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

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          <p>
            <s xml:id="echoid-s4097" xml:space="preserve">Sit A. </s>
            <s xml:id="echoid-s4098" xml:space="preserve">Polygonum regulare ſectori inſcriptum. </s>
            <s xml:id="echoid-s4099" xml:space="preserve">B eidem
              <lb/>
            ſimile circumſcriptum; </s>
            <s xml:id="echoid-s4100" xml:space="preserve">continetur ſeries convergens poly-
              <lb/>
            gonorum &</s>
            <s xml:id="echoid-s4101" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4102" xml:space="preserve">ut ſit ejus terminatio ſeu circuli ſector Z: </s>
            <s xml:id="echoid-s4103" xml:space="preserve">ſit
              <lb/>
            X eodem modo compoſita à terminis C, D, quo Z à ter-
              <lb/>
            minis A, B; </s>
            <s xml:id="echoid-s4104" xml:space="preserve">dico Z & </s>
            <s xml:id="echoid-s4105" xml:space="preserve">X eſſe indefinitè æquales; </s>
            <s xml:id="echoid-s4106" xml:space="preserve">ſi non ſint
              <lb/>
            indefinitè æquales, ſit inter illas indefinita differentia a, & </s>
            <s xml:id="echoid-s4107" xml:space="preserve">
              <lb/>
            continuetur ſeries convergens in terminos convergentes I, K,
              <lb/>
            ita ut eorum differentia ſit minor quam a; </s>
            <s xml:id="echoid-s4108" xml:space="preserve">hoc
              <lb/>
              <note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve">
                <lb/>
              A # B
                <lb/>
              C # D
                <lb/>
              E # F
                <lb/>
              G # H a
                <lb/>
              I # K
                <lb/>
              L # M
                <lb/>
              # Z
                <lb/>
              # X
                <lb/>
              </note>
            enim abſque dubio concipi poteſt, etiamſi hic
              <lb/>
            omnes quantitates ſint indefinitæ, quoniam
              <lb/>
            definitis quantitatibus A, B, definitur etiam a,
              <lb/>
            ſed adhuc reſtat K-1 quantitas indeterminata
              <lb/>
            in infinitum decreſcens. </s>
            <s xml:id="echoid-s4109" xml:space="preserve">Manifeſtum eſt, ſe-
              <lb/>
            ctorem Z eſſe indefinitè minorem quam K, & </s>
            <s xml:id="echoid-s4110" xml:space="preserve">
              <lb/>
            majorem quam I: </s>
            <s xml:id="echoid-s4111" xml:space="preserve">item quoniam Zeodem mo-
              <lb/>
            do componitur ex quantitatibus A, B, quo X. </s>
            <s xml:id="echoid-s4112" xml:space="preserve">è quantita-
              <lb/>
            tibus C, D, & </s>
            <s xml:id="echoid-s4113" xml:space="preserve">Z indefinitè minor eſt quam K & </s>
            <s xml:id="echoid-s4114" xml:space="preserve">major
              <lb/>
            quam I, patet ex Proprietatibus ſerierum convergentium,
              <lb/>
            X etiam eſſe indefinitè majorem quàm I, & </s>
            <s xml:id="echoid-s4115" xml:space="preserve">minorem quàm
              <lb/>
            K (eſt enim revera indefinitè major quàm L & </s>
            <s xml:id="echoid-s4116" xml:space="preserve">minor quam
              <lb/>
            M) & </s>
            <s xml:id="echoid-s4117" xml:space="preserve">proinde ſunt quatuor quantitates indefinitæ, quarum
              <lb/>
            maxima & </s>
            <s xml:id="echoid-s4118" xml:space="preserve">minima ſunt I, K, intermediæ autem Z & </s>
            <s xml:id="echoid-s4119" xml:space="preserve">X,
              <lb/>
            & </s>
            <s xml:id="echoid-s4120" xml:space="preserve">ideo differentia extremarum K-I major eſt quàm a diffe-
              <lb/>
            rentia mediarum, quod eſt abſurdum, ponitur enim minor:
              <lb/>
            </s>
            <s xml:id="echoid-s4121" xml:space="preserve">quantitates ergò Z & </s>
            <s xml:id="echoid-s4122" xml:space="preserve">X non ſunt indefinitè inæquales, & </s>
            <s xml:id="echoid-s4123" xml:space="preserve">
              <lb/>
            ideo ſunt indefinitè æquales, quod demonſtrandum erat. </s>
            <s xml:id="echoid-s4124" xml:space="preserve">
              <lb/>
            Manifeſtum eſt hanc demonſtrationem eodem modo appli-
              <lb/>
            cabilem eſſe omni ſeriei convergenti.</s>
            <s xml:id="echoid-s4125" xml:space="preserve"/>
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            <s xml:id="echoid-s4126" xml:space="preserve">In objectionibus 2, 3, & </s>
            <s xml:id="echoid-s4127" xml:space="preserve">4, contra ſuas ipſius imaginatio-
              <lb/>
            nes argumentatur Hugenius: </s>
            <s xml:id="echoid-s4128" xml:space="preserve">Ego enim ſatis dilucidè affir-
              <lb/>
            mo in Scholio propoſit. </s>
            <s xml:id="echoid-s4129" xml:space="preserve">5. </s>
            <s xml:id="echoid-s4130" xml:space="preserve">& </s>
            <s xml:id="echoid-s4131" xml:space="preserve">in fine prop. </s>
            <s xml:id="echoid-s4132" xml:space="preserve">9. </s>
            <s xml:id="echoid-s4133" xml:space="preserve">Septimam & </s>
            <s xml:id="echoid-s4134" xml:space="preserve">no-
              <lb/>
            nam propoſitionem eſſe Particularem, unamquamque ſuo ca-
              <lb/>
            ſui; </s>
            <s xml:id="echoid-s4135" xml:space="preserve">item in Prop. </s>
            <s xml:id="echoid-s4136" xml:space="preserve">decima (quàm ergo pro generali ſubſti-
              <lb/>
            tuo) evidenter ſuppono, & </s>
            <s xml:id="echoid-s4137" xml:space="preserve">non quæro, illam quantitatem
              <lb/>
            eo modo compoſitam ex primis, quo ex ſecundis terminis
              <lb/>
            convergentibus; </s>
            <s xml:id="echoid-s4138" xml:space="preserve">ſatis enim ſcio, talem methodum </s>
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