Barrow, Isaac - Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

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24148 linea OOO talis, ut ſi è D utcunque ducatur recta DO, ſecans
anguli latera punctis M, N, habeat DM ad NO ſemper eandem
11Fig. 43. rationem (puta X ad Y) erit etiam linea OOO hyperbola.

Nam ducatur DL ad BC parallela; ſitque DL. DE: : X. Y;
& per E ducatur ER ad AB parallela; ſecans BC in Z; de-
mum per O ducatur OH ad BA parallela.

Eſt jam DL. DE: : DM. NO: : LM. GO (ob ſimilia tri-
angula DLM, NGO): : LM x DH. GO x DH item DL x
HO = LM x DH (ob DL. LM: : DH. HO) quare DL. DE: :
DL x HO. GO x DH hoc eſt DL x HO. DE x HO: : DL x HO.
GO x DH adeóq; DE x HO = GO x DH. hoc eſt DE x HG + DE x
GO = GO x DE + GO x EH quare (communi ſublato) eſt
DE x HG = GO x EH; ſeu DE x HG = GO x ZG. Pa-
tet itaque curvam OOO eſſe _hyperbolam_ cujus _aſymptoti_ ZR
ZC.

_Coroll_. Si ratio data ſit æqualitatis (ceu DM = NO,) ipſæ AB,
CB aſymptoti erunt.

Sequentia quædam, quia magìs id perſpicuum videtur, Alge-
bricè monſtrabimus.

X. Eſto poſitione data recta ID, in qua punctum deſignatum D,
ſit item curva DNN talis ut in ID ſumpto quopiam puncto G, &
22Fig. 44. ductâ rectâ GN ad poſitionem datam IK paràllelá; tum adſumptis
determinatis rectis _m, b_; poſitiſq; DG = _x_, & GN = _y_; ſit
conſtantèr _m y_ + _x y_ = {_m_/_b_}_x x_; erit linea DNN _hyperbola_; quæ
ſic determinatur; ſumantur DM, & DO (hinc indè) pares ipſi _m_;
& per M ducatur M L@ad IK parallela, factóq; _b. m_: : _m_. MQ; ſit
MZ = 2 MQ = {2_mm_; /_b_} tum per Z, O traducatur recta ZT; erunt
ZM, ZT aſymptoti.

Ducatur enim ZS ad MO parallela, cui occurrat N Gin R (quæ
& ipſam ZT ſect in P). & connectatur DQ. Eſt ergò PN = RG
+ GN - RP. Verùm eſt MD. MQ: : ZR (MG). RP; hoc
eſt _m_. {_mm_/_b_}: : _m_ + _x_. RP = {_mm_/_b_} + {_mx._ /_b_} adeóq; RG - RP
= {_mm_/_b_} - {_mx._ /_b_} ergò PN = {_mm_ - _mx_/_b_} + _y_. Unde PN x MG
= {_m_3/_b_} + _my_ + _xy_ - {_mxx._ /_b_} Verùm (ex hypotheſi) eſt _m

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