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

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        <div xml:id="echoid-div231" type="section" level="1" n="115">
          <p>
            <s xml:id="echoid-s4234" xml:space="preserve">
              <pb o="474" file="0194" n="204" rhead="HUGENII EXCEPTIO"/>
            & </s>
            <s xml:id="echoid-s4235" xml:space="preserve">in ſequentibus dicam de Circulo debet intelligi pariter de
              <lb/>
            ſectore Circuli.</s>
            <s xml:id="echoid-s4236" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4237" xml:space="preserve">Præter hanc approximationem D. </s>
            <s xml:id="echoid-s4238" xml:space="preserve">Gregorius aliam pro-
              <lb/>
            ponit in fine ſuæ
              <emph style="sc">XXV</emph>
            Propoſitionis, quam admirandam di-
              <lb/>
            cit, cujus demonſtrationem ſe ignorare fatetur; </s>
            <s xml:id="echoid-s4239" xml:space="preserve">hæc eſt, in-
              <lb/>
            ter duos terminos, ſtatim memoratos {1/3} a + {2/3} d & </s>
            <s xml:id="echoid-s4240" xml:space="preserve">{4/3} c - {1/3} a,
              <lb/>
            inventis quatuor mediis quantitatibus in proportione Arith-
              <lb/>
            metica, aſſerit maximam harum quantitatum adeo Circuli
              <lb/>
            magnitudini vicinam eſſe, ut, ſi in numeris, qui deſignant
              <lb/>
            Polygona ſimilia a & </s>
            <s xml:id="echoid-s4241" xml:space="preserve">d, prima notarum triens ſit eadem,
              <lb/>
            error ad unitatem non pertingat.</s>
            <s xml:id="echoid-s4242" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4243" xml:space="preserve">Sed invenio hanc approximationem in Circulo veram non
              <lb/>
            eſſe licet in Hyperbola locum habeat, &</s>
            <s xml:id="echoid-s4244" xml:space="preserve">, dum in hac utimur
              <lb/>
            maxima quatuor mediarum Arithmeticarum proportiona-
              <lb/>
            lium, minimam pro approximatione Circuli adhibendam
              <lb/>
            eſſe.</s>
            <s xml:id="echoid-s4245" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4246" xml:space="preserve">Ita minima quatuor mediarum proportionalium inter
              <lb/>
            terminos dictos primæ approximationis erit {16 c + 2 d - 3 a/15}, uti
              <lb/>
            facile eſt videre per Calculum; </s>
            <s xml:id="echoid-s4247" xml:space="preserve">& </s>
            <s xml:id="echoid-s4248" xml:space="preserve">probare poſſum non ſo-
              <lb/>
            lum experientiâ, ſed & </s>
            <s xml:id="echoid-s4249" xml:space="preserve">per demonſtrationem quantitatem
              <lb/>
            hanc, poſitis numeris quorum prima notarum triens eadem
              <lb/>
            eſt, ex primentibus polygonis a & </s>
            <s xml:id="echoid-s4250" xml:space="preserve">d, a vera Circuli ma-
              <lb/>
            gnitudine non aberrare niſi in duabus ultimis notis, & </s>
            <s xml:id="echoid-s4251" xml:space="preserve">ple-
              <lb/>
            rumque in omnibus notis & </s>
            <s xml:id="echoid-s4252" xml:space="preserve">ulterius cum vera magnitudi-
              <lb/>
            ne conincidere, quam tamen ſemper ſuperat, cum e contra-
              <lb/>
            rio maxima quatuor Mediarum, qua utitur D
              <emph style="super">us</emph>
            Gregorius in
              <lb/>
            Hyperbola deficiat.</s>
            <s xml:id="echoid-s4253" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4254" xml:space="preserve">Inveni præterea, approximationem hanc pro Circulo non
              <lb/>
            æque accuratam eſſe, ac eſt illa quam dedi in Tractatu deCircu-
              <lb/>
            li magnitudine, juxta quam, quando a, c & </s>
            <s xml:id="echoid-s4255" xml:space="preserve">d deſignant eadem
              <lb/>
            Polygona, ac ſupra, terminus excedens contentum Circuli
              <lb/>
            eſt a + {10cc - 10aa /6c + 9a} Neque demonſtratio difficilis eſt,
              <note symbol="* it" position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve">vide ſupra
                <lb/>
              p. 383. in
                <lb/>
              ſ
                <unsure/>
              ine.</note>
            ſi neges illum terminum eſſe minorem ideoque magis </s>
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