Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

multiplicando) 2 _tn_ + _sn_ = _mt_ + _ms_. vel tranſponendo 2 _nt_ -
_mt_ = _ms_ - _ns_. unde _m_ - _n_. 2 _n_ - _m_: : _t. s_: : CA. AD. er-
gò patet ex antecedente.

XIV. Stante duodecimæ hypotheſi, _paraboliformis_ AFB intra hy-
perbolam AEB tota cadet.

Nam utcunque ducatur EFG ad BD parallela; & recta ER _hy-_
_perbolam_, recta FS _paraboliformem_ tangant. Eſtque SG. AG: : _m. n_: : TD. AD & lt; RG. AG. unde RG & gt; SG. unde curva AEB extra curvam AFB tota cadet.

42.1.

Fig. 141.
2. _hujus ap._
6. _hujus ap._
3. _hujus ap._

XV. Etiam, ſi reliquis perſtantibus, ad baſin GE, axin AG con-
ſtitutam imagineris ejuſdem ordinis _paraboliformem_; hæc ad partes
ipsâ GE ſuperiores intra _hyperbolam_ tota cadet.

42.1.

Fig. 141.

Nam ſi in _curva hyperbolica_ AE ſumatur ubicunque punctum M, & ordinetur MP, ducatúrque hyperbolam tangens MV; erit VP. AP & gt; _m. n._ adeoque rurſus è tertia liquet Propoſitum.

XVI. Quinetiam ſi hæc altera coordinata _paraboliformis_, ad baſin
EG conſtituta, ad DB protracta concipiatur, ejus ipſis EG, BD in-
tercepta pars extra _hyperbolam_ tota cadet.

42.1.

Fig. 141.

Nam quòd extra _hyperbolam_ infra EG cadit, exinde patet, quòd
ipſa cum ipſius tangente recta ES angulum efficit minorem eo, quem
eadem recta ES efficit cum recta RE hyperbolam tangente. quòd au-
tem eadem alibi, velut ad N, _hyperbolæ_ non occurrit, patet; quoniam
hoc poſito, ipſa intra _hyperbolam_ AN tota conſiſteret, contra quàm mox oſtenſum eſt.

42.1.

3. _hujus ap._

XVII. Habeant _Circulus_ AEB, & _parabola_ AFB communem
axem AD, & baſin DB; _parabola_ ad partes ſupra BD intra _Circu-_
_lum_; at infra BD extra _circulum_ cadet.

Sit enim _Circuli Diameter_ AZ, & eiæqualis A Had BD paralle-
la, & connectatur ZH; & huic BD producta ad I; ergo DI eſt
_Parameter parabolæ_ AFB. quòd ſi ſupra BD utcunque ducatur recta
EF GK ad BD parallela circulum ſecans in E, parabolam in F, rectas
AZ, HZ, in G, & K, patet eſſe GEq = AG x GK & gt; AG x DI
= GFq. unde GE & gt; GF. Item, ſi infra BD utcunque ducatur
recta MN OL ad BD parallela _parabolam_ ſecans in M, _circu-_
_lum_ in N, rectas AZ, HZ in O, & L, itidem patet eſſe MO q
= AO x DI & gt; AO x OL = NO q. & ideò M O

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