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XXV.
Nè ſpeculatio præſens, _ob bujuſmodi complures metbodos Cy-_
_clometricas indies promulgatas_, aſpernanda videatur, adjungemus con-
ſectarium unum vel alterum, quibus fortè ſolis hæc paucula meruerant
11Fig. 148. impendi; à quibus nempe _Maxima, Minimaque_ ſui generis innume-
ra determinantur.
_clometricas indies promulgatas_, aſpernanda videatur, adjungemus con-
ſectarium unum vel alterum, quibus fortè ſolis hæc paucula meruerant
11Fig. 148. impendi; à quibus nempe _Maxima, Minimaque_ ſui generis innume-
ra determinantur.
Sit _Semicirculus_ ABZ, cujus centrum C;
ſitque _ſegmentum_
ADB; & huic adſcripta _paraboliformis_ AFB, cujus exponens {_u_/_m_};
ſit item AD = {_m_ - 2 _n_/_m_ - _n_} CA; _paraboliform@s_ autem _parameter_ (hoc
eſt recta, cujus aliqua poteſtas in poteſtatem ſegmenti axis, ſeu AD,
ducta conficit _poteſtatem_ ordinatæ, ceu D B) nominetur _p_; erit _p_ in ſuo
genere _maximum_.
ADB; & huic adſcripta _paraboliformis_ AFB, cujus exponens {_u_/_m_};
ſit item AD = {_m_ - 2 _n_/_m_ - _n_} CA; _paraboliform@s_ autem _parameter_ (hoc
eſt recta, cujus aliqua poteſtas in poteſtatem ſegmenti axis, ſeu AD,
ducta conficit _poteſtatem_ ordinatæ, ceu D B) nominetur _p_; erit _p_ in ſuo
genere _maximum_.
Nam utcunque ducatur GE ad DB parallela, &
ad GE poſita
concipiatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter_ di-
catur _q_. quum ergò _paraboliformis_ AFB _circulum_ extrorſum contin-
gat, erit GF & gt; GE; adeóque GF {_m_/ } & gt; GE {_m_/ }; hoc eſt _p_ {_m_ - _n_/ } x
AG {_n_/ } & gt; q {_m_ - _n_/ } x AG {_n_/ }; quare _p_ & gt; _q_.
concipiatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter_ di-
catur _q_. quum ergò _paraboliformis_ AFB _circulum_ extrorſum contin-
gat, erit GF & gt; GE; adeóque GF {_m_/ } & gt; GE {_m_/ }; hoc eſt _p_ {_m_ - _n_/ } x
AG {_n_/ } & gt; q {_m_ - _n_/ } x AG {_n_/ }; quare _p_ & gt; _q_.
Notandum eſt eſſe _p_ {2 _m_ - 2 _n_/ } = ZD_m_ x AD {_m_ - 2 _n_/ }.
&
q {2 _m_ - 2 _n_/ }
= ZG {_m_/ } x AG {_m_ - 2 _n_/ }. unde ZD {_m_/ } x AD {_m_ - 2 _n_/ } & gt; ZG {_m_/ } x AG {_m_ - 2 _n_/ }.
quare ZD {_m_/ } x AD {_m_ - 2 _n_/ } eſt maximum.
= ZG {_m_/ } x AG {_m_ - 2 _n_/ }. unde ZD {_m_/ } x AD {_m_ - 2 _n_/ } & gt; ZG {_m_/ } x AG {_m_ - 2 _n_/ }.
quare ZD {_m_/ } x AD {_m_ - 2 _n_/ } eſt maximum.
Exemp.
1.
Sit _n_ = 1, &
_m_ = 3.
erit ideò _p_ {4/ } = ZD {3/ } x AD =
ZD_q_ x BD_q_; vel _p_2 = ZD x BD. Item AD =
{1/2} CA.
ZD_q_ x BD_q_; vel _p_2 = ZD x BD. Item AD =
{1/2} CA.
2.
Sit _n_ = 3, &
_m_ = 10.
erit p {14/ } = ZD10 x AD4.
vel p {7/ } = ZD {5/ } x AD2 = ZD3 x BD4. & AD
= {4/7} CA.
vel p {7/ } = ZD {5/ } x AD2 = ZD3 x BD4. & AD
= {4/7} CA.
XXVI.
Sit item _hyperbola_ (æquilatera) cujus centrum C, axis
ZA; & huic inſcripta _paraboliformis_ AFB cujus expo-
22Fig. 149. nens {_n_/_m_} _parameter p_; ſitque AD = {2_n_ - _m_/_m_ - _n_} CA; erit _p_ ſui gene-
neris maximum.
ZA; & huic inſcripta _paraboliformis_ AFB cujus expo-
22Fig. 149. nens {_n_/_m_} _parameter p_; ſitque AD = {2_n_ - _m_/_m_ - _n_} CA; erit _p_ ſui gene-
neris maximum.
Nam utcunque ducatur EG ad BD parallela;
&
ad EG conſtituta
intelligatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter q._
quum ergo _paraboliformis_ AFB _hyperbolam_ introrſum contingat,
erit GF {_m_/ } & lt; GE {_n_/ }; hoc eſt _p_ {_m_ - _n_/ } x AGn & lt; _q_ {_m_ - _n_/ } x AGn;
quare _p_ & lt; _q_.
intelligatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter q._
quum ergo _paraboliformis_ AFB _hyperbolam_ introrſum contingat,
erit GF {_m_/ } & lt; GE {_n_/ }; hoc eſt _p_ {_m_ - _n_/ } x AGn & lt; _q_ {_m_ - _n_/ } x AGn;
quare _p_ & lt; _q_.