24249
+ _xy_ - {_mxx_/_b_} = _o_.
ergò PN x MG = {_m_3/_b_} = MD x ZQ.
vel PN. ZQ: : (MD. MG: :) QD. ZP. Quapropter eſt
11Fig. 44. PN x ZP = ZQ x QD. Unde palàm eſt curvam DNN eſſe hy-
perbolam, cujus aſymptoti ZM, ZT.
vel PN. ZQ: : (MD. MG: :) QD. ZP. Quapropter eſt
11Fig. 44. PN x ZP = ZQ x QD. Unde palàm eſt curvam DNN eſſe hy-
perbolam, cujus aſymptoti ZM, ZT.
XI.
Notetur;
ſi æquatio ſit _my_ - _xy_ = {_m_/_b_}_xx_;
eadem ha-
bebitur _hyperbola_; tunc ſolùm puncta G ad partes DM ſumuntur.
Quin & fi æquatio ſit _xy_ - _my_ = {_m_/_b_} _xx_; puncta G ultra M
capiendo, proveniet _hyperbola_, huic ipſi _conjugata_.
bebitur _hyperbola_; tunc ſolùm puncta G ad partes DM ſumuntur.
Quin & fi æquatio ſit _xy_ - _my_ = {_m_/_b_} _xx_; puncta G ultra M
capiendo, proveniet _hyperbola_, huic ipſi _conjugata_.
XII.
Sit Triangulum BDF;
&
linea DNN talis, ut ductâ ut-
22Fig. 45. cunque RN ad BD parallelâ (quæ lineas BF, DF, DNN ſecet
punctis R, G, N) connexâque rectâ DN; ſit perpetuò DN propor-
tione media inter RN, NG; erit linea DNN _hyperbola_.
22Fig. 45. cunque RN ad BD parallelâ (quæ lineas BF, DF, DNN ſecet
punctis R, G, N) connexâque rectâ DN; ſit perpetuò DN propor-
tione media inter RN, NG; erit linea DNN _hyperbola_.
Per D ducatur DK ad DB perpendicularis (ſecans ipſam RN in E)
& ſit FH ad DB parallela; vocentúrque DB = _b_; DF = _d_; FH
= _f_; tum DG = _x_; & GN = _y_; Eſtque _d. f: : x._ {_fx_/_d_} = _GE_;
unde {_zfxy_/_d_} + _xx_ + _yy_ = 2 EG x GN + DGq + GNq
= DN q. Porrò eſt _d. b_: : FG. GR: : _d_ - _x_. RG = _b_ - {_bx._ /_d_} Un-
de RN = _b_ - {_bx_/_d_} + _y_. Et ideò _by_ - {_bxy_/_d_} + _yy_ = RN x
NG = DNq = {2_fxy_/_d_} + _xx_ + _yy_. quare _by_ - {_bxy_/_d_} =
{2_fxy_/_d_} + _xx_. quam æquationem ordinando fit {_db_/2_f_+_b_}_y_ - _yx_ =
{_d_/2_f_+_b_}_xx_. quòd ſi ponatur _m_ = {_db_/2_f_+_b_3} erit _my_ - _xy_ =
{_m_/_b_}_xx_. Unde liquet DNN eſſe _hyperbolam_, qualis habetur in præ-
cedente determinata,
& ſit FH ad DB parallela; vocentúrque DB = _b_; DF = _d_; FH
= _f_; tum DG = _x_; & GN = _y_; Eſtque _d. f: : x._ {_fx_/_d_} = _GE_;
unde {_zfxy_/_d_} + _xx_ + _yy_ = 2 EG x GN + DGq + GNq
= DN q. Porrò eſt _d. b_: : FG. GR: : _d_ - _x_. RG = _b_ - {_bx._ /_d_} Un-
de RN = _b_ - {_bx_/_d_} + _y_. Et ideò _by_ - {_bxy_/_d_} + _yy_ = RN x
NG = DNq = {2_fxy_/_d_} + _xx_ + _yy_. quare _by_ - {_bxy_/_d_} =
{2_fxy_/_d_} + _xx_. quam æquationem ordinando fit {_db_/2_f_+_b_}_y_ - _yx_ =
{_d_/2_f_+_b_}_xx_. quòd ſi ponatur _m_ = {_db_/2_f_+_b_3} erit _my_ - _xy_ =
{_m_/_b_}_xx_. Unde liquet DNN eſſe _hyperbolam_, qualis habetur in præ-
cedente determinata,