Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

GZ, GK media GL, bimedia GM, trimedia GN; propoſitas æ-
quationes explicabunt hæ lineæ. Nam ſi AG (vel GZ) vocetur _a_; erit BG (vel GK) = _a_ - _b_; & GLq = _aa_ - _ba_; & GM cub. = _a_ 3 - _baa_; & GN _qq_ = _a_ 4 - _ba_ 3 .

78.1.

Fig. 207.

79. Not.

1. Ductâ AD ad AI perpendiculari, & EF ad AI parallelâ, ſi
AE ponatur æqualis ipſi _n_; erunt EK, EL, EM, EN radices æqua-
tionum reſpectivæ, ſeu æquales quæſitis _a_.

2. Quoniam ordinatæ GK, GL, GM, GN à termino B verſus I
infinitè excreſcunt, ſemper habetur una vera radix, & unica.

3. Curva BLL eſt _hyperbola æquilatera_, cujus _axis_ AB, reliquæ
curvæ ſunt _hyperboliformes._

4. Si AB biſecetur in O, triſecetur in P, quadriſecetur in Q, du-
cantúrque ad AR parallelæ OT, PV, QX, erunt hæ curvarum BLL,
BMM, BNN _aſymptoti._

5. Hinc ſeqiutur in ſecundo gradu fore _a_ & gt; _n_ + {_b_/2}; in tertio
_a_ & gt;_ n_ + {_b_/3}; in quarto _a_ & gt; _n_ + {_b_/4}; quòd ſi _n_ ſatis magna ſit,
iſtæ inæqualitates ad æqualitatem proximè accedunt.

6. Verarum in his radicum habetur _minima;_ ſcilicet ipſa AB, vel _b_.

80. Series tertia.

_b_ - _a_ = _n_.

_ba_ - _aa_ = _nn_.

_baa_ - _a_ 3 = _n_ 3 .

_ba_ 3 -_a_ 4 = _n_ 4 . & c.

Sit AB = _b_, & anguli RAB, SBA ſemirecti; tum curvæ
ALB, AMB, ANB tales, ut ductâ rectâ GK ad AB utcunque
perpendiculari (quæ lineas expoſitas ſecet, ut vides) ſit inter AG
(ſeu GZ) & GK _media_ GL, _bimedia_ GM, _trimedia_ GN; pro-
poſitas æquationes explicatas dabunt hæ lineæ. Nam poſito fore AG
= _a_, erit GK = _b_ - _a_; & GLq = _ba_ - _aa_; & GMq =
_baa_ - _a_ 3 . & GNq = _ba_ 3 - _a_ 4 .

80.1.

Fig. 280.

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