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

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        <div xml:id="echoid-div190" type="section" level="1" n="91">
          <pb o="441" file="0159" n="168" rhead="ET HYPERBOLÆ QUADRATURA."/>
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        <div xml:id="echoid-div192" type="section" level="1" n="92">
          <head xml:id="echoid-head128" xml:space="preserve">SCHOLIUM.</head>
          <p>
            <s xml:id="echoid-s3554" xml:space="preserve">NOn opus eſt ut hic demonſtrem majorem duarum me-
              <lb/>
            diarum arithmeticè continuè proportionalium inter duas
              <lb/>
            inæquales quantitates majorem eſſe quam major duarum me-
              <lb/>
            diarum Geometricè continuè proportionalium inter eaſdem,
              <lb/>
            & </s>
            <s xml:id="echoid-s3555" xml:space="preserve">igitur hujus propoſitionis approximationem præcedentis
              <lb/>
            eſſe exactiorem, quod etſi fiat; </s>
            <s xml:id="echoid-s3556" xml:space="preserve">præcedente tamen ob facilita-
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            tem potius utimur.</s>
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        <div xml:id="echoid-div193" type="section" level="1" n="93">
          <head xml:id="echoid-head129" xml:space="preserve">PROP. XXIII. THEOREMA.</head>
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            <s xml:id="echoid-s3558" xml:space="preserve">Sint duo polygona complicata
              <lb/>
              <note position="right" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve">
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              A B # A
                <lb/>
              C D # C
                <lb/>
              E F # G
                <lb/>
              K L # H
                <lb/>
              Z # X
                <lb/>
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            A, B, nempè A extra hyperbolæ
              <lb/>
            ſectorem, B intra. </s>
            <s xml:id="echoid-s3559" xml:space="preserve">continuetur ſe-
              <lb/>
            ries convergens horum polygono-
              <lb/>
            rum complicatorum ſecundum me-
              <lb/>
            thodum noſtram ſubduplam de-
              <lb/>
            ſcriptorum, ita ut polygona extra
              <lb/>
            hyperbolam ſint A, C, E, K, &</s>
            <s xml:id="echoid-s3560" xml:space="preserve">c, & </s>
            <s xml:id="echoid-s3561" xml:space="preserve">intra hyperbolam B,
              <lb/>
            D, F, L, &</s>
            <s xml:id="echoid-s3562" xml:space="preserve">c; </s>
            <s xml:id="echoid-s3563" xml:space="preserve">Sitque ſeriei convergentis terminatio ſeu hy-
              <lb/>
            perbolæ ſector Z. </s>
            <s xml:id="echoid-s3564" xml:space="preserve">dico Z majorem eſſe quam C dempto tri-
              <lb/>
            ente exceſſus A ſupra C. </s>
            <s xml:id="echoid-s3565" xml:space="preserve">ſit exceſſus C ſupra G quarta pars
              <lb/>
            exceſſus A ſupra C, & </s>
            <s xml:id="echoid-s3566" xml:space="preserve">exceſſus G ſupra H quarta pars ex-
              <lb/>
            ceſſus C ſupra G, continueturque hæc ſeries in infinitum ut
              <lb/>
            ejus terminatio ſit X. </s>
            <s xml:id="echoid-s3567" xml:space="preserve">exceſſus A ſupra C major eſt quadru-
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            plo exceſſus C ſupra E, & </s>
            <s xml:id="echoid-s3568" xml:space="preserve">ideo exceſſus C ſupra E minor
              <lb/>
            eſt exceſſus C ſupra G, eſt ergo E major quam G. </s>
            <s xml:id="echoid-s3569" xml:space="preserve">Deinde
              <lb/>
            exceſſus C ſupra E major eſt quadruplo exceſſus E ſupra K,
              <lb/>
            & </s>
            <s xml:id="echoid-s3570" xml:space="preserve">ideo exceſſus C ſupra G multò major eſt quadruplo ex-
              <lb/>
            ceſſu E ſupra K, & </s>
            <s xml:id="echoid-s3571" xml:space="preserve">igitur exceſſus G ſupra H major eſt
              <lb/>
            exceſſu E ſupra K; </s>
            <s xml:id="echoid-s3572" xml:space="preserve">cumque E major ſit quam G, ma-
              <lb/>
            nifeſtum eſt K etiam majorem eſſe quam H: </s>
            <s xml:id="echoid-s3573" xml:space="preserve">eodem pror-
              <lb/>
            ſus modo demonſtratur in omni ſerierum A, C, E, K; </s>
            <s xml:id="echoid-s3574" xml:space="preserve">A,
              <lb/>
            C, G, H, continuatione, terminum quemcumque ſeriei A,
              <lb/>
            C, E, majorem eſſe quam idem numero terminus ſeriei </s>
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