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VI.
_Hyperbolam_ AEB, cujus _Centrum_ C, tangant duæ rectæ
BT, ES, & reliqua ponantur ut in proximè præcedente; erit T D:
11Fig. 137. A D & lt; SP. AP.
BT, ES, & reliqua ponantur ut in proximè præcedente; erit T D:
11Fig. 137. A D & lt; SP. AP.
Nam eſt CA.
CD:
: CT.
CA.
unde CA - CT.
CD -
CA: : CT. CA; hoceſt TA. AD: : CT. CA. ſuppare diſ-
curſu, eſt SA. AP: : CS. CA. Verùm eſt CT. CA & lt; CS.
CA. quare TA. AD & lt; SA. AP; ſeu componendo TD. AD
& lt; SP. AP.
CA: : CT. CA; hoceſt TA. AD: : CT. CA. ſuppare diſ-
curſu, eſt SA. AP: : CS. CA. Verùm eſt CT. CA & lt; CS.
CA. quare TA. AD & lt; SA. AP; ſeu componendo TD. AD
& lt; SP. AP.
VII.
_Circali_ AEB (cujus _Centrum_ C) &
_paraboliformis_ AFB
communes ſint axis AD, & baſis BD; ſit autem _paraboliformis_ ex-
ponens {_n_/_m_}; & AD = {_m_ - 2 _n_/_m_ - _n_} CA (vel _m_ - _n_. _m_ - 2 _n_: :
CA. AD) _circulum_ verò tangat recta BT; hæc quoque _paraboli-_
_formem_ AFB continget.
communes ſint axis AD, & baſis BD; ſit autem _paraboliformis_ ex-
ponens {_n_/_m_}; & AD = {_m_ - 2 _n_/_m_ - _n_} CA (vel _m_ - _n_. _m_ - 2 _n_: :
CA. AD) _circulum_ verò tangat recta BT; hæc quoque _paraboli-_
_formem_ AFB continget.
Nam quia BT _circulum_ tangit, eſt CT CA:
: CA.
CD;
unde TA.
22Fig. 138.AD: :. CACD. componendóque TD. AD: : CA + CD. CD Item, quo-
uiam eſt (ex hypotheſi) CA. AD: : _m_ - _n_. _m_ -2 _n_; erit per ratio-
nis converſionem CA. CD: : _m_ - _n. n_. & componendo CA +
CD. CD: : _m. n_. hoc eſt TD. AD: : _m. n_. unde palàm 332 _hujus ap._ quòd BT _paraboliformem_ AFB tangit.
22Fig. 138.AD: :. CACD. componendóque TD. AD: : CA + CD. CD Item, quo-
uiam eſt (ex hypotheſi) CA. AD: : _m_ - _n_. _m_ -2 _n_; erit per ratio-
nis converſionem CA. CD: : _m_ - _n. n_. & componendo CA +
CD. CD: : _m. n_. hoc eſt TD. AD: : _m. n_. unde palàm 332 _hujus ap._ quòd BT _paraboliformem_ AFB tangit.
VIII.
Subnotetur, quòd inversè, datâ ratione ipſius AD ad CA’
deſignabitur hinc _paraboliformis_; quæ _Circulum_ AEB ad B contin-
get. Nempe, ſi AD = {_s_/_t_}, erit {_t_ - _s_/2 _t_ - _s_} dictæ _paraboliformis ex-_
ponens. Nam poſito fore {_t_ - _s_/2 _t_ - _s_} = {_n_/_m_}; erit ideò (juxta crucem
multiplicando) _mt_ - _ms_ = 2 _tn_ - _sn_; & tranſponendo _mt_ -
2 _nt_ = _ms_ - _ns_. ac ideò (æqualitatem ad analogiſmum redigendo)
_m_ - _n. m_ - 2 _n_: : _t. s_: : CA. AD. itaque conſtat ex anteceden-
te Propoſitum.
deſignabitur hinc _paraboliformis_; quæ _Circulum_ AEB ad B contin-
get. Nempe, ſi AD = {_s_/_t_}, erit {_t_ - _s_/2 _t_ - _s_} dictæ _paraboliformis ex-_
ponens. Nam poſito fore {_t_ - _s_/2 _t_ - _s_} = {_n_/_m_}; erit ideò (juxta crucem
multiplicando) _mt_ - _ms_ = 2 _tn_ - _sn_; & tranſponendo _mt_ -
2 _nt_ = _ms_ - _ns_. ac ideò (æqualitatem ad analogiſmum redigendo)
_m_ - _n. m_ - 2 _n_: : _t. s_: : CA. AD. itaque conſtat ex anteceden-
te Propoſitum.