Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica
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              <pb o="338" file="0044" n="47" rhead="ΕΞΕΤΑΣΙΣ CYCLOM."/>
            tro Π P. </s>
            <s xml:id="echoid-s798" xml:space="preserve">Ducto deinde plano per A Π P, & </s>
            <s xml:id="echoid-s799" xml:space="preserve">alio huic pa-
              <lb/>
            rallelo D Φ S ſecundum lineam Φ S, erit jam pars ungulæ
              <lb/>
            hiſce duobus planis terminata, æqualis ſolido quod fit ex du-
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            ctu plani S Φ Π P in ſe ipſum; </s>
            <s xml:id="echoid-s800" xml:space="preserve">& </s>
            <s xml:id="echoid-s801" xml:space="preserve">pars ungulæ E D S Φ, æ-
              <lb/>
            qualis ei ſolido quod fit ex ductu plani E Φ S in ſe ipſum.
              <lb/>
            </s>
            <s xml:id="echoid-s802" xml:space="preserve">Quare nunc demonſtrandum erit duntaxat, partem E D S Φ
              <lb/>
            eſſe ad partem Φ A P ut 53 ad 203, Sit Φ N parallela E P,
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            & </s>
            <s xml:id="echoid-s803" xml:space="preserve">N C parallela Π A. </s>
            <s xml:id="echoid-s804" xml:space="preserve">Ergo quoniam ex proprietate Para-
              <lb/>
            boles, P N eſt {3/4} Π P, erit quoque P C {3/4} A P. </s>
            <s xml:id="echoid-s805" xml:space="preserve">Verùm & </s>
            <s xml:id="echoid-s806" xml:space="preserve">
              <lb/>
            S D æquatur {3/4} A P, quum ſit huic parallela , ſitque
              <note symbol="*" position="left" xlink:label="note-0044-01" xlink:href="note-0044-01a" xml:space="preserve">11. 16.
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              Elem.</note>
            rabola E A F: </s>
            <s xml:id="echoid-s807" xml:space="preserve">Itaque junctâ C D, ea parallela & </s>
            <s xml:id="echoid-s808" xml:space="preserve">æqualis
              <lb/>
            erit lineis P S, N Φ. </s>
            <s xml:id="echoid-s809" xml:space="preserve">Ducatur ſecundùm D C planum
              <lb/>
            D B C parallelum baſi E Π F, fietque ſemiparabola B D C
              <lb/>
            æqualis & </s>
            <s xml:id="echoid-s810" xml:space="preserve">ſimilis ſemiparabolæ Π Φ N; </s>
            <s xml:id="echoid-s811" xml:space="preserve">& </s>
            <s xml:id="echoid-s812" xml:space="preserve">erit Φ B N di-
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            midius cylindrus parabolicus: </s>
            <s xml:id="echoid-s813" xml:space="preserve">D A C B verò dimidiata un-
              <lb/>
            gula. </s>
            <s xml:id="echoid-s814" xml:space="preserve">Hæc autem æquatur ſicut antea oſtendimus, duabus
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            quintis cylindri dimidiati, baſin habentis D B C & </s>
            <s xml:id="echoid-s815" xml:space="preserve">altitudi-
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            nem B A. </s>
            <s xml:id="echoid-s816" xml:space="preserve">Ergo quum ſemicylindrus Φ B N habeat altitu-
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            dinem B Π triplam ipſius B A, erit ungula dimid. </s>
            <s xml:id="echoid-s817" xml:space="preserve">D A C B
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            ad ſemicyl. </s>
            <s xml:id="echoid-s818" xml:space="preserve">Φ B N, ut 2 ad 15, hoc eſt, ut 8 ad 60.</s>
            <s xml:id="echoid-s819" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s820" xml:space="preserve">Junctâ porro Φ Π, conſtat ſemiparabolam Π Φ N ad trian-
              <lb/>
            gulum Π Φ N eſſe ut 4 ad@ 3; </s>
            <s xml:id="echoid-s821" xml:space="preserve">ſed triangulus Π Φ N eſt ad
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            rectangulum Φ P ut 1 ad 6, (eſt enim baſis Π N tertia pars
              <lb/>
            ipſius N P) hoc eſt, ut 3 ad 18. </s>
            <s xml:id="echoid-s822" xml:space="preserve">Ergo ex æquo erit ſemi-
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            parab. </s>
            <s xml:id="echoid-s823" xml:space="preserve">Π Φ N ad rectang. </s>
            <s xml:id="echoid-s824" xml:space="preserve">Φ P ut 4 ad 18. </s>
            <s xml:id="echoid-s825" xml:space="preserve">Itaque & </s>
            <s xml:id="echoid-s826" xml:space="preserve">ſemi-
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            cylindrus Φ B N eſt ad parallelepipedum ejuſdem altitudinis
              <lb/>
            ſuper baſi Φ P, ut 4 ad 18. </s>
            <s xml:id="echoid-s827" xml:space="preserve">Dictiautem parallelepipedi di-
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            midium eſt priſma D N S; </s>
            <s xml:id="echoid-s828" xml:space="preserve">ergo ſemicylindrus Φ B N eſt ad
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            priſma D N S, ut 4 ad 9, hoc eſt, ut 60 ut 135. </s>
            <s xml:id="echoid-s829" xml:space="preserve">Qua-
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            lium igitur partium dimidiata ungula D A C B erat 8, ta-
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            lium ſemicylindrus parab. </s>
            <s xml:id="echoid-s830" xml:space="preserve">Φ B N erat 60, (ut ſuprà oſten-
              <lb/>
            ſum eſt) taliumque priſma D N S erit 135. </s>
            <s xml:id="echoid-s831" xml:space="preserve">Ac proinde ſo-
              <lb/>
            lidum A Π S D, quod ex iſtis tribus componitur, erit 203.
              <lb/>
            </s>
            <s xml:id="echoid-s832" xml:space="preserve">Eſt autem ungula dimidiata A D C B ad dimidiatam un-
              <lb/>
            gulam E A P Π, ut 1 ad 32, ſicut Cl. </s>
            <s xml:id="echoid-s833" xml:space="preserve">Vir. </s>
            <s xml:id="echoid-s834" xml:space="preserve">demonſtravit in
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            prop. </s>
            <s xml:id="echoid-s835" xml:space="preserve">95. </s>
            <s xml:id="echoid-s836" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s837" xml:space="preserve">9. </s>
            <s xml:id="echoid-s838" xml:space="preserve">Ergo qualium partium ungula dimid.</s>
            <s xml:id="echoid-s839" xml:space="preserve"/>
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