Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/050.jpg" pagenum="380"/>
              made of the Exceſs by which the Axis is greater than Seſquialter
                <lb/>
              of the Semi-parameter, hath to the Square made of the Axis, being
                <lb/>
              demitted into the Liquid, ſo as hath been ſaid, it ſhall ſtand erect,
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              or
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              erpendicular.</s>
            </p>
            <p type="margin">
              <s>
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              F</s>
            </p>
            <p type="margin">
              <s>
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              G</s>
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            <p type="margin">
              <s>
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              H</s>
            </p>
            <p type="margin">
              <s>
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              K</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1305"/>
              L</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1306"/>
              M</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1307"/>
              N</s>
            </p>
            <p type="margin">
              <s>
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              O</s>
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            <p type="margin">
              <s>
                <margin.target id="marg1309"/>
              P</s>
            </p>
            <p type="margin">
              <s>
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              Q</s>
            </p>
            <p type="margin">
              <s>
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              R</s>
            </p>
            <p type="head">
              <s>COMMANDINE.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The particulars contained in this Tenth Propoſition, are divided by
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              Archimedes
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                <emph type="italics"/>
              into five Parts and Concluſions, each of which he proveth by a diſtinct Demonſtration.
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                <lb/>
                <arrow.to.target n="marg1312"/>
              </s>
            </p>
            <p type="margin">
              <s>
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              A</s>
            </p>
            <p type="main">
              <s>It ſhall ſometimes ſtand perpendicular.]
                <emph type="italics"/>
              This is the firſt Concluſion, the
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              Demonstration of which he hath ſubjoyned to the Propoſition.
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              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1313"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1313"/>
              B</s>
            </p>
            <p type="main">
              <s>And ſometimes ſo inclined, as that its Baſe touch the Surface
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              of the Liquid, in one Point only.]
                <emph type="italics"/>
              This is demonſtrated in the third Con­
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              cluſion.
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              </s>
            </p>
            <p type="main">
              <s>Sometimes, ſo that its Baſe be moſt ſubmerged in the Liquid.] </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1314"/>
                <lb/>
                <emph type="italics"/>
              This pertaineth unto the fourth Concluſion.
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              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1314"/>
              C</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              nd, ſometimes, ſo as that it doth not in the leaſt touch the Sur­
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                <arrow.to.target n="marg1315"/>
                <lb/>
              face of the Liquid.]
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              This it doth hold true two wayes, one of which is explained is
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              the ſecond, and the other in the fifth Concluſion.
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              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1315"/>
              D</s>
            </p>
            <p type="main">
              <s>According to the proportion, that it hath to the Liquid in Gra­
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                <arrow.to.target n="marg1316"/>
                <lb/>
              vity. </s>
              <s>Every one of which Caſes ſhall be anon demonſtrated.]
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                <emph type="italics"/>
              In
                <emph.end type="italics"/>
              Tartaglia's
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              Verſion it is rendered, to the confuſion of the ſence,
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              Quam autem pro­
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              portionem habeant ad humidum in Gravitate fingula horum demonſtrabuntur.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1316"/>
              E</s>
            </p>
            <p type="main">
              <s>It is manifeſt, therefore, that K C is greater than the Semi­
                <lb/>
                <arrow.to.target n="marg1317"/>
                <lb/>
              parameter]
                <emph type="italics"/>
              For, ſince B D hath to K C the ſame proportion, as fifteen to four, and
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              hath unto the Semi-parameter greater proportion; (a) the Semi-parameter ſhall be leſs
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1318"/>
                <lb/>
                <emph type="italics"/>
              than K C.
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              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1317"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1318"/>
              (a)
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              By 10. of the
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              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let the Semi-parameter be equall to KR.]
                <emph type="italics"/>
              We have added theſe words,
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1319"/>
                <lb/>
                <emph type="italics"/>
              which are not to be found in
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              Tartaglia.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1319"/>
              G</s>
            </p>
            <p type="main">
              <s>But S B is alſo Seſquialter of BR.]
