Clavius, Christoph - In Sphaeram Ioannis de Sacro Bosco commentarius

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137100Comment. in I. Cap. Sphæræ des (nempe corpus A B C D, ex illis compoſitum) æquales ſolido rectangu-
lo L R. Quamobrem area cuiuſlibet corporis planis ſuperficiebus contenti,
& c. quod demonſtrandum erat.

THEOR. 14. PROPOS. 16.

Area cuiuslibet ſphærę æqualis eſt ſolido rectangulo comprehenſo
11Sphę ra q̃li
bet cui pa-
rallel epipe
do ſit ęqua
lis. ſub ſemidiametro ſphæræ, & tertia parte ambitus ſphæræ.

Esto ſphæra A B C, cuius centrum D, ſemidiameter A D: Solidum au-
tem rectangulum E, contentum ſub ſemidiametro A D, & tertia parte ambi-
tus ſpæræ A B C. Dico corpus E, ſphæræ A B C, eſſe æquale. Nam ſi non eſt
æquale; ſit, ſi fieri poteſt, primum maius, ſitq́ue exceſfus corporis E, ſupra
ſphęram A B C, quantitas F. Intelligatur circa ccntrum D, deſcripta ſphæ-
ra GHK, maior quàm ſphæra A B C, ita tamen, ut exceſſus ſphęrę G H K,
ſupra ſphęram A B C, non ſit maior quantitate F, ſed uel æqualis, uel mi-
nor, hoc eſt, vt ſphæra G H K, ſit uel ęqualis ſolido E, quando nimirum

38[Figure 38] ipſa excedit ſphæram A B C, præciſe
quantitate F; uel minor, ſi nimirum
ipſa excedit ſphęram A B C, mino-
ri quantitate, quàm F. Neceſlario
enim aliqua ſphæra erit, quæ uel
æqualis ſit magnitudini E, atque
adeo maior, quàm ſphæra A B C;
uel maior quidem quã ſphęra A B C,
minor vero quàm magnitudo E, quæ
maior ponitur, quàm ſphæra A B C.
Inſcribatur deinde intra ſphæram
G H K, corpus, quod non tangat
ſphæram A B C, ita ut unaquæque
2237. duod. perpendicularium ex centro D, ad
baſes iſtius corporis eductarum ma-
ior fit ſemidiametro A D. Si igitur
à centro D, ad omnes angulos di-
cti corporis ducantur lineæ rectæ,
ut totum corpus in pyramides di-
uidatur, quarum baſes ſunt eædem,
quæ corporis G H K, uertex au-
tem communis centrum D; erit quæ
libet pyramis (per 14. propoſ. hu-
ius) æqualis ſolido rectangulo contento ſub eius perpendiculari, & tertia
parte baſis; A tque idcirco ſolidum rectangulum contentum ſub ſemidiame-
tro A D & tertia parte baſis cuiuſlibet pyramidis, minus ipſa pyramide
erit. Et quoniam omnia ſolida rectangula contenta ſub ſingulis perpendi-
cularibus ex centro D, ad baſes corporis dicti protractis, & ſingulis ter-
tijs partibus baſium, ſimul ęqualia ſunt toti corpori, efficiunt autem om-
@es tertiæ partes baſium ſimul tertiam partem ambitus corporis, erit

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