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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149422VERA CIRCULI eſſe termino a, quoniam termino a additur {bd - ad/c} ut fiat termi-
nus {ca + bd - ad/c}:
manifeſtum quoque eſt terminum {ca + bd - ad/c} mi-
norem eſſe termino b, quoniam differentia inter a &
b eſt ad
differentiam inter a &
{ca + bd - ad/c} in ratione majoris inæqualita-
tis:
evidens quoque eſt terminum {bc - be + ae/c} minorem eſſe ter-
mino b.
quoniam ex b ſubſtrahitur {be - ae/c} ut fiat {bc - be + ae/c}; ma-
nifeſtum etiam eſt terminum {bc - be + ae/c} majorem eſſe termino a,
quoniam differentia inter a &
b eſt ad differentiam inter
{bc - be + ae/c} &
b in ratione majoris inæqualitatis: evidens igitur
eſt differentiam inter terminos convergentes a &
b majorem
eſſe differentiâ inter terminos convergentes {ca + bd - ad/c} &
{bc - be + ae/c}.
fed quoniam termini convergentes a & b ponuntur indefiniti,
poſſunt a &
b ſumi loco quorumlibet terminorum convergen-
tium totius hujus ſeriei;
& poſitis a & b pro terminis hujus
ſeriei convergentibus quibuſcunque, ſequitur neceſſario ex
ſeriei compoſitione {ca + bd - ad.
/c}, {bc - be + ae/c} eſſe terminos conver-
gentes immediatè ſequentes:
cumque differentia terminorum
a &
b major ſit differentia terminorum {ca + bd - ad/c} & {bc - be + ae/c},
evidens eſt differentiam terminorum convergentium priorum
ſemper eſſe majorem differentia terminorum convergentium
immediatè ſequentium;
& igitur quò magis continuatur hæc
ſeries convergens eò minor fit differentia terminorum con-
vergentium:
& quoniam hæc differentiarum diminutio ſem-
per fit proportionaliter nempe in ratione b-a ad {bc - be + ae - ca - bd + ad;
/c}
igitur poſſunt inveniri hujus ſeriei termini convergentes quo-
rum differentia ſit omni exhibita quantitate minor;
& igitur
imaginando hanc ſeriem in infinitum continuari, poſſumus
imaginari ultimos terminos convergentes eſſe

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