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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            eſſe termino
              <emph style="sub">a</emph>
            , quoniam termino
              <emph style="super">a</emph>
            additur {bd - ad/c} ut fiat termi-
              <lb/>
            nus {ca + bd - ad/c}: </s>
            <s xml:id="echoid-s2993" xml:space="preserve">manifeſtum quoque eſt terminum {ca + bd - ad/c} mi-
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            norem eſſe termino b, quoniam differentia inter
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s2994" xml:space="preserve">
              <emph style="super">b</emph>
            eſt ad
              <lb/>
            differentiam inter a & </s>
            <s xml:id="echoid-s2995" xml:space="preserve">{ca + bd - ad/c} in ratione majoris inæqualita-
              <lb/>
            tis: </s>
            <s xml:id="echoid-s2996" xml:space="preserve">evidens quoque eſt terminum {bc - be + ae/c} minorem eſſe ter-
              <lb/>
            mino b. </s>
            <s xml:id="echoid-s2997" xml:space="preserve">quoniam ex
              <emph style="sub">b</emph>
            ſubſtrahitur {be - ae/c} ut fiat {bc - be + ae/c}; </s>
            <s xml:id="echoid-s2998" xml:space="preserve">ma-
              <lb/>
            nifeſtum etiam eſt terminum {bc - be + ae/c} majorem eſſe termino
              <emph style="super">a</emph>
            ,
              <lb/>
            quoniam differentia inter
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s2999" xml:space="preserve">
              <emph style="sub">b</emph>
            eſt ad differentiam inter
              <lb/>
            {bc - be + ae/c} & </s>
            <s xml:id="echoid-s3000" xml:space="preserve">
              <emph style="super">b</emph>
            in ratione majoris inæqualitatis: </s>
            <s xml:id="echoid-s3001" xml:space="preserve">evidens igitur
              <lb/>
            eſt differentiam inter terminos convergentes
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3002" xml:space="preserve">
              <emph style="super">b</emph>
            majorem
              <lb/>
            eſſe differentiâ inter terminos convergentes {ca + bd - ad/c} & </s>
            <s xml:id="echoid-s3003" xml:space="preserve">{bc - be + ae/c}.
              <lb/>
            </s>
            <s xml:id="echoid-s3004" xml:space="preserve">fed quoniam termini convergentes
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3005" xml:space="preserve">
              <emph style="super">b</emph>
            ponuntur indefiniti,
              <lb/>
            poſſunt
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3006" xml:space="preserve">
              <emph style="super">b</emph>
            ſumi loco quorumlibet terminorum convergen-
              <lb/>
            tium totius hujus ſeriei; </s>
            <s xml:id="echoid-s3007" xml:space="preserve">& </s>
            <s xml:id="echoid-s3008" xml:space="preserve">poſitis
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3009" xml:space="preserve">
              <emph style="super">b</emph>
            pro terminis hujus
              <lb/>
            ſeriei convergentibus quibuſcunque, ſequitur neceſſario ex
              <lb/>
            ſeriei compoſitione {ca + bd - ad.</s>
            <s xml:id="echoid-s3010" xml:space="preserve">/c}, {bc - be + ae/c} eſſe terminos conver-
              <lb/>
            gentes immediatè ſequentes: </s>
            <s xml:id="echoid-s3011" xml:space="preserve">cumque differentia terminorum
              <lb/>
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3012" xml:space="preserve">
              <emph style="super">b</emph>
            major ſit differentia terminorum {ca + bd - ad/c} & </s>
            <s xml:id="echoid-s3013" xml:space="preserve">{bc - be + ae/c},
              <lb/>
            evidens eſt differentiam terminorum convergentium priorum
              <lb/>
            ſemper eſſe majorem differentia terminorum convergentium
              <lb/>
            immediatè ſequentium; </s>
            <s xml:id="echoid-s3014" xml:space="preserve">& </s>
            <s xml:id="echoid-s3015" xml:space="preserve">igitur quò magis continuatur hæc
              <lb/>
            ſeries convergens eò minor fit differentia terminorum con-
              <lb/>
            vergentium: </s>
            <s xml:id="echoid-s3016" xml:space="preserve">& </s>
            <s xml:id="echoid-s3017" xml:space="preserve">quoniam hæc differentiarum diminutio ſem-
              <lb/>
            per fit proportionaliter nempe in ratione
              <emph style="super">b-a</emph>
            ad {bc - be + ae - ca - bd + ad;</s>
            <s xml:id="echoid-s3018" xml:space="preserve">/c}
              <lb/>
            igitur poſſunt inveniri hujus ſeriei termini convergentes quo-
              <lb/>
            rum differentia ſit omni exhibita quantitate minor; </s>
            <s xml:id="echoid-s3019" xml:space="preserve">& </s>
            <s xml:id="echoid-s3020" xml:space="preserve">igitur
              <lb/>
            imaginando hanc ſeriem in infinitum continuari, poſſumus
              <lb/>
            imaginari ultimos terminos convergentes eſſe </s>
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