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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              <pb o="371" file="0081" n="86" rhead="DE CIRCULI MAGNIT. INVENTA."/>
            jor quam 5176 {3/8}. </s>
            <s xml:id="echoid-s1512" xml:space="preserve">cujus dupla F H major quam 10352 {3/4}. </s>
            <s xml:id="echoid-s1513" xml:space="preserve">unde
              <lb/>
            G H major quam 352 {3/4}; </s>
            <s xml:id="echoid-s1514" xml:space="preserve">& </s>
            <s xml:id="echoid-s1515" xml:space="preserve">H I major quam 117 {7/12}. </s>
            <s xml:id="echoid-s1516" xml:space="preserve">Tota igi-
              <lb/>
            tur F I major quam 10470 {1/3}. </s>
            <s xml:id="echoid-s1517" xml:space="preserve">Arcus autem C D, ſextans pe-
              <lb/>
            ripheriæ, minor eſt quam 10472. </s>
            <s xml:id="echoid-s1518" xml:space="preserve">Ergo deficiunt lineæ F I
              <lb/>
            partium earundem pauciores quam 1 {2/3}. </s>
            <s xml:id="echoid-s1519" xml:space="preserve">Quæ non æquant {1/6000}
              <lb/>
            F I. </s>
            <s xml:id="echoid-s1520" xml:space="preserve">Porro cum arcus quadrante major datus erit, dividen-
              <lb/>
            dus eſt in partes æquales 4 vel 6 vel plures, prout accura-
              <lb/>
            tiori dimenſione uti voluerimus; </s>
            <s xml:id="echoid-s1521" xml:space="preserve">ſed numero pares: </s>
            <s xml:id="echoid-s1522" xml:space="preserve">Earum-
              <lb/>
            que partium ſubtenſis ſimul ſumptis adjungendus eſt triens
              <lb/>
            exceſſus quo ipſæ ſuperant aggregatum earum quæ arcubus
              <lb/>
            duplis ſubtenduntur. </s>
            <s xml:id="echoid-s1523" xml:space="preserve">Ita namque componetur longitudo ar-
              <lb/>
            cus totius. </s>
            <s xml:id="echoid-s1524" xml:space="preserve">Vel hac etiam ratione eadem habebitur, ſi arcus
              <lb/>
            reliqui ad ſemicircumferentiam longitudo inveniatur aut ſu-
              <lb/>
            pra eandem exceſſus, aut reliqui ad circumferentiam totam,
              <lb/>
            ſi dodrante major fuerit datus; </s>
            <s xml:id="echoid-s1525" xml:space="preserve">eaque longitudo adjungatur
              <lb/>
            vel auferatur à dimidiæ vel totius circumferentiæ longitudi-
              <lb/>
            ne, quam antea invenire docuimus.</s>
            <s xml:id="echoid-s1526" xml:space="preserve"/>
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        <div xml:id="echoid-div81" type="section" level="1" n="38">
          <head xml:id="echoid-head61" xml:space="preserve">
            <emph style="sc">Theor</emph>
          . X.
            <emph style="sc">Prop</emph>
          . XIII.</head>
          <p style="it">
            <s xml:id="echoid-s1527" xml:space="preserve">
              <emph style="bf">L</emph>
            Atus Polygoni æquilateri circulo inſcripti, pro-
              <lb/>
            portione medium eſt inter latus polygoni ſimi-
              <lb/>
            lis circumſcripti, & </s>
            <s xml:id="echoid-s1528" xml:space="preserve">dimidium latus polygoni in-
              <lb/>
            ſcriptiſub duplo laterum numero.</s>
            <s xml:id="echoid-s1529" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1530" xml:space="preserve">IN circulo cujus centrum A, radius A B, ſit latus inſcri-
              <lb/>
              <note position="right" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve">TAB. XXXIX.
                <lb/>
              Fig. 4.</note>
            pti polygoni æquilateri B C; </s>
            <s xml:id="echoid-s1531" xml:space="preserve">& </s>
            <s xml:id="echoid-s1532" xml:space="preserve">latus circumſcripti ſimilis
              <lb/>
            polygoni D E ipſi B C parallelum. </s>
            <s xml:id="echoid-s1533" xml:space="preserve">Ergo producta A B trans-
              <lb/>
            ibit per D, & </s>
            <s xml:id="echoid-s1534" xml:space="preserve">A C per E. </s>
            <s xml:id="echoid-s1535" xml:space="preserve">Et ſi ducatur C F ipſi A B ad
              <lb/>
            angulos rectos, ea erit dimidium latus polygoni inſcripti ſub-
              <lb/>
            duplo numero laterum. </s>
            <s xml:id="echoid-s1536" xml:space="preserve">Itaque oſtendendum eſt, B C me-
              <lb/>
            diam eſſe proportione inter E D & </s>
            <s xml:id="echoid-s1537" xml:space="preserve">C F. </s>
            <s xml:id="echoid-s1538" xml:space="preserve">Ducatur A G, quæ
              <lb/>
            dividat E D bifariam, itaque erit ipſa quoque circuli ſemi-
              <lb/>
            diameter & </s>
            <s xml:id="echoid-s1539" xml:space="preserve">æqualis A B. </s>
            <s xml:id="echoid-s1540" xml:space="preserve">Et quoniam eſt ut E D ad C B,
              <lb/>
            ſic D A ad A B, hoc eſt, D A ad A G; </s>
            <s xml:id="echoid-s1541" xml:space="preserve">ſicut autem D </s>
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