Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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          <pb o="381" file="0093" n="99" rhead="DE CIRCULI MAGNIT. INVENTA."/>
          <p>
            <s xml:id="echoid-s1809" xml:space="preserve">SIt portio ſemicirculo minor, cui inſcriptum triangulum
              <lb/>
              <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">TAB. XL.
                <lb/>
              Fig. 4.</note>
            maximum A B C. </s>
            <s xml:id="echoid-s1810" xml:space="preserve">Diameter autem portionis ſit B D; </s>
            <s xml:id="echoid-s1811" xml:space="preserve">& </s>
            <s xml:id="echoid-s1812" xml:space="preserve">
              <lb/>
            diameter circuli à quo portio reſecta eſt, B F, centrum E.
              <lb/>
            </s>
            <s xml:id="echoid-s1813" xml:space="preserve">Oſtendendum eſt primo, portionis A B C ad triangulum in-
              <lb/>
            ſcriptum majorem eſſe rationem quam ſeſquitertiam. </s>
            <s xml:id="echoid-s1814" xml:space="preserve">Eſto
              <lb/>
            portionis A B C centrum grav. </s>
            <s xml:id="echoid-s1815" xml:space="preserve">punctum G, & </s>
            <s xml:id="echoid-s1816" xml:space="preserve">ſecetur D F
              <lb/>
            in H, ut ſit H D dupla reliquæ H F.</s>
            <s xml:id="echoid-s1817" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1818" xml:space="preserve">Quoniam igitur F B eſt dupla E B; </s>
            <s xml:id="echoid-s1819" xml:space="preserve">D B autem minor
              <lb/>
            quam dupla G B. </s>
            <s xml:id="echoid-s1820" xml:space="preserve">Erit major ratio F B ad B D, quam E B
              <lb/>
            ad B G. </s>
            <s xml:id="echoid-s1821" xml:space="preserve">Et per converſionem rationis, minor B F ad F D,
              <lb/>
            quam B E ad E G. </s>
            <s xml:id="echoid-s1822" xml:space="preserve">Et permutando minor B F ad B E,
              <lb/>
            (quæ proportio dupla eſt) quam F D ad E G. </s>
            <s xml:id="echoid-s1823" xml:space="preserve">Igitur F D
              <lb/>
            major eſt quam dupla E G. </s>
            <s xml:id="echoid-s1824" xml:space="preserve">Ipſius autem F D duas tertias
              <lb/>
            continet H D. </s>
            <s xml:id="echoid-s1825" xml:space="preserve">Ergo H D major eſt quam ſeſquitertia E G.
              <lb/>
            </s>
            <s xml:id="echoid-s1826" xml:space="preserve">Sicut autem H D ad E G, ita eſt portio A B C ad inſcri-
              <lb/>
            ptum ſibi triangulum: </s>
            <s xml:id="echoid-s1827" xml:space="preserve">hoc enim antehac demonſtravimus in
              <lb/>
            Theorematis de Hyperboles Ellipſis & </s>
            <s xml:id="echoid-s1828" xml:space="preserve">Circuli quadratura .</s>
            <s xml:id="echoid-s1829" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">Vide ſupra
                <lb/>
              p. 324.</note>
            Itaque major eſt ratio portionis ad inſcriptum triangulum
              <lb/>
            A B C quam ſeſquitertia.</s>
            <s xml:id="echoid-s1830" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1831" xml:space="preserve">Quod autem ad triangulum A B C portio minorem ha-
              <lb/>
            beat rationem quam tripla ſeſquitertia ipſius D F ad diame-
              <lb/>
            trum circuli B F unà cum tripla E D, id nunc oſtendemus.
              <lb/>
            </s>
            <s xml:id="echoid-s1832" xml:space="preserve">Secetur diameter portionis in R, ut B R ſit ſeſquialtera re-
              <lb/>
            liquæ R D. </s>
            <s xml:id="echoid-s1833" xml:space="preserve">Ergo cadit R punctum inter G & </s>
            <s xml:id="echoid-s1834" xml:space="preserve">D
              <note symbol="*" position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">per præced.</note>
            niam poſitum fuit G centrum gravitatis in portione A B C.
              <lb/>
            </s>
            <s xml:id="echoid-s1835" xml:space="preserve">Quumque portionis ad inſcriptum triangulum eadem ſit ra-
              <lb/>
            tio, quæ H D ad E G, ut modo dictum fuit; </s>
            <s xml:id="echoid-s1836" xml:space="preserve">minor au-
              <lb/>
            tem ſit ratio H D ad E G, quam H D ad E R: </s>
            <s xml:id="echoid-s1837" xml:space="preserve">Erit
              <lb/>
            propterea minor quoque portionis ad inſcriptum triangulum
              <lb/>
            ratio quam H D ad E R, ſive quam H D quinquies ſum-
              <lb/>
            pta ad quintuplam E R. </s>
            <s xml:id="echoid-s1838" xml:space="preserve">Atqui H D, (cum ſit æqualis
              <lb/>
            duabus tertiis D F) quinquies ſumpta æquabitur decem ter-
              <lb/>
            tiis, hoc eſt, triplæ ſeſquitertiæ D F. </s>
            <s xml:id="echoid-s1839" xml:space="preserve">E R verò quæ con-
              <lb/>
            tinet E D & </s>
            <s xml:id="echoid-s1840" xml:space="preserve">duas quintas ipſius D B, ſi quinquies ſuma-
              <lb/>
            tur, æquabitur duplæ B D & </s>
            <s xml:id="echoid-s1841" xml:space="preserve">quintuplæ E D; </s>
            <s xml:id="echoid-s1842" xml:space="preserve">hoc eſt,
              <lb/>
            duplæ totius E B atque inſuper triplæ E D. </s>
            <s xml:id="echoid-s1843" xml:space="preserve">Igitur </s>
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