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              For, D B is ſuppoſed Seſquialter of
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1320"/>
                <lb/>
                <emph type="italics"/>
              B K; and D S alſo is Seſquialter of K R: Wherefore as
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              (b)
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              the whole D B, is to the whole
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              B K, ſo is the part D S to the part K R. Therefore, the Remainder S B, is alſo to the
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1321"/>
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                <emph type="italics"/>
              Remainder B R, as D B is to B K.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1320"/>
              H</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1321"/>
              (b)
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              By 19 of the
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              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              A
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              nd let them be like to the
                <emph type="italics"/>
              P
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              ortion
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              A B L.
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              ] Apollonius
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              thus defineth
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                <lb/>
                <arrow.to.target n="marg1322"/>
                <lb/>
                <emph type="italics"/>
              like Portions of the Sections of a Cone, in
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              Lib. 6. Conicornm,
                <emph type="italics"/>
              as
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              Eutocius
                <emph type="italics"/>
              writeth
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              ^{*};
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                <arrow.to.target n="marg1323"/>
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                <foreign lang="grc">ὄν οἱ̄ς ἀχ δεισω̄ν ὄν ἑχάσῳ ωαραλλήλων τη̄ <35>ὰσει, ἵσων τὸ πλη̄ο<34>, αἱ παράλληλος, καὶ ἁι <35>άσεις ωρὸς τάς αποτρμ
                  <gap/>
                  <lb/>
                νομένας ἀπὸ
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                διαμέτσων τω̄ς κορυφαῑς ἐν τοῑς ἀντοῑς λόγοις εἰσι, καὶ αἱ ἀποτεμνόμεναι ωρὸς τὰς ἀ τεμνομίνασ
                  <gap/>
                </foreign>
                <lb/>
                <emph type="italics"/>
              that is,
                <emph.end type="italics"/>
              In both of which an equall number of Lines being drawn parallel to the
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              Baſe; the parallel and the Baſes have to the parts of the Diameters, cut off from
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              the Vertex, the ſameproportion: as alſo, the parts cut off, to the parts cut off.
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                <emph type="italics"/>
              Now the Lines parallel to the Baſes are drawn, as I ſuppoſe, by making a Rectilineall Figure (cal-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1324"/>
                <lb/>
                <emph type="italics"/>
              led)
                <emph.end type="italics"/>
              Signally inſcribed [
                <foreign lang="grc">χη̄μα γιωρίμως ἐγν̀<36>ρόμενον</foreign>
              ]
                <emph type="italics"/>
              in both portions, having an equall num­
                <lb/>
              ber of Sides in both. </s>
              <s>Therefore, like Portions are cut off from like Sections of a Cone; and
                <lb/>
              their Diameters, whether they be perpendicular to their Baſes, or making equall Angles with their
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              Baſes, have the ſame proportion unto their Baſes.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1325"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1322"/>
              K</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1323"/>
              *
                <emph type="italics"/>
              Upon prop. 3 lib.
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              2
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              Archim.
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              Æqui­
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              pond.
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              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1324"/>
                <emph type="italics"/>
              Vide
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              Archim,
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              ante
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              prop. 2. lib. 2.
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              Æquipond.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1325"/>
              L</s>
            </p>
            <p type="main">
              <s>Now the Section of the Cone
                <emph type="italics"/>
              A E I
                <emph.end type="italics"/>
              ſhall paſs thorow K.]
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                <emph type="italics"/>
              For, if it be poſſible, let it not paſs thorow K, but thorow ſome other Point of the Line D B, as
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              thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, whoſe
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              Diameter is E Z, A E is drawn and prolonged; and D B parallel unto the Diameter, cutteth
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              both A E and A I; A E in B, and A I in D; D B ſhall have to B V, the ſame proportion
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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