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

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4. Conſequentèr in harum ſecundo gradu ſin & gt;_ c_; in tertio, ſi _n_3
& gt; _cc_√{_cc_/3} - {_cc_/3} √ {_cc_/3} = {2/3}_cc_ √ {_cc_/3}; vel _n_6& gt; {@@/27}_c_6; in quar-
to ſi _n_4& gt; {_c_4/4} - {_c_4/16} = {3/16}_c_4; nulla radix habetur; unam in iſtis
caſibus recta EF curvas ſupergreditur; nec iis occurrit.

5. Itidem in his omnibus maxima poſſibilis radix eſt AH = AC.

6. Curva CYH eſt _Circuli quadrans_, reliquæ AMH, ANH
quodammodo κυχλο{ει}δ{ετ}ς.

7. Ad ſextam ſeriem pertinentium curva HLL eſt _byperbola æqui_-
_latera_, cujus axis AH; reliquæ ſunt _Hyperboliformes_. Unde quoad
hanc ſeriem liquent cætera.

Series ſeptima.

_a_ + _b_ + {_cc_/_a_} = _n_.

_aa_ + _ba_ + _cc_ = _nn._

_a_3 + _baa_ + _cca_ = _n_3.

_a_4 + _ba_3 + _ccaa_ = _n_4, & c.

In recta BAH indefinitè protensâ capiatur AB = _b_; & in AD
11Fig. 214. ad BH perpendiculari ſit AC = _c_; ſint etiam anguli HAR, HBS Semi-
recti; tum arbitrariè ductâ GY ad AH perpendiculari quæ ipſam
BS ſecet in Y; fiat AG. AC: : AC. YK; & per K intra angulum
DVS deſcribatur _hyperbola_ KKK; ſint demum curvæ CLL, AMM,
ANN tales, ut inter AG (vel GZ) & GK ſit _media_ GL, _bime_-
_dia_ GM, _trimedia_ GN; hæ ſatisfacient negotio. Nam eſt GK = _a_
+ _b_ + {_cc_/_a_}; & GLq = _aa_ + _ba_ + _cc_; & GMcub = _a_3 + _baa_
+ _cca_; & GNqq = _a_4 + _ba_3 + _ccaa_.

Not.

1. Secundi gradûs curva CLL eſt pars _hyperbolæ æquilateræ_, cujus
_centrum_ O, ipſam AB biſecans; & ſiquidem AC& gt; AO, eſt OH
(ad AB perpendicularis, &) = √ ACq - AO qejus _ſemiaxis_;
ſin AC& lt; AO, ejus axis eſt OI = √ AOq - ACq. reliquæ
verò curvæ AMM, ANN ſunt _hyperboliformes_.

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