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dinatam BDin I, &
lineam RSin X) ſit MP.
ME:
: VG.
IX;
vel, ſit linea AL talis, ut ductâ MPY ad BDparallelâ (quæ ſecet
axem ADin P, & lineam ALin Y) ſit PE. ME: : VG. PY; erit
11Fig. 177. tunc utrumque _ſpatium_ (ſingillatim) BRS D, vel ADL duplum _ſu-_
_perfici@i conicœ_, quod ex recta per V & curvam AMB mota progene-
ratur.
vel, ſit linea AL talis, ut ductâ MPY ad BDparallelâ (quæ ſecet
axem ADin P, & lineam ALin Y) ſit PE. ME: : VG. PY; erit
11Fig. 177. tunc utrumque _ſpatium_ (ſingillatim) BRS D, vel ADL duplum _ſu-_
_perfici@i conicœ_, quod ex recta per V & curvam AMB mota progene-
ratur.
Nam ſumatur MNindefinita curvæ particula;
&
per N ducantur
rectæ NOKTad ipſam AD, & NQZ ad BDparallelæ (quæ li-
neas expoſitas, ut _Schema_ monſtrat, ſecent) connectantúrque rectæ
VM, VN. eſtque MO. MN: : MP. MF: : VG. IX. quare
MN x VG = MO x IX = IK x IX. Item eſt NO. MN: : PE.
ME: : VG. PY. unde MN x VG = NO x PY = QP x PY.
Eſt autem MN x VG duplum trianguli MVN. quapropter tam IK
x IX, quàm QP x PY duplum eſt _trianguli_ MVN. pariter autem
ubique fit. ergò conſtat Propoſitum.
rectæ NOKTad ipſam AD, & NQZ ad BDparallelæ (quæ li-
neas expoſitas, ut _Schema_ monſtrat, ſecent) connectantúrque rectæ
VM, VN. eſtque MO. MN: : MP. MF: : VG. IX. quare
MN x VG = MO x IX = IK x IX. Item eſt NO. MN: : PE.
ME: : VG. PY. unde MN x VG = NO x PY = QP x PY.
Eſt autem MN x VG duplum trianguli MVN. quapropter tam IK
x IX, quàm QP x PY duplum eſt _trianguli_ MVN. pariter autem
ubique fit. ergò conſtat Propoſitum.
Exemplum.
Sit curva AMB _byperbola æquilatera_, cujus _Centrum_ C, ſitque
22Fig. 177. CV = CA = _r._ & CP = _x_ (nam hujuſmodi _calculo_ plerunque
rem expedit peragere) tum connexâ MC; patet eſſe EC = {_rr_/_x_};
& MCq = 2 _xx_ - _rr_ (nam PMq = _xx_ - _rr_) item eſt MCq.
CPq: : MEq. MPq; hoc eſt MCq. CPq: : ECq. CGq. hoc
eſt 2 _xx_ - _rr_. _xx_: : {_r_4/_xx_}. CGq = {_r_4/2 _xx_ - _rr_}. quare VGq = {_r_4/2 _xx_ - _rr_} +
_rr_ = {2 _rrxx_/2 _xx_ - _rr_} = {VAq x CPq/MCq}. vel VG = {VA x CP/MC}. quare
VG. VA: : (CP. MC): : MP. ME. hinc conſectatur in hoc
caſu, quum ubique ſit IX = VA, _lineam_ RS fore _rectam_; & _rectan-_
_gulum_ BRSD _ſuperficiei conicœ_ AMBV _duplum eſſe._
22Fig. 177. CV = CA = _r._ & CP = _x_ (nam hujuſmodi _calculo_ plerunque
rem expedit peragere) tum connexâ MC; patet eſſe EC = {_rr_/_x_};
& MCq = 2 _xx_ - _rr_ (nam PMq = _xx_ - _rr_) item eſt MCq.
CPq: : MEq. MPq; hoc eſt MCq. CPq: : ECq. CGq. hoc
eſt 2 _xx_ - _rr_. _xx_: : {_r_4/_xx_}. CGq = {_r_4/2 _xx_ - _rr_}. quare VGq = {_r_4/2 _xx_ - _rr_} +
_rr_ = {2 _rrxx_/2 _xx_ - _rr_} = {VAq x CPq/MCq}. vel VG = {VA x CP/MC}. quare
VG. VA: : (CP. MC): : MP. ME. hinc conſectatur in hoc
caſu, quum ubique ſit IX = VA, _lineam_ RS fore _rectam_; & _rectan-_
_gulum_ BRSD _ſuperficiei conicœ_ AMBV _duplum eſſe._
Cæterùm hoc _elegans exemplum_ ſuppeditavit Generoſus, ingenio ac
eruditione præſtans, Vir (_Collegii noſtri, quod olim Sociorum Com-_
_menſalis incoluit_, ornamentum) D. _Franciſcus Feſſopius_, Armiger;
cujus in hanc rem perquam ingenioſo mihi comiter impertito ſcripto
(ipſius injuſſu quidem, at ſpero non ingratiis) ſeu _Gemmâ_ quâdam au-
debo mea condecorare.
eruditione præſtans, Vir (_Collegii noſtri, quod olim Sociorum Com-_
_menſalis incoluit_, ornamentum) D. _Franciſcus Feſſopius_, Armiger;
cujus in hanc rem perquam ingenioſo mihi comiter impertito ſcripto
(ipſius injuſſu quidem, at ſpero non ingratiis) ſeu _Gemmâ_ quâdam au-
debo mea condecorare.