1huius ſeſquialtera BEF: & ſumpta axis BD quarta par
te DF, & tertia DG: qua ratione erit FG duodecima
pars axis BD quarta ab ea, cuius terminus D; fiat vt
IB ad BD, ita FH ad HG. Dico conoidis, vel portio
nis ABC centrum grauitatis eſſe H. Nam vt eſt EB
ad BD ita fiat DK ad KA: & ponatur KDY ſeſqui
altera ipſius DK, & ex AK abſcindatur KM ſubſeſ
quialtera ipſius AK: & ipſis DK DM, DA, æquales
eodem ordine abſcindantur DL, DN, DC: & deſcri
bantur triangula, KBL, MBN: & per puncta ABC
vertice communi B, tranſeant duæ ſectiones parabolæ
AOB, & BPC, ita vt contingat recta BK parabolam
AOB, recta autem BL parabolam BPC; ſit autem
AKLC, parabolarum diametris parallela,. Deinde
ſecto axe BD bifariam, & ſingulis eius partibus rurſus bi
fariam in quotlibet partes æquales, ſint ex illis duæ
partes DQ, QF: & per puncta QF planis quibuſdam
baſi parallelis ſecentur vnà ſolidum & hyperbole ABC:
ſintque hyperboles ſectiones, quæ continent ſectiones trian
gulorum ABC mixti, & rectilinei KBL, rectæ RTX
ZVS: αγεζδβ. ſolidi autem ABC ſectiones erunt cir
culi, vel ellipſes ſimiles baſi circa diametros RS, αβ.
Quoniam igitur eſt vt ΥK ad KD, ita AK ad KM;
vtrobique enim eſt proportio ſeſquialtera: erit permutan
do vt YK ad AK, hoc eſt vt IB ad BD, vel FH, ad
HG, ita DK ad KM, hoc eſt triangulum BDK ad
triangulum BKM, hoc eſt ad æquale huic ex demon
ſtratis triangulum AKB mixtum: hoc eſt in duplis ita,
triangulum BKL ad duo mixta rriangula AKB, BLC
ſimul. ſed duorum triangulorum AKB, BLC ſimul eſt
centrum grauitatis F, vt in hoc tertio libro demonſtra
uimus: trianguli autem BKL, vt in primo, centrum gra
uitatis G; totius igitur trianguli ABC centrum graui
tatis erit H. Rurſus quoniam eſt vt BD ad BQ hoc
te DF, & tertia DG: qua ratione erit FG duodecima
pars axis BD quarta ab ea, cuius terminus D; fiat vt
IB ad BD, ita FH ad HG. Dico conoidis, vel portio
nis ABC centrum grauitatis eſſe H. Nam vt eſt EB
ad BD ita fiat DK ad KA: & ponatur KDY ſeſqui
altera ipſius DK, & ex AK abſcindatur KM ſubſeſ
quialtera ipſius AK: & ipſis DK DM, DA, æquales
eodem ordine abſcindantur DL, DN, DC: & deſcri
bantur triangula, KBL, MBN: & per puncta ABC
vertice communi B, tranſeant duæ ſectiones parabolæ
AOB, & BPC, ita vt contingat recta BK parabolam
AOB, recta autem BL parabolam BPC; ſit autem
AKLC, parabolarum diametris parallela,. Deinde
ſecto axe BD bifariam, & ſingulis eius partibus rurſus bi
fariam in quotlibet partes æquales, ſint ex illis duæ
partes DQ, QF: & per puncta QF planis quibuſdam
baſi parallelis ſecentur vnà ſolidum & hyperbole ABC:
ſintque hyperboles ſectiones, quæ continent ſectiones trian
gulorum ABC mixti, & rectilinei KBL, rectæ RTX
ZVS: αγεζδβ. ſolidi autem ABC ſectiones erunt cir
culi, vel ellipſes ſimiles baſi circa diametros RS, αβ.
Quoniam igitur eſt vt ΥK ad KD, ita AK ad KM;
vtrobique enim eſt proportio ſeſquialtera: erit permutan
do vt YK ad AK, hoc eſt vt IB ad BD, vel FH, ad
HG, ita DK ad KM, hoc eſt triangulum BDK ad
triangulum BKM, hoc eſt ad æquale huic ex demon
ſtratis triangulum AKB mixtum: hoc eſt in duplis ita,
triangulum BKL ad duo mixta rriangula AKB, BLC
ſimul. ſed duorum triangulorum AKB, BLC ſimul eſt
centrum grauitatis F, vt in hoc tertio libro demonſtra
uimus: trianguli autem BKL, vt in primo, centrum gra
uitatis G; totius igitur trianguli ABC centrum graui
tatis erit H. Rurſus quoniam eſt vt BD ad BQ hoc