Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/059.jpg" pagenum="389"/>
              with their Diameters; as alſo, the Angles at Y and G being equall;
                <lb/>
              Therefore, the Lines Y B and G B, & B C & B S, ſhall alſo be equall.]
                <lb/>
                <emph type="italics"/>
              Let the Line A Q cut the Diameter D B in
                <emph.end type="italics"/>
                <foreign lang="grc">γ,</foreign>
                <emph type="italics"/>
              and let it cut A O in
                <emph.end type="italics"/>
                <foreign lang="grc">δ.</foreign>
                <emph type="italics"/>
              Now becauſe that in
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.059.1.jpg" xlink:href="073/01/059/1.jpg" number="57"/>
                <lb/>
                <emph type="italics"/>
              the equall and like Portions A P O L & A M Q L,
                <lb/>
              from the Extremities of their Baſes, A O and
                <lb/>
              A Q are drawn, that contain equall Angles with
                <lb/>
              thoſe Baſes; and ſince the Angles at D, are both
                <lb/>
              Right; Therefore, the Remaining Angles A
                <emph.end type="italics"/>
                <foreign lang="grc">δ</foreign>
                <emph type="italics"/>
              D
                <lb/>
              and A
                <emph.end type="italics"/>
                <foreign lang="grc">γ</foreign>
              D
                <emph type="italics"/>
              ſhall be equall to one another: But
                <lb/>
              the Line P G is parallel unto the Line A O; alſo
                <lb/>
              M Y is parallel to A
                <expan abbr="q;">que</expan>
              and P S and M C to
                <lb/>
              A D: Therefore the Triangles P G S and M Y C,
                <lb/>
              as alſo the Triangles A
                <emph.end type="italics"/>
                <foreign lang="grc">δ</foreign>
                <emph type="italics"/>
              D and A
                <emph.end type="italics"/>
                <foreign lang="grc">γ</foreign>
                <emph type="italics"/>
              D, are all
                <lb/>
              alike to each other
                <emph.end type="italics"/>
              : (b)
                <emph type="italics"/>
              And as A D is to A
                <emph.end type="italics"/>
                <foreign lang="grc">δ,</foreign>
                <lb/>
                <arrow.to.target n="marg1368"/>
                <lb/>
                <emph type="italics"/>
              ſo is A D to A
                <emph.end type="italics"/>
                <foreign lang="grc">γ</foreign>
                <emph type="italics"/>
              : and,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              the Lines
                <lb/>
              A D and A D are equall to each other: Therefore,
                <lb/>
              A
                <emph.end type="italics"/>
                <foreign lang="grc">δ</foreign>
                <emph type="italics"/>
              and A
                <emph.end type="italics"/>
                <foreign lang="grc">γ</foreign>
                <emph type="italics"/>
              are alſo equall: But A O and
                <lb/>
              A Q are equall to each other; as alſo their halves
                <lb/>
              A T and A N: Therefore the Remainders T
                <emph.end type="italics"/>
                <foreign lang="grc">δ</foreign>
                <emph type="italics"/>
              and N
                <emph.end type="italics"/>
                <foreign lang="grc">γ</foreign>
                <emph type="italics"/>
              ; that is, TG and MY, are alſo
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1369"/>
                <lb/>
                <figure id="id.073.01.059.2.jpg" xlink:href="073/01/059/2.jpg" number="58"/>
                <lb/>
                <emph type="italics"/>
              equall. </s>
              <s>And, as
                <emph.end type="italics"/>
              (c)
                <emph type="italics"/>
              P G is to G S, ſo is M Y to
                <lb/>
              Y C: and
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              as P G is to M Y, ſo is
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              G S to Y C: And, therefore, G S and Y C are
                <lb/>
              equall; as alſo their halves B S and B C: From
                <lb/>
              whence it followeth, that the Remainders S R and C R
                <lb/>
              are alſo equall: And, conſequently, that P Z and
                <lb/>
              M V, and V N and Z T, are lkiewiſe equall to one
                <lb/>
              another.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1367"/>
              H</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1368"/>
              (b)
                <emph type="italics"/>
              By 4. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1369"/>
              (c)
                <emph type="italics"/>
              By 34 of the
                <lb/>
              firſt,
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Since, therefore, that N V is leſſer
                <lb/>
                <arrow.to.target n="marg1370"/>
                <lb/>
              than double of V N.]
                <emph type="italics"/>
              For M H is double of
                <lb/>
              H N, and M V is leſſer than M H: Therefore, M V
                <lb/>
              is leſſer than double of H N, and much leſſer than
                <lb/>
              double of V N.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1370"/>
              K</s>
            </p>
            <p type="main">
              <s>Therefore, the Portion ſhall not abide, but ſhall turn about,
                <lb/>
                <arrow.to.target n="marg1371"/>
                <lb/>
              ſo, as that its Baſe do not in the leaſt touch the Surface of
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              the Liquid; in regard that now when it toucheth in but one Point
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              only, it moveth upwards on the part towards A.] Tartaglia's
                <emph type="italics"/>
              his Tranſla­
                <lb/>
              tion hath it thus,
                <emph.end type="italics"/>
              Non ergo manet Portio ſed inclinabitur ut Baſis ipſius, nec ſecundum
                <lb/>
              unum tangat Superficiem Humidi, quon am nunc ſecundum unum tacta ipſa reclina­
                <lb/>
              tur
                <emph type="italics"/>
              : Which we have thought fit in this manner to correct, from other Places of
                <emph.end type="italics"/>
                <lb/>
              Archimedes,
                <emph type="italics"/>
              that the ſenſe might be the more perſpicuous. </s>
              <s>For in the ſixth Propoſition of this,
                <lb/>
              he thus writeth (as we alſo have it in the Tranſlation,)
                <emph.end type="italics"/>
              The Solid A P O L, therefore, ſhall
                <lb/>
              turn about, and its Baſe ſhall not in the leaſt touch the Surface of the Liquid.
                <emph type="italics"/>
              Again,
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              in the ſeventh Propoſition
                <emph.end type="italics"/>
              ; From whence it is manifeſt, that its Baſe ſhall turn about in
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              ſuch manner, a that its Baſe doth in no wiſe touch the Surface of the Liquid; For
                <lb/>
              that now when it toucheth but in one Point only, it moveth downwards on the part
                <lb/>
              towards L.
                <emph type="italics"/>
              And that the Portion moveth upwards, on the part towards A, doth plainly ap­
                <lb/>
              pear: For ſince that the Perpendiculars unto the Surface of the Liquid, that paſs thorow
                <foreign lang="grc">ω</foreign>
              , de
                <lb/>
              fall on the part towards A, and thoſe that paſs thorow E, on the part towards L; it is neceſſary
                <lb/>
              that the Centre
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              do move upwards, and the Centre E downwards.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1371"/>
              L</s>
            </p>
            <p type="main">
              <s>It is therefore perſpicuous, that the Portion ſhall conſiſt, ſo, as that
                <lb/>
              its Axis ſhall make an Angle with the Liquids Surface greater than
                <lb/>
              the Angle
                <emph type="italics"/>
              X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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