Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/035.jpg" pagenum="365"/>
              the ſaid K
                <foreign lang="grc">ω</foreign>
              in H, and A S is parallel unto the Line that toucheth
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              in P; It is neceſſary that P I hath unto P H either the ſame propor­
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              tion that N
                <foreign lang="grc">ω</foreign>
              hath to
                <foreign lang="grc">ω</foreign>
              O, or greater; for this hath already been
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              demonſtrated.]
                <emph type="italics"/>
              Where this is demonſtrated either by
                <emph.end type="italics"/>
              Archimedes
                <emph type="italics"/>
              himſelf, or by
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              any other, doth not appear; touching which we will here inſert a Demonſtration, after that
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              we have explained ſome things that pertaine thereto.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1223"/>
              C</s>
            </p>
            <p type="head">
              <s>LEMMA I.</s>
            </p>
            <p type="main">
              <s>Let the Lines A B and A C contain the Angle B A C; and from
                <lb/>
              the point D, taken in the Line A C, draw D E and D F at
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              pleaſure unto A B: and in the ſame Line any Points G and L
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              being taken, draw G H & L M parallel to D E, & G K and
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              L N parallel unto F D: Then from the Points D & G as farre
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              as to the Line M L draw D O P, cutting G H in O, and G Q
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              parallel unto B A. </s>
              <s>I ſay that the Lines that lye betwixt the Pa­
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              rallels unto F D have unto thoſe that lye betwixt the Par­
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              allels unto D E (namely K N to G Q or to O P; F K to D O;
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              and F N to D P) the ſame mutuall proportion: that is to ſay,
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              the ſame that A F hath to A E.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For in regard that the Triangles A F D, A K G, and A N L
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                <lb/>
                <figure id="id.073.01.035.1.jpg" xlink:href="073/01/035/1.jpg" number="31"/>
                <lb/>
                <emph type="italics"/>
              are alike, and E F D, H K G, and M N L are alſo alike: There-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1224"/>
                <lb/>
                <emph type="italics"/>
              fore,
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              as A F is to F D, ſo ſhall A K be to K G; and as F D is to
                <lb/>
              F E, ſo ſhall K G be to K H: Wherefore,
                <emph.end type="italics"/>
              ex equali,
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              as A F is to F
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              E, ſo ſhall A K be to K H: And, by Converſion of proportion, as
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              A F is to A E, ſo ſhall A K be to K H. </s>
              <s>It is in the ſame manner
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              proved that, as A F is to A E, ſo ſhall A N be to A M. </s>
              <s>Now A
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1225"/>
                <lb/>
                <emph type="italics"/>
              N being to A M, as A K is to A H; The
                <emph.end type="italics"/>
              (b)
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              Remainder K N ſhall
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              be unto the Remainder H M, that is unto G Q, or unto O P, as
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              A N is to A M; that is, as A F is to A E: Again, A K is to
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              A H, as A F is to A E; Therefore the Remainder F K ſhall be to
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              the Remainder E H, namely to D O, as A F is to A E. </s>
              <s>We might in
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              like manner demonstrate that ſo is F N to D P: Which is that that
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              was required to be demonstrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1224"/>
              (a)
                <emph type="italics"/>
              By 4. of the
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              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1225"/>
              (b)
                <emph type="italics"/>
              By 5. of the
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              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA II.</s>
            </p>
            <p type="main">
              <s>In the ſame Line A B let there be two Points R and S, ſo diſpo­
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              ſed, that A S may have the ſame Proportion to A R that
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              A F hath to A E; and thorow R draw R T parallel to E D,
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              and thorow S draw S T parallel to F D, ſo, as that it may
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              meet with R T in the Point T. </s>
              <s>I ſay that the Point T fall­
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              eth in the Line A C.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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