D
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                <arrow.to.target n="marg1309"/>
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              is to D I, as five to one:
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              B
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              ut the proportion compounded of the
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              proportion of two to five, and of the proportion of five to one, is
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                <arrow.to.target n="marg1310"/>
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              the ſame with that of two to one: and two is to one, in double
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              proportion: Therefore, O G is double of GX: and, in the ſame
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              manner is P Y proved to be double of Y F: Therefore, ſince that
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              D S is Seſquialter of K R;
                <emph type="italics"/>
              B S
                <emph.end type="italics"/>
              ſhall be the Exceſs by which the
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              Axis is greater than Seſquialter of the Semi-parameter. </s>
              <s>If there­
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              fore, the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion have the ſame proportion in Gravity unto the
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              Liquid, as the Square made of the Line
                <emph type="italics"/>
              B S,
                <emph.end type="italics"/>
              hath to the Square
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              made of
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              B D,
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              or greater, being demitted into the Liquid, ſo as hat
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              its
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              B
                <emph.end type="italics"/>
              aſe touch not the Liquid, it ſhall ſtand erect, or perpendicular:
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              For it hath been demonſtrated above, that the
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              P
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              ortion whoſe
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                <arrow.to.target n="marg1311"/>
                <lb/>
              Axis is greater than Seſquialter of the Semi-parameter, if it have
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              not leſser proportion in Gravity unto the Liquid, than the Square </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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