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

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            G H, nempe X; </s>
            <s xml:id="echoid-s3624" xml:space="preserve">atque ex hujus 7 terminatio ſeriei A B,
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            G H, nempe X, æqualis eſt minori duarum mediarum arith-
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            meticè continuè proportionalium inter A & </s>
            <s xml:id="echoid-s3625" xml:space="preserve">B, & </s>
            <s xml:id="echoid-s3626" xml:space="preserve">ideo Z ea-
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            dem minor eſt, quod demonſtrare oportuit.</s>
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        <div xml:id="echoid-div197" type="section" level="1" n="95">
          <head xml:id="echoid-head131" xml:space="preserve">PROP. XXV. THEOREMA.</head>
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            <s xml:id="echoid-s3628" xml:space="preserve">Iisdem poſitis; </s>
            <s xml:id="echoid-s3629" xml:space="preserve">dico Z ſeu ſectorem
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              A B # A B
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              C D # G H
                <lb/>
              E F # M N
                <lb/>
              K L # O P
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              Z # X
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              </note>
            hyperbolæ minorem eſſe quam mi-
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            nor duarum mediarum geometricè con-
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            tinuè proportionalium inter A & </s>
            <s xml:id="echoid-s3630" xml:space="preserve">B.
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            </s>
            <s xml:id="echoid-s3631" xml:space="preserve">Inter A & </s>
            <s xml:id="echoid-s3632" xml:space="preserve">B ſit media geometrica G,
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            & </s>
            <s xml:id="echoid-s3633" xml:space="preserve">inter G & </s>
            <s xml:id="echoid-s3634" xml:space="preserve">B media geometrica H; </s>
            <s xml:id="echoid-s3635" xml:space="preserve">
              <lb/>
            Item inter G & </s>
            <s xml:id="echoid-s3636" xml:space="preserve">H media geometrica M, & </s>
            <s xml:id="echoid-s3637" xml:space="preserve">inter M & </s>
            <s xml:id="echoid-s3638" xml:space="preserve">H media
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            geometriea N; </s>
            <s xml:id="echoid-s3639" xml:space="preserve">continueturque hæc ſeries convergens AB, GH,
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            MN, OP, &</s>
            <s xml:id="echoid-s3640" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3641" xml:space="preserve">in infinitum ut fiat ejus terminatio X. </s>
            <s xml:id="echoid-s3642" xml:space="preserve">ſatis patet
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            ex prædictis C & </s>
            <s xml:id="echoid-s3643" xml:space="preserve">G eſſe inter ſe æquales, & </s>
            <s xml:id="echoid-s3644" xml:space="preserve">H majorem eſſe
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            quam D; </s>
            <s xml:id="echoid-s3645" xml:space="preserve">atque ob hanc rationem M media geometrica inter G
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            & </s>
            <s xml:id="echoid-s3646" xml:space="preserve">H major eſt quam E media geometrica inter C & </s>
            <s xml:id="echoid-s3647" xml:space="preserve">D. </s>
            <s xml:id="echoid-s3648" xml:space="preserve">Deinde
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            N media geometrica inter M & </s>
            <s xml:id="echoid-s3649" xml:space="preserve">H major eſt media harmonica
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            inter eaſdem; </s>
            <s xml:id="echoid-s3650" xml:space="preserve">& </s>
            <s xml:id="echoid-s3651" xml:space="preserve">quoniam M major eſt quam E & </s>
            <s xml:id="echoid-s3652" xml:space="preserve">H quam D, erit
              <lb/>
            media harmonica inter M & </s>
            <s xml:id="echoid-s3653" xml:space="preserve">H major quam F media harmo-
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            nica inter E & </s>
            <s xml:id="echoid-s3654" xml:space="preserve">D; </s>
            <s xml:id="echoid-s3655" xml:space="preserve">proinde N media geometrica inter M & </s>
            <s xml:id="echoid-s3656" xml:space="preserve">H
              <lb/>
            major eritquam F. </s>
            <s xml:id="echoid-s3657" xml:space="preserve">eadem methodo utramque ſeriem in in-
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            finitum continuando, ſemper demonſtratur terminum quem-
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            libet ſeriei A B, C D, minorem eſſe quam idem numero ter-
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            minus ſeriei A B, G H; </s>
            <s xml:id="echoid-s3658" xml:space="preserve">& </s>
            <s xml:id="echoid-s3659" xml:space="preserve">igitur terminatio ſeriei A B, C D,
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            nempe Z minor erit quam terminatio ſeriei A B, G H, nem-
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            pe X; </s>
            <s xml:id="echoid-s3660" xml:space="preserve">atque ex hujus 9 terminatio ſeriei A B, G H, ſeu X, æqua-
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            lis eſt minori duarum mediarum geometricè continuè propor-
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            tionalium inter A & </s>
            <s xml:id="echoid-s3661" xml:space="preserve">B; </s>
            <s xml:id="echoid-s3662" xml:space="preserve">& </s>
            <s xml:id="echoid-s3663" xml:space="preserve">ideo Z eadem minor eſt, quod
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            demonſtrare oportuit.</s>
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            <s xml:id="echoid-s3665" xml:space="preserve">Ex dictis manifeſtum eſt hanc approximationem exactio-
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            rem eſſe illa, in antecedenti propoſitione, demonſtrata, et-
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            iamſi hæc ſit paulò laborioſior. </s>
            <s xml:id="echoid-s3666" xml:space="preserve">ſed non diſſimulandum
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            eſt duas poſſe eſſe ſeries æquales terminationes habentes, </s>
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