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

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8466 centrum C) datáque ſint duo puncta A, X; reperiendum eſt in
circumferentia punctum aliquod; à quo ductæ ad A, X rectæ, altera ſit
alterius reflexa. ‖ Hocità perficimus:

Conjungantur rectæ AC, XC; & fiat (ſeorſim) ang. δ = {1/2} ang.
11Fig. 92, 93. AC X. & in ξ δ crure anguli δ ſumpto liberè puncto π ducatur π V
ad ξ δ perpendicularis alterum crus ſecans in V; & in V π protracta
capiatur π γ = π V; tum dividatur γ V in φ, ut ſit γ φ. φ V: : XC. CA;
perque punctum φ trajiciatur κ ξ ſic ut ſit κξ. κ v: : CX. CN. denique
22L@m. pr@ced. fiat angulus XCNæ qualis angulo ξ κ v; erit punctum N quale deſi-
deramus. Nam ducantur XN, ξ v; & fiat ang. CNG = ang.
κ v γ. adſumatúrque PG = PN; & connectatur XG. liquet jam
trangula XCN, ξ κ v ſimilia fore; nec non ipſa CNF, κ v φ;
& ipſa XP F, ξ π φ; ipsáque demùm XFN, ξ φ v aſſimilari. quare
PF. XF: : π φ. ξ φ. & XF. FN: : ξ φ. φ V. & ex æquo PF.
FN: : π φ. φ V. & antecedentes duplando 2 PF. FN : : 2 π φ. φ V.
componendóque 2 PF + FN. FN: : 2 π φ + φ V. φ V. hoc eſt
GF. FN: : γ φ. φ V (hoc eſt): : XC. CA. ducatur jam NL ad
XG parailela; quare eſt ang. ING = ang. G = ang. XNG;
& XG (XN). NL: : GF. FN: : XC. CA. porro fiàt ang.
LNH = ang. XCA; & HN protracta ipſi CA occurrat in M;
cſtque proptereà triangulum HNLſimile triangulo HCM; idcir-
cóque HC. CM: : HN. NL. ducatur denuò tangens NQ; eſt-
que tum ang. PNQ = rect - CNP = rect - κ V π = ang.
δ = {1/2} XC A; vel 2 ang. PNQ = ang XCA = ang LNH.
verùm erat priùs 2 ang. XNF = ang. XNL. ergo 2 ang XNF
- 2 ang PNQ = ang XNL- ang. LNH. hoc eſt 2 ang XNQ
= ang. XNH. ergo tangens NQ biſecat angulum XNH; indéque
conſectatur fore rectam HM ipſius XN reflexam; ac ideò eſſe XC.
HC: : XN. HN. atqui fuit priùs HC. CM: : HN. NL quare
jam erit ex æquo XC. CM: : XN. NL (h@c eſt etiam è præmon-
ſtratis): : XC. CA. unde CM = CA. quapropter HM, ipſius
XN reflexa tranſit per A: Quod propoſitum erat efficere. ‖

VIII. Hujuſce _Problematis_ ità generaliùs propoſiti varii quidem
caſus ſunt (etenim vel data puncta jacent ambo extra circuh@ re-
flectentem; vel utrumque poſitum eſt intra circulum; vel unum intra
jacet, alterum extra; quinetiam in horum caſuum unoquoque pluries
conficitur negotium) aſt ubique non abſimilis erit conſtructio; ſanè
nimius eſſem; meámque pariter ac veſtram patientiam macerarem
omnes intricati _Problematis_ nodos evolvendo; ſuffecerit ejuſce ſpeci-
men aliquod protuliſſe.

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