Archimedes, Natation of bodies, 1662

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    <archimedes>
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              <s>
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              the Portion hath to the Liquid of equall Maſſe, the ſame hath the
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              Magnitude of the Portion ſubmerged unto the whole Portion; as
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              hath been demonſtrated in the firſt Propoſition; The Magnitude
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              ſubmerged, therefore, ſhall not have greater proportion to the
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              whole
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              (b)
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              Portion, than that which hath been mentioned: ^{*}And
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              therefore the whole Portion hath not greater proportion unto that
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              which is above the Liquid, than the Square N O hath to the Square
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              M O: But the
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              (c)
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              whole Portion hath the ſame proportion unto
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              that which is above the Liquid that the Square N O hath to the
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              Square P F: Therefore the Square N O hath not greater propor­
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              tion unto the Square P F, than it hath unto the Square M O: ^{*}And
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              hence it followeth that P F is not leſſe than O M, nor P B than O
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              H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O
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              ſhall meet with B P betwixt P and B: Let it be in T: And be­
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              cauſe that in the Section of the Rectangled Cone P F is parallel unto
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              the Diameter N O; and H T perpendicular unto the ſaid Diame­
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              ter; and R H equall to the Semi-parameter: It is manifeſt that
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              R T prolonged doth make Right Angles with K P
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              : And there­
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              fore doth alſo make Right Angles with I S: Therefore R T is per­
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              pendicular unto the Surface of the Liquid; And if thorow the
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              Points B and G Lines be drawn parallel unto R T, they ſhall be
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              perpendicular unto the Liquids Surface. </s>
              <s>The Portion, therefore,
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              which is above the Liquid ſhall move downwards in the Liquid ac­
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              cording to the Perpendicular drawn thorow B; and that part
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              which is within the Liquid ſhall move upwards according to the
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              Perpendicular drawn thorow G; and the Solid Portion A P O L
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              ſhall not continue ſo inclined, [
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              as it was at its demerſion
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              ], but ſhall
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              move within the Liquid untill ſuch time that N O do ſtand accor­
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              ding to the Perpendicular.</s>
            </p>
            <p type="margin">
              <s>
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              (a)
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              In 4. Prop. of
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              this.
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              </s>
            </p>
            <p type="margin">
              <s>
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              (a)
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              By 11. of the
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              fifth.
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              </s>
            </p>
            <p type="margin">
              <s>
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              A</s>
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            <p type="margin">
              <s>
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              (b)
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              By 26. of the
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              Book
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              De Conoid.
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              </s>
              <s>& Sphæroid.</s>
            </p>
            <p type="margin">
              <s>
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              B</s>
            </p>
            <p type="margin">
              <s>
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              C</s>
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            <p type="head">
              <s>COMMANDINE.
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              </s>
            </p>
            <p type="margin">
              <s>
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              A</s>
            </p>
            <p type="main">
              <s>And therefore the whole Portion hath not greater proportion
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              unto that which is above the Liquid, than the Square N O hath to
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              the Square M O.]
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              For in regard that the Magnitude of the Portion demerged
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              within the Liquid hath not greater proportion unto the whole Portion than the Exceſſe by which
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              the Square N O is greater than the Square M O hath to the ſaid Square N O; Converting of
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              the Proportion, by the 26. of the fifth of
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              Euclid,
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              of
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              Campanus
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              his Tranſlation, the whole
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              Portion ſhall not have leſſer proportion unto the Magnitude ſubmerged, than the Square N O
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              hath unto the Exceſſe by which N O is greater than the Square M O. </s>
              <s>Let a Portion be taken;
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              and let that part of it which is above the Liquid be the firſt Magnitude; the part of it which
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              is ſubmerged the ſecond: and let the third Magnitude be the Square M O; and let the Exceſſe
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              by which the Square N O is greater than the Square M O be the fourth. </s>
              <s>Now of theſe Mag­
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              nitudes, the proportion of the firſt and ſecond, unto the ſecond, is not leſſe than that of the third &
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              fourth unto the fourth: For the Square M O together with the Exceſſe by which the Square
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              N O exceedeth the Square M O is equall unto the ſaid Square N O: Wherefore, by Converſi­
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              on of Proportion, by 30 of the ſaid fifth Book, the proportion of the firſt and ſecond unto the
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              firſt, ſhall not be greater than that of the third and fourth unto the third: And, for the ſame
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              </s>
            </p>
          </chap>
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      </text>
    </archimedes>
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