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

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322129 am puncto M, ductíſque rectis MPX ad BD, & MQY ad AD
parallelis, poſitóque rectam MT tangere curvam AMB, ſit TP.
11Fig.201.PM: : QY. PX; erunt figuræ ADKE, DBLG ſibimet æqua-
les.

Valet hoc converſum. Nempe ſi figuræ ADKE, DBLG æ-
quentur, & MT curvam AMB tangat, erit TP. PM: : QY.
PX:

_Not_. Omnium hactenus Propoſitorum fœcundiſſimum eſt hoc
_Tbeorema_; præcedentium quippe complura vel in eo continentur, aut
ab eo facilè conſectantur. Nam poſito lineam AMB indeterminatam
eſſe naturâ, ſi ipſarum EXK, GYL alterutra pro tuo arbitratu de-
terminetur, exinde reſultabit Theorema quoddam ejuſmodi, qualia
ſuperiùs exhibentur aliquammulta. Si _e. g_. linea GYL ponatur recta
cum ipſa BD ſemi-rectum conſtituens angulum (quo caſu concipiun-
tur puncta D, G coincidere) proveniet indè prima _Lectionis_ XI. Si
GYL ſit recta ad DB parallela, emerget _Lectionis ejuſdem._ Rur-
22Fig. 202. ſus ſi PM = PX (vel lineæ AMB, EXK ſint eædem) conſeque-
tur hinc _decima_ ejuſdem. Exhinc porrò liquet adſumpto cuilibet ſpa-
tio _infinita, genere diverſa, ſpatia æqualia_ facilè deſignari veluti ſi _ſpa-_
_tium_ DGLB ponatur _circuli quadrans_, cujus _centrum_ D; & curva
AMB ſit _parabola_, cujus _axis_ AD, emerget curvæ EXK hæc pro-
prietas, ut (ſi dicatur DB = r; AP = x; PX = y; & _k_ (vel
{DB q/2 AD}) ſit _parabolæ ſemipar ameter_) ſit {_rrk_/2} = _kkx_ + _xyy_. Sin
AMB ponatur _hyperbola_, procreabitur alterius generis curva EXK.
his autem expenſis ἀβλεφιαν meam incuſo, qui non hoc _Theorema_ (ſi-
cut & ea quæ ſubſequuntur, quorum ferè ratio conſimilis eſt, & ſup-
par uſus) primo loco poſuerim, & ex eo (nec non è reliquis mox
ſubjiciendis) quod fieri poſſe video, reliqua deduxerim. Veruntamen
hujuſmodi _Phrygiam ſapientiam_ juxta mecum pleriſque familiarem au-
tumo, literas has tractantibus.

_Theor_. V.

Sit ſpatium quodpiam ADB (rectis DA, DB, & curva AMB
33Fig. 203. comprehenſum) ſint item curvæ EXK, GYL ità relatæ, ut ſi in curva
AMB liberè ſumatur punctum M, ducatur DMX, ſit DQ = DM,
ducatur QY ad DB perdendicularis, ſit DT ad DM perpendicula-
ris, recta MT curvam AMB contingat; ſi, his inquam ſuppoſitis, ſit
TD. DM: : DM x QY. DXq; erit ſpatium DGLB ſpatii EDK
duplum.

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