Full text: Clavius, Christoph: Christopheri Clavii Bambergensis ex Societate Iesv In Sphaeram Ioannis de Sacro Bosco commentarius

76. THEOR. 4. PROPOS. 4.

Area cuiuslibet circuli æqualis eſt rectangulo comprehenſo [?] ſub ſe-
midiametro, & dimidiata circumferentia circuli.

76.1.

Circulus
quicunque
cui rectan [?] -
gulo æqua-
lis ſit.

Esto circulus A B C, cuius ſemidiameter D B: Rectangulum autem
D B E F, comprehenſum ſub D B, ſemidiametro circuli, & B E, recta, quæ
æqualis ſit dimidiatæ circunferentiæ circuli. Dico aream circuli A B C, æqua
lem eſſe rectangulo D B E F. Producatur enim B E, in continuum, ponatur-
q́ue E G, æqualis ipſi B E, ut ſit B G, recta æqualis toti circunferentiæ circu-
li. Coniungantur denique puncta D, G, recta D G. Quoniam igitur (per 1. propoſ. Archimedis de Dimenſione circuli) circulus A B C, æqualis eſt trian
gulo D B G: Eſt autem triangulum D B G, rectangulo D B E F, æquale, ut in
ſcholio propoſ. 41. lib. 1. Eucl. demonſtrauimus, quòd baſis trianguli dupla ſit
baſis rectanguli, (Id quod etiam ex demonſtratione antecedentis propoſ. li-
quet, ubi oſtendimus, triangulum D E F, æquale eſſe rectangulo D E H I:) erit quoque circulus A B C, rectangulo D B E F, æqualis. Area ergo cuius-
libet circuli æqualis eſt rectangulo, & c. quod oſtendendum erat.

76.1.

121-01

77. THEOR. 5. PROPOS. 5.

Proprietas
quædã triã-
guli rectan
guli.

In omni triangulo rectangulo, ſi ab uno acutorum angul orum ut-
cunque ad latus oppoſitum linea recta ducatur, erit maior proportio
huius lateris ad eius ſegmentum, quod prope angulum rectum exi-
ſtit, quàm anguli acuti prędicti ad eius partem dicto ſegmento late-
ris oppoſitam.

Sit triangulum rectangulum A B C, cuius angulus C, ſit rectus; duca-

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