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

Table of contents

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[41.] Theor. XII. Prop. XV.
Page: 91 (376)
[42.] Theor. XIII. Prop. XVI.
Page: 95 (377)
[43.] Theorema XIV. Propos. XVII.
Page: 96 (378)
[44.] Theor. XV. Propos. XVIII.
Page: 98 (380)
[45.] Theor. XVI. Propos. XIX.
Page: 100 (382)
[46.] Problema IV. Propos. XX.
Page: 102 (384)
[47.] Christiani Hugenii C. F. ILLVSTRIVM QVORVNDAM PROBLEMATVM CONSTRVCTIONES. Probl. I. Datam ſphæram plano ſecare, ut portiones inter ſe rationem habeant datam.
Page: 109
[48.] LEMMA.
Page: 111 (390)
[49.] Probl. II. Cubum invenire dati cubi duplum.
Page: 112 (391)
[50.] Probl. III. Datis duabus rectis duas medias propor-tionales invenire.
Page: 114 (393)
[51.] ALITER.
Page: 115 (394)
[52.] ALITER.
Page: 116 (395)
[53.] Probl. IV.
Page: 117 (396)
[54.] Probl. V.
Page: 118 (397)
[55.] Probl. VI.
Page: 118 (397)
[56.] Probl. VII.
Page: 123 (399)
[57.] Utrumque præcedentium Aliter.
Page: 125 (401)
[58.] Probl. VIII. In Conchoide linea invenire confinia flexus contrarii.
Page: 127 (403)
[59.] FINIS.
Page: 128 (404)
[60.] DE CIRCULI ET HYPERBOLÆ QUADRATURA CONTROVERSIA.
Page: 132
[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)
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111390CHRISTIANI HUGENII ad H Q. Et dividendo, portio K A L ad portionem
K C L ut E Q ad Q H, hoc eſt ut S ad T. Quod erat fa-
ciendum. Quod autem dictum fuit O Q duplam eſſe ipſius
M N, ſic fiet manifeſtum. Quia enim ut quadratum C M
ad quadr. M N, ita eſt M N ad N O longitudine: Eſt au-
tem Q M æqualis ſubtenſæ arcus A R cujus trienti ſubten-
ditur M N. Erunt propterea duæ ſimul Q M & N O æqua-
les triplæ M N, uti ſequenti lemmate demonſtratur. Quam-
obrem ablata communi O N, erit ſola Q M æqualis du-
plæ N M & ipſi M O. Sed eadem Q M æqualis eſt dua-
bus ſimul his Q O, O M, ergo apparet duplam M N æqua-
ri ipſi O Q.
LEMMA.
SI Circumferentiæ arcus in tria æqualia ſecetur, tres ſimul
11TAB. XLI.
Fig. 2. rectæ quæ æqualibus partibus ſubtenduntur, æquantur ſub-
tenſæ arcus totius & ei quæ ad ſubtenſam trientis ſeſe habe-
at, ſicut hujus quadratum ad quadratum ſemidiametri. Ar-
cus ſectoris A B C in tria æqualia diviſus ſit punctis D, E.
Et ſubtendantur partibus rectæ B D, D E, E C; & toti
arcui linea B C. Porro jungantur D A, E A, atque inter-
ſecent ſubtenſam B C in punctis G & H. Sitque H F pa-
rallela G D.
Quoniam igitur circumferentia B D E dupla eſt circum-
ferentiæ E C, angulus autem huic inſiſtens E A C ad cen-
trum conſtitutus, qui vero illi inſiſtit angulus B C E ad
circumferentiam. Erit propterea angulus B C E, hoc eſt,
angulus H C E in triangulo H C E æqualis angulo C A E
in triangulo C A E. Sed angulus ad E utrique eſt commu-
nis; itaque ſimiles inter ſe ſunt dicti trianguli: Eritque ut
A E ad E C ita E C ad E H. Ratio igitur A E ad E H
hoc eſt D E ad E F, duplicata eſt rationis A E ad E C,
ac proinde cadem quæ quadrati A E ad quadr. E C ſeu
quadr. E D. Erit igitur invertendo F E ad E D ſicut
quadr. E D ad quadr. E A. Quamobrem oſtendendum
eſt, tres ſimul B D, D E, E C æquari ſubtenſæ B

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