547527LIBER VII.
hoc eſt quadratis ſolidis, BGEN, BNEO, additis communibus duobus
adbuc rectangulis ſub, BGEN, BIEN, ſeu, BNEO, quæ ſuperſunt, fi-
eut quatuor rectangula ſolida ſub, BGEN, BIEN, cum quadrato ſo-
lido, BGEI, æqualia duobus quadratis ſolidis, BGEN, BNEO, cum
duobus rectangulis ſolidis ſub, BGEN, BNEO, hoc eſt quadrato ſoli-
do, BGEO, per Cor. Tertium.
adbuc rectangulis ſub, BGEN, BIEN, ſeu, BNEO, quæ ſuperſunt, fi-
eut quatuor rectangula ſolida ſub, BGEN, BIEN, cum quadrato ſo-
lido, BGEI, æqualia duobus quadratis ſolidis, BGEN, BNEO, cum
duobus rectangulis ſolidis ſub, BGEN, BNEO, hoc eſt quadrato ſoli-
do, BGEO, per Cor. Tertium.
COROLLARIVM VIII.
_C_olligitur octauò, ſi figuræ, BGEO, ſecetur vt in Cor.
4.
quadrata
ſolida figurarum, BGEN, BNEO, dupla eſſe quadratorum ſoli-
dorum, BGEI, BIEN. Nam quad. ſolidum, BGEN, æquatur quadra-
tis ſolidis, BGEI, BIEN cum duobus rectangulis ſolidis ſub, BGEI, BI
EN, per Cor. Tertium, ideſt cum duobus rectangulis ſolidis ſub, BIEO,
(homologa ipſi, BGEI,) & , BIEN, quibus ſi addatur reſiduum qua-
dratum ſolidum, BNEO fiunt duo rect angula ſolida ſub, BIEO, BIE
N, cum quadrato ſolido, BNEO, æqualia quadrato ſolido, BIEO, ſeu,
BGEI, cum quadrato ſolido, BIEN, igitur quadrata ſolida, BGEN, B
NEO, dupla ſunt quadratorum ſolidorum; BGEI, BIEN.
ſolida figurarum, BGEN, BNEO, dupla eſſe quadratorum ſoli-
dorum, BGEI, BIEN. Nam quad. ſolidum, BGEN, æquatur quadra-
tis ſolidis, BGEI, BIEN cum duobus rectangulis ſolidis ſub, BGEI, BI
EN, per Cor. Tertium, ideſt cum duobus rectangulis ſolidis ſub, BIEO,
(homologa ipſi, BGEI,) & , BIEN, quibus ſi addatur reſiduum qua-
dratum ſolidum, BNEO fiunt duo rect angula ſolida ſub, BIEO, BIE
N, cum quadrato ſolido, BNEO, æqualia quadrato ſolido, BIEO, ſeu,
BGEI, cum quadrato ſolido, BIEN, igitur quadrata ſolida, BGEN, B
NEO, dupla ſunt quadratorum ſolidorum; BGEI, BIEN.
COROLLARIVM IX.
_C_olligitur nonò, ſuppoſitis in figura ſectionibus ipſius Cor.
5.
qua-
drata ſolida, BGEO, BNEO, dupla eſſe quadratorum ſolidorum,
BGEI, BIEO. Etenim quadratum ſolidum, BGEO, æquatur per Cor 3.
quadratis ſolidis, BGEI, BIEO, cum duobus rectangulis ſolidis ſub, BG
EI, ſeu, BIEN, illi homologa, & , BIEO, quæ duo rectangula ſolida fa-
ciunt cum quadrato ſolido, BNEO, reſiduo, quadrata ſolida, BIEO, BI
EN, ſeu, BGEI, BIEO, ergo quadrata ſolida, BGEO, BNEO, dupla
ſunt quadratorum ſolidorum, BGEI, BIEO.
drata ſolida, BGEO, BNEO, dupla eſſe quadratorum ſolidorum,
BGEI, BIEO. Etenim quadratum ſolidum, BGEO, æquatur per Cor 3.
quadratis ſolidis, BGEI, BIEO, cum duobus rectangulis ſolidis ſub, BG
EI, ſeu, BIEN, illi homologa, & , BIEO, quæ duo rectangula ſolida fa-
ciunt cum quadrato ſolido, BNEO, reſiduo, quadrata ſolida, BIEO, BI
EN, ſeu, BGEI, BIEO, ergo quadrata ſolida, BGEO, BNEO, dupla
ſunt quadratorum ſolidorum, BGEI, BIEO.
_C_olligitur decimò, &
vltimò, ſi tandem ex.
g.
lineæ, BNE, ſecet
quaſcumq; intra figuram, BNE, ipſi, DF, æquidiſtantes, ſecun-
dum extremam, ac mediam rationem, ita vt maior portio cuiuſcumq;
ſectæ lineæ ſit ex. g. in figura, BGEN, rectangulum ſolidum ſub, BGE
O, BNEO, æquari quadrato ſolido, BGEN, bæc enim ſolida erunt
æqualiter analogaiuxta regulam planum, DFH, ex eo quod in vnoquo-
que eidem parallelorũ planorum ipſa ſolida ſecantiũ, ac capientiũ vnũ
rectangulum, & vnum quadratum, ſemper rectangulum eſt æquale,
quadrato in eodem plano exiſtenti.
quaſcumq; intra figuram, BNE, ipſi, DF, æquidiſtantes, ſecun-
dum extremam, ac mediam rationem, ita vt maior portio cuiuſcumq;
ſectæ lineæ ſit ex. g. in figura, BGEN, rectangulum ſolidum ſub, BGE
O, BNEO, æquari quadrato ſolido, BGEN, bæc enim ſolida erunt
æqualiter analogaiuxta regulam planum, DFH, ex eo quod in vnoquo-
que eidem parallelorũ planorum ipſa ſolida ſecantiũ, ac capientiũ vnũ
rectangulum, & vnum quadratum, ſemper rectangulum eſt æquale,
quadrato in eodem plano exiſtenti.