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-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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          <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
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              A B # A
                <lb/>
              C D # C
                <lb/>
              E F # G
                <lb/>
              K L # H
                <lb/>
              Z # X
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            A, B, nempè A extra hyperbolæ
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            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,
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            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-
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            perbolæ ſector Z. </s>
            <s xml:id="echoid-s3564" xml:space="preserve">dico Z majorem eſſe quam C dempto tri-
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            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-
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            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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