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

Table of contents

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[61.] VERA CIRCULI ET HYPERBOLÆ QUADRATURA AUTHORE JACOBO GREGORIO. LECTORI GEOMETRÆ SALUTEM.
Page: 134
[62.] DEFINITIONES.
Page: 139
[63.] PETITIONES.
Page: 141 (414)
[64.] VERA CIRCULI ET HYPERBOLÆ QUADRATURA.
Page: 142 (415)
[65.] PROP. I. THEOREMA. Dico trapezium B A P I eſſe medium propor-tionale inter trapezium B A P F, & triangulum B A P.
Page: 142 (415)
[66.] PROP. II. THEOREMA. Dico trapezia A B F P, A B I P ſimul, eſſe ad du- plum trapezii A B I P, ſicut trapezium A B F P ad polygonum A B D L P.
Page: 143 (416)
[67.] PROP. III. THEOREMA. Dico triangulum B A P, & trapezium A B I P ſimul, eſſe ad trapezium A B I P, ut duplum trapezii A B I P ad polygonum A B D L P.
Page: 144 (417)
[68.] PROP. IV. THEOREMA. Dico polygonum A B E I O P eſſe medium pro- portionale inter polygonum A B D L P & trapezium A B I P.
Page: 144 (417)
[69.] PROP. V. THEOREMA.
Page: 145 (418)
[70.] SCHOLIUM.
Page: 146 (419)
[71.] PROP. VI. THEOREMATA.
Page: 147 (420)
[72.] SCHOLIUM.
Page: 148 (421)
[73.] PROP. VII. PROBLEMA. Oportet prædictæ ſeriei terminationem invenire.
Page: 150 (423)
[74.] PROP. VIII. PROBLEMA.
Page: 152 (425)
[75.] PROP. IX. PROBLEMA.
Page: 153 (426)
[76.] PROP. X. PROBLEMA.
Page: 154 (427)
[77.] CONSECTARIUM.
Page: 155 (428)
[78.] PROP. XI. THEOREMA.
Page: 156 (429)
[79.] SCHOLIUM.
Page: 159 (432)
[80.] PROP. XII. THEOREMA.
Page: 161 (434)
[81.] PROP. XIII. THEOREMA.
Page: 161 (434)
[82.] PROP. XIV. THEOREMA.
Page: 162 (435)
[83.] PROP. XV. THEOREMA.
Page: 162 (435)
[84.] PROP. XVI. THEOREMA.
Page: 163 (436)
[85.] PROP. XVII. THEOREMA.
Page: 164 (437)
[86.] PROP. XVIII. THEOREMA.
Page: 164 (437)
[87.] PROP. XIX. THEOREMA.
Page: 165 (438)
[88.] CONSECTARIUM.
Page: 165 (438)
[89.] PROP. XX. THEOREMA.
Page: 165 (438)
[90.] PROP. XXI. THEOREMA.
Page: 166 (439)
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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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