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

Table of contents

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[91.] PROP. XXII. THEOREMA.
[92.] SCHOLIUM.
[93.] PROP. XXIII. THEOREMA.
[94.] PROP. XXIV. THEOREMA.
[95.] PROP. XXV. THEOREMA.
[96.] PROP. XXVI. THEOREMA.
[97.] PROP. XXVII. THEOREMA.
[98.] PROP. XXVIII. THEOREMA.
[99.] PROP. XXIX. PROBLEMA. Dato circulo æquale invenire quadratum.
[100.] PROP. XXX. PROBLEMA. Ex dato ſinu invenire arcum.
[101.] PROP. XXXI. PROBLEMA. Ex dato arcu invenire ſinum.
[102.] PROP. XXXII. PROBLEMA. Invenire quadratum æquale ſpatio hyperbolico con-tento à curva hyperbolica, uno aſymptoto & dua-bus rectis alteri aſymptoto parallelis; quod ſpatium æquale eſt ſectori hyperbolico cujus baſis eſt eadem curva.
[103.] PROP. XXXIII. PROBLEMA. Propoſiti cujuscunque numeri logorithmum invenire.
[104.] SCHOLIUM.
[105.] PROP. XXXIV. PROBLEMA. Ex dato logorithmo invenire ejus numerum.
[106.] Tom. II. Mmm
[107.] PROP. XXXV. PROBLEMA. Rectâ per datum punctum in diametro ductâ, ſemicirculum in ratione data dividere.
[108.] SCHOLIUM.
[109.] FINIS.
[110.] II. HUGENII OBSERVATIONES IN LIBRUM JACOBI GREGORII, DE VERA CIRCULI ET HYPERBOLÆ QUADRATURA.
[111.] III. DOMINI GREGORII RESPONSUM AD ANIMADVERSIONES DOMINI HUGENII, IN EJUS LIBRUM, DE VERA CIRCULI ET HYPERBOLÆ QUADRATURA.
[112.] PROP. X. PROBLEMA.
[113.] Tom. II. Nnn
[114.] CONSECTARIUM.
[115.] IV. EXCERPTA EX LITERIS Dni. HUGENII DE RESPONSO, QUOD Dnus. GREGORIUS DEDIT AD EXAMEN LIBRI, CUI TITULUS EST, VERA CIRCULI ET HYPERBOLÆ QUADRATURA.
[116.] V. EXCERPTA EX EPISTOLA D. JACOBI GREGORII, CONTINENTE QUASDAM EJUS CONSIDERATIO-NES, SUPER EPISTOLA D. HUGENII, IMPRESSA IN VINDICATIONEM EXAMINIS SUI LIBRI, DE VERA CIRCULI ET HY-PERBOLÆ QUADRATURA.
[117.] FINIS.
[118.] CHRISTIANI HUGENII GEOMETRICA VARIA. Tom. II. Ppp
[119.] I. CONSTRUCTIO LOCI AD HYPERBOLAM PER ASYMPTOTOS.
[120.] DEMONSTRATIO.
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        <div xml:id="echoid-div146" type="section" level="1" n="64">
          <head xml:id="echoid-head95" xml:space="preserve">VERA
            <lb/>
          CIRCULI ET HYPERBOLÆ
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          QUADRATURA.</head>
          <p>
            <s xml:id="echoid-s2778" xml:space="preserve">Sit circuli, ellipſeos vel hyperbolæ ſegmentum B I P
              <lb/>
              <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">TAB. XLIII.
                <lb/>
              Fig. 1. 2. 3.</note>
            cujus centrum A: </s>
            <s xml:id="echoid-s2779" xml:space="preserve">compleatur triangulum A B P, & </s>
            <s xml:id="echoid-s2780" xml:space="preserve">
              <lb/>
            ſegmentum in punctis, B, P, tangentes ducantur re-
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            ctæ B F, P F, ſe invicem ſecantes in puncto F; </s>
            <s xml:id="echoid-s2781" xml:space="preserve">pro-
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            ducatur (ſi opus ſit) recta A F ſegmentum interſecans in
              <lb/>
            puncto I & </s>
            <s xml:id="echoid-s2782" xml:space="preserve">rectam B P in puncto Q; </s>
            <s xml:id="echoid-s2783" xml:space="preserve">deinde jungantur re-
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            ctæ B I, P I.</s>
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          <head xml:id="echoid-head96" xml:space="preserve">PROP. I. THEOREMA.</head>
          <head xml:id="echoid-head97" style="it" xml:space="preserve">Dico trapezium B A P I eſſe medium propor-
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          tionale inter trapezium B A P F, &
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          triangulum B A P.</head>
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            <s xml:id="echoid-s2785" xml:space="preserve">Quoniam recta A Q ducitur per F concurſum duarum re-
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            ctarum F B, F P, ſegmentum in punctis B, P, tan-
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            gentium; </s>
            <s xml:id="echoid-s2786" xml:space="preserve">igitur recta A Q rectam B P contactuum
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            puncta jungentem bifariam ſecabit in puncto Q; </s>
            <s xml:id="echoid-s2787" xml:space="preserve">& </s>
            <s xml:id="echoid-s2788" xml:space="preserve">proinde
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            triangulum A B Q eſt æquale triangulo A Q P, & </s>
            <s xml:id="echoid-s2789" xml:space="preserve">trian-
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            gnlum F B Q triangulo F Q P; </s>
            <s xml:id="echoid-s2790" xml:space="preserve">& </s>
            <s xml:id="echoid-s2791" xml:space="preserve">igitur triangulum A B F
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            æquale eſt triangulo A P F; </s>
            <s xml:id="echoid-s2792" xml:space="preserve">eſt ergo triangulum A B F di-
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            midium trapezii A B F P: </s>
            <s xml:id="echoid-s2793" xml:space="preserve">eodem modo probatur triangu-
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            lum A B I eſſe dimidium trapezii A B I P; </s>
            <s xml:id="echoid-s2794" xml:space="preserve">& </s>
            <s xml:id="echoid-s2795" xml:space="preserve">triangulum
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            A B Q eſt dimidium trianguli A B P: </s>
            <s xml:id="echoid-s2796" xml:space="preserve">cumque triangula
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            A B F, A B I, A B Q, eandem habeant altitudinem, in-
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            ter ſe ſunt ut baſes, ſed eorum baſes nempe A F, A I, A Q,
              <lb/>
            ſunt continuè proportionales; </s>
            <s xml:id="echoid-s2797" xml:space="preserve">& </s>
            <s xml:id="echoid-s2798" xml:space="preserve">igitur ipſa quoque triangu-
              <lb/>
            la ſunt continuè proportionalia; </s>
            <s xml:id="echoid-s2799" xml:space="preserve">& </s>
            <s xml:id="echoid-s2800" xml:space="preserve">proinde eorum dupla ni-
              <lb/>
            mirum trapezia A B F P, A B I P, & </s>
            <s xml:id="echoid-s2801" xml:space="preserve">triangulum A B P
              <lb/>
            ſunt continuè proportionalia in ratione A F ad A I, quod
              <lb/>
            demonſtrare oportuit.</s>
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