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

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29299& gt; NO. quare liquent ea, quæ Propoſita ſunt.

Si _Circulo_ ſubſtituatur _Ellipſis_, eadem concluſio valet idem diſcur-
ſus probat; pofitâ AH _Ellipſis parametro_.

XVIII Habeant _hyperbola_ AEB (cujus axis AZ, parameter AH)
& _parabola_ AFB axin eundem AD, & baſin DB, _parabola_ ſupra
DB tota extra _hyperbolam_ cadet, extra verò, ſi infra DB protraha-
11Fig. 143.tur.

Nam connexæ ZH occurrat BD in I; ergò DI eſt _parabolæ pa-_
_rameter_. Quòd ſi ſupra BD utcunque ducatur recta FEGK ad BD
parallela, ſecans hyperbolam in E, parabolam in F, rectas AD, ZH
punctis G, K, erit FGq = AG x DI & gt; AG x GK = EGq. qua-
re FG & gt; EG. Quòd ſiinfra BD, utcunque ducatur recta MNOL
ſecans hyperbolam in N, parabolam in M, rectas AD, ZH in O, &
L, erit NO q = AO x OL & gt; AO x DI = MOq. & indè NO
& gt; MO. unde conſtant ea quæ propoſita ſunt.

XIX. E dictis eliciuntur hæ _ad Circuli dimenſionem pertinentes regu-_
_la._ Sit BAE circuli portio, cujus axis AD, baſis BE; ſitque C
22Fig. 144. centrum circuli, & EH ſinus rectus arcus BAE; item, ſit AD =
{_s_/_t_} CA; erit 1. {2 _t_ - _s_/3 _t_ - 2 _s_} AD x BE & gt; port. BAE.

2. EH + {4 _t_ - 2 _s_/3 _t_ - 2 _s_} BH & gt; arc. BAE.

3. {2/3} AD x BE & lt; port. BAE.

4. EH + {4/3} BH & lt; arc. BAE.

XX. Itidem hæ deducuntur ad _hyperbolæ dimenſionem ſpectantes re-_
_gulæ_. Sit _hyperbolæ_ (cujus centrum C) ſegmentum ADB, habens
33Fig. 145. axin AD = {_s_/_t_} CA; & baſin DB;

erit 1. {2 _t_ + _s_/3 _t_ + 2 _s_} AD x DB & lt; ſegm. ADB. &

2. {2/3} AD x DB & gt; ſegm. ADB.

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