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

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3. Curva HLLI eſt _ſemicirculus_; reliquas itidem oſtentat
Schema.

4. Si A ζ = {_cc_/_b_}; A Ψ = {_b_/4} - √ {_bb_/16} - {_cc_/2}; & A φ = {_b_/4} +
√ {_bb_/16} - {_cc_/2}; ordinentúrque rectæ ζ V, ψ X, φ Y; erunt puncta V,
X, Y _nodi_ curvarum (ſi _b_ & lt; √ 8 _c c_, deerunt _nodi_ X, Y; ſi _b_ = √
8 _c c_; ii coaleſcent).

5. Ordinatarum ad curvam CL H _maxima_ eſt ipſa AC ; ſin AP
= {_b_/3} - √ {_bb_/9} - {_cc_/3}, & ordinetur P γ ad curvam AM H; erit
P γ _maxima_; item ſi AQ = {3/8} _b_ - √ {@9/64} _b b_ - {_cc_/2}; & ordinetur
Q δ ad curvam AN H, erit Q δ _maxima_.

6. Ordinatarum ad curvam HLLI _maxima_ eſt ipſa OT ; ſin AP
= {_b_/3} + √ {_bb_/9} - {_cc_/3}, & ad curvam HM I ordinetur _p g_, erit _p g_
_maxima_; item ſi A q = {3/8} _b_ + √ {9/64} _b b_ - {_cc_/2}; & ordinetur _q d_
ad curvam HN I, erit _q d maxima_.

7. Hinc radicum limites dignoſcentur, ut innuitur in iis, quæ ad
octavam ſeriem animadverſa ſunt.

8. Patet in Serie duodecima nunc tres, modo duas, ſemper unam
radicem haberi; in decima tertia verò ſubinde duas, aliquando tantùm
unam, interdum nullam haberi.

9. Et hæc quidem conſtant poſito fore {_b_/2}& gt; _c_; at ſi {_b_/2} = β;
evaneſcet Series decima tertia; coaleſcent puncta H, O, I; recta AB
_byperbolam_ KK K tanget; curvæque CL H, IL λ in rectas lineas
degenerabunt.

10. Sin {_b_/2} & lt; _c_; etiam evaneſcit Series decima tertia; _byperbola_ KKK
tota infra rectam AB jacente; quo caſu curva CL L erit hyperbola
æquilatera, habens centrum O, ſemiaxem (ipſi AB perpendicula-
rem) OT = √ AC q - AO q; tunc & curvæ AM M, AN N
11Fig. 218. ad infinitum procurrent, ſic ut æquationes, quæ in Serie duodecima,
unam ſemper, & unicam radicem obtineant. Hæc ſuffecerit inſinu-
âſſe; quin & rem totam hactenus particulatim attigiſſe. Subnecte-
mus autem notas quaſdam magìs generales.

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