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

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10082 habeatur determinatus; ſuccedit ut breviter etiam ipſiſſimum in ſin-
gulo tali refracto punctum oſtendamus, ad quod illa conſiſtit. in cu-
jus rei gratiam hoc quaſi _Lemma_ præſternemus.

VI. In circulo AN B, cujus centrum C, ſint Semidiametro CA
perpendiculares NE, RF; item Semidiametro CB ſint perpendicu-
lares NG, XH; ſint autem CE, EF ipſis CG, GH proportiona-
11Fig. 122. les; & arcus NR, NX indefinitè parvi; ſeu quaſi minimi dictâ
conditione præditi; dicimus arcum NR ad arcum NX rationem ha-
bere conflatam è rationibus ipſarum CE ad CG, & NG ad NE; vel eſſe
arc. NR, NX: : CE x NG. CG x NE. Nam per N ducatur VT
tangens circulum, ipſiſque FR, HX occurrens punctis T, V. eſt
itaque (propter Summam ex Hypotheſi parvitatem dictorum arcuum)
arc NR. CN: : NT. CN: : EF. EN. item CN. arc NX: :
CN. NV: : NG. GH. quapropter erit arc NR. CN+ CN.
arc NX = (EF. EN + NG. GH = EF. GH+ NG. EN
= ) CE. CG + NG. EN. hoc eſt arc NR. arc NX = CE.
CG + NG. EN: Q. E. D. (vel arc NR. NX = CE x NG.
CG x EN.)

VII. Sit jam radii cujuſvis talis MNP, refringentem interſecantis
punctis N, P, refractus N π (refringentem nempe denuò ſecans
in π) huic autem indeſinitè vicinus (& quaſi proximus) adjaceat
radius QR S, cujus itidem refractus R σ (refringenti nempe rurſus
occurrens in σ), priorem N π decuſſans in Z; biſecentur autem
ſubtenſæ NP, N π punctis G, E: Dico rationem NZ ad GZ com-
poni è rationibus NG ad NE, & CE ad CG.

22Fig. 123,
124.

Nam ducantur rectæ CE (hæc ipſam RS quoque ſecans in F) &
C G; nec non CI ad R σ perpendicularis, & in protracta CG ſu-
@@@@@r CH = CI; & per H ducatur XY ad N π parallela, ſeu per-
pendicularis ad CH; unde eſt XY = R π; & arc NX = Y π &
arc XY = arc R σ adeóque arc NR ±: σ π = 2 arc NX. Eſt-
que prætereà CG. CE: : R. I: : CI. CF: : CH. CF; adeóque
permutatim CG. CH: : CE. CF. ergò (juxta præcedentem) eſt
arc. NR. NX = NG. NE + CE. CG. ad hæc ob illam (quæ
ponitur) arcuum NR, SP, π σ exiquitatem, erit arc NR. π σ : :
ſubtenſa NR. π σ : : NZ. Zσ : : NZ. Zπ. ergò (inverſè compo-
nendo, vel dividendo, tum & conſequentes ſubduplando) arc NR.
{arc NR±: π σ/2}: : NZ. {NZ±Zπ/2}. atqui velut modò

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