Clavius, Christoph - In Sphaeram Ioannis de Sacro Bosco commentarius

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13497Ioan. de Sacro Boſco.

THEOR. 11. PROPOS. 13.

IRCVLVS omnibus figuris rectilineis regularibus ſibi iſoperime-
11Circulus
omnium
figurarũ re
cti linearũ
regulariũ
ſibi iſoperi
metrarum
maximus
eſt. tris eſt.

Esto circulus A B C, figura autem regularis quotcunque laterum ei iſo-
perimetra D E F. Dico circulum A B C, eſſe maiorem figura D E F. Sit enim G,
centrum circuli A B C; & H, centrum figuræ D E F; Deſcribaturq́. circa cir-
culum A B C, figura B I K C, tot laterum, & angulorum ęqualium, quot con-
tinet figura D E F, id eſt, ſimilis figurę D E F, per ea, quæ ex Campano docui-
mus in ſcholio 1. propoſ. 16. lib. 4. Eucl. Deinde ex puncto contactus A, ad cen
trum G, ducatur recta A G, quæ perpendicularis erit ad I K. Ducatur rur-
2218. tertĳ. ſus H D, ad L M, perpendicularis; Diuidentq́. rectæ G A, H D, rectas I K, L M,
333. tertĳ. bifariam, ut conſtat, ſi figuris B I K C, D E F, circunſcribantur circuli. Du-
cantur quoque recte G I, H L, quæ diuident angulos I, & L, bifariam, ut ma-
nifeſtum eſt ex demonſtratione propoſ. 12. lib. 4. Eucl. Quoniam igitur toti
anguli I, & L, ſunt æquales, propter ſimilitudinem figurarum, erunt etia@

35[Figure 35] ipſorum dimidia, uidelicet anguli A I G, D L H, ęqualia. Cum ergo & an-
4432. primi. guli I A G, L D H, ſint ęquales, vtpote recti, erunt triangula A I G, D L H,
ęquiangula. Quia uero ambitus figuræ B I K C, maior eſt (per 1. propoſ. lib. 1.
Archimedis de ſphæra, & cylindro) ambitu circuli A B C; Ambitus autem cir-
culi æqualis ponitur ambitui figuræ D E F; erit quoq. ambitus figurę B I K C.
maior ambitu figurę D E F. Cum igitur figuræ ſint regulares, & ſimiles, erit
etiam latus I K, latere L M, maius, & ideo I A, dimidium lateris I K, maius,
554. ſexti. quàm L D, dimidium lateris L M. Rurſus, quoniam eſt, vt I A, ad A G, ita L D,
6614. quinti. ad D H; Et eſt I A, maior quàm L D, erit quoq. A G, maior, quàm D H. Quam
obrem rectangulum contentum ſub A G, & dimidio ambitu circu li A B G,
quod (per 4. propoſ. huius) circulo A B C, eſt æquale, maius eſt, quàm rectangu
lum contentum ſub D H, & dimidio ambitu figurę D E F, hoc eſt, (per 2. pro-
poſ. huius) quàm area figurę D E F. Circulus igitur omnibus figuris rectilineis
regularibus ſibi iſoperimetris maior eſt, quod oſtendendum erat.

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