24350
_d_ = _m_;
vel _dy_ - _xy_ = {_d_/_b_}_x x_.
Aliaquædam hîc (nonnulla forſan παρέργως)
inſeremus.
inſeremus.
XIII.
Sit poſitione data recta ID;
ſit item curva DNN talis,
11Fig. 46. utin ID ſumpto puncto quopiam G, ductâque rectâ GN ad poſitio-
nem datam IK parallelâ; ſumptiſque determinatis lineis _g, m, r_;
poſitíſque DG = _x_, & GN = _y_; ſit perpetim _y x_ + _gx_ - _my_ =
{_m_/_r_}_x x_; linea DNN erit _hyperbola_, ſic determinabilis: Sumatur
DM = _m_; & per M ducatur ML ad IK parallela; & in hac acci-
piatur MQ = {_mm_/_r_}; & ſit QY = MQ; & ab MY auferatur
YZ = _g_; connexâque QD, ducatur ZT ad QD parallelâ; erunt
ZM, ZT _aſymptoti_.
11Fig. 46. utin ID ſumpto puncto quopiam G, ductâque rectâ GN ad poſitio-
nem datam IK parallelâ; ſumptiſque determinatis lineis _g, m, r_;
poſitíſque DG = _x_, & GN = _y_; ſit perpetim _y x_ + _gx_ - _my_ =
{_m_/_r_}_x x_; linea DNN erit _hyperbola_, ſic determinabilis: Sumatur
DM = _m_; & per M ducatur ML ad IK parallela; & in hac acci-
piatur MQ = {_mm_/_r_}; & ſit QY = MQ; & ab MY auferatur
YZ = _g_; connexâque QD, ducatur ZT ad QD parallelâ; erunt
ZM, ZT _aſymptoti_.
Nam ducatur ZS ad MD parallela;
cui occurrat GN producta in
R (ſed & GR ipſam ZT ſecet in P). Eſtque jam PN = RG -
RP - GN = {_mm_/_r_} - _g_ + {_mx_/_r_} - _y_. adeoque PN x MG = {_m_3/_r_}
- _mg_ + _yx_ + _gx_ - _my_ - {_m_/_r_}_x x_ = {_m_3/_r_} - _mg_ + _o_. = {_m_3/_r_}
- _mg_ = DM x ZQ. unde PN. ZQ: : (DM. MG: :) QD.
ZP. ergo PN x ZP = ZQ x QD. Liquetigitur curvam DNN
eſſe _hyperbolam_, cujus _aſymptoti_ ZM, ZT.
R (ſed & GR ipſam ZT ſecet in P). Eſtque jam PN = RG -
RP - GN = {_mm_/_r_} - _g_ + {_mx_/_r_} - _y_. adeoque PN x MG = {_m_3/_r_}
- _mg_ + _yx_ + _gx_ - _my_ - {_m_/_r_}_x x_ = {_m_3/_r_} - _mg_ + _o_. = {_m_3/_r_}
- _mg_ = DM x ZQ. unde PN. ZQ: : (DM. MG: :) QD.
ZP. ergo PN x ZP = ZQ x QD. Liquetigitur curvam DNN
eſſe _hyperbolam_, cujus _aſymptoti_ ZM, ZT.
Siæquatiò ſit - _yz_ + _gx_ + _my_ = {_m_/_r_} _xx_;
eadem erit _hyper-_
_bola_. Sed puncta G inter B, M tunc accipiuntur; & ità prout aliis
ac aliis locis puncta G deſignantur, æquationis ſigna variantur; at
non eſt ea jam exponendi locus.
_bola_. Sed puncta G inter B, M tunc accipiuntur; & ità prout aliis
ac aliis locis puncta G deſignantur, æquationis ſigna variantur; at
non eſt ea jam exponendi locus.
XIV.
Poſitione datæ ſint rectæ DB, BA;
pérque rectam DB
feratur recta CX ad BA parallela; item per punctum D rotando
22Fig. 47. tranſeat recta DY, ſic ipſam BA ſecans in E, ut ſit inter rectas BE,
DC eadem ſemper proportio (puta quæ cujuſdam aſſignatæ R ad DB)
rectæ verò DE, CX ſe interſecent punctis N; erit linea DNN _Pa-_
_rabola_.
feratur recta CX ad BA parallela; item per punctum D rotando
22Fig. 47. tranſeat recta DY, ſic ipſam BA ſecans in E, ut ſit inter rectas BE,
DC eadem ſemper proportio (puta quæ cujuſdam aſſignatæ R ad DB)
rectæ verò DE, CX ſe interſecent punctis N; erit linea DNN _Pa-_
_rabola_.
Nam ſit R.
DB:
: DB.
P.
Eſt ergò BE.
DC:
: DB.
P.
Item
eſt DB. BE: : DC. CN. ergò DB. BE + BE. DC = DC.
eſt DB. BE: : DC. CN. ergò DB. BE + BE. DC = DC.
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