Barrow, Isaac
,
Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur
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<
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xml:space
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">In _pramiſſas explicationes_ animadvertatur generatim.</
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<
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">1. </
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<
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xml:space
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">Propoſitam quamvis æquationem explicans _@μγνα_ deſignatur
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hoc modo: </
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<
s
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xml:space
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">proponatur, exempli causâ, _æquatio a
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_ + _ba
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_ + _cca
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_
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- _d
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aa_ - _f
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a_ = _n
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_; </
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<
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xml:space
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">In recta indefinitè protenſa HI deſignetur pun-
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<
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">Fig. 219.</
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ctum A, pro radicum termino, vel origine; </
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<
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xml:space
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">tum arbitrariè ſumptâ
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AG pro indeterminatâ radice _a_; </
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<
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xml:space
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">fiat GK æqualis primo feriei pro-
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poſitam æquationem continentis gradu; </
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<
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xml:space
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">nempe ſit hîc GK = _a_ + _b_
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+ {_cc_/_a_} - {_d
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_/_aa_} - {_f
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_/_a
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_} (utique rationem _a_ ad _c_ ſemel continuando fit
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{_cc_/_a_}; </
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<
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xml:space
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">rationem _a_ ad _d_ bis continuando fit {_d
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_/_aa_}; </
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<
s
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xml:space
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">acità porrò) tum inter
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AG, GK tot mediarum proportionalium, quot æquationis propoſitæ
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gradus exigit (is autem à pura quæſitæ radicis poteſtate indicatur) in
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hoc nempe caſu quatuor mediarum proportionalium prima ſit GO;
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</
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<
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">per ejuſmodi puncta O traducta curva AOO propoſito deſerviet.</
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<
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">2. </
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<
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">De radicibus falſis, ſeu negativis nihil attigimus ſuprà; </
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<
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rùm eæ reperiuntur hoc modo. </
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">Æquationi propoſitæ ſubrogetur
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altera, cujus in locis paribus (etiam vacuos locos adnumerando)
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ſigna ſunt illis contraria, quæ habet æquatio propoſita; </
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<
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">erunt hu-
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juſce _ſubdititiæ æquationis_ radices veræ, ſeu poſitivæ ipſius propoſitæ
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æquationis radices falſæ, ſeu negativæ. </
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<
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">_Exemplo_ ſit _æquatio a
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_ + _baa_
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= _n
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_; </
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<
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">vel _a
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_ + _baa*_ - _n
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_ = _o_. </
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<
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">Subrogetur _a
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_ - _baa
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_ + _n
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_ = _o_; </
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">& </
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hujus, + utì ſuprà edoctum, veræ radices deſignentur, hæ _propoſitæ_
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_aquationis_ falſæ erunt. </
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<
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_ - _baa_ = _n
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_; </
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<
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">vel _a
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_ - _baa_ - _n
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_
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= _o_; </
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<
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">ſubſtituatur æquatio _a
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_ + _baa_ + _n
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_ = _o_; </
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<
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">hæc nullam veram
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radicem obtinet; </
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<
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">ergò nec _æquatio propoſita_ falſam admittit.</
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<
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<
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">Quinimò datâ verâ radice quâpiam, depreſſioris gradûs æqua-
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tio quædam ſalſis reperiendis inſerviet, qualis ità determinatur. </
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<
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">Pro-
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ponatur æquatio quævis, puta _a
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_ + _baa_ = _n
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_; </
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<
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">cujus nota ſit radix una,
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quæ vocetur _f_. </
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<
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">Conſtruatur æquatio planè ſimilis propoſitæ, eáſ-
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demque _coefficientes_ habens, tantum pro _a_ ſubſtituendo _f_; </
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<
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_f
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_ + _bff_ = _n
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_. </
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<
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">ergo _a
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_ + _baa_ = _n
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_ = _f
<
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_ + _bff_; </
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<
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">adeóque
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_a
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_ + _baa_ - _f
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_ - _bff_ = _o_. </
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<
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">dividatur hæc æquatio (id quod ſem-
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per fieri poteſt) per _a_ - _f_; </
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<
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xml:space
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">proveniet _a a_ { + _ba_ + _bf_ + _fa_ + _ff_} = _o_; </
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<
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">cujus æ-
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quationes eædem erunt cum reliquis æquationis propoſitæ radicibus;
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</
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<
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">quæ proinde duas colligitur radices falſas habere; </
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<
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">itaque mutatis loco-
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rum parìum ſignis, ut ità fiat _a a_ { - _ba_ + _bf_/ - _fa_ + _ff_} = _o_; </
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<
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