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

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4325 obliquiùs quàm DB.) Horum verò refracti ſint B _a_, Bδ; dico an-
11Fig. 21. gulum β B _a_ majorem eſſe angulo HB δ. Nam ad BP in perpendicu-
lari liberè ſumptam diametrum conſtituatur ſemicirculus BGP; cui
occurrant ipſæ AB, DB protractæ ad G, H; nec non ipſæ B _a_, B δ
punctis _a_, δ. Fiat autem angulus GBK æqualis angulo HBδ, vel
arcus GK arcui Hδ; connectatur etiam rècta δ G, ſecans ipſam PK
in X; ducatnurque denuò ſubtenſæ G δ, H δ. Jam ob angulos PG δ,
PH δ pares (arcui quippe P δ inſiſtentes ambos) & angulos GPK,
22Fig. 22. HP δ ex conſtructione quoque pares, erunt triangula GPX,
HP δ inter ſe ſimilia. Quapropter erit PG. PX : : PH. P δ. eſt
autem, è lege refractionum PH. P δ : : PG. P _a_. quare PG. PX : :
PG. P _a_: unde PX = P _a_. eſt autem PX minor quàm PK (quia
tota ſubtenſa G δ intra circulum jacet.) Quare P _a_ minor eſt quàm
PK; adeóque PK ſecabit angulum GP _a_. quamobrem arcùs G _a_ ma-
jor erit arcu GK, hoc eſt arcu H δ. & idcircò major erit angulus
GB _a_ angulo HB δ: Q. E. D.

Procedit hæc demonſtratio quoad caſum, ubi I & gt; R (vel cùm ra-
dius è medio rariori denſius ingreditur) at exinde quoad alterum quo-
que caſum facilè deducitur concluſio. Nam ſi viciſſim _a_ B, δ B con-
cipiantur incidentes, erunt ipſæ BA, BD earum refractæ; ac etiam-
num anguli _a_ BG, δ BH erunt anguli refracti.

Hujuſce Theorematis apud _Herigonium_ habetur alia demonſtra-
33_Diopt. Prop@.4_. tio. Confer ſodes, & utramvis elige. No3 quam res obtulit
poſuimus.

VII. In iſto refractionis caſu, quum I minor eſt quàm R, ſi anguli
44Fig. 23. incidentiæ, puta anguli DBQ, rectus ſinus PH, ad ſinum totum ſe
habeat ut I ad R; nullus incidente DB obliquior radius medium EF
refractus ingredietur, aut penetrabit.

Nam penerret (ſi fieri poteſt) obliquioris alicujus ABG refractus
B _a_. Erit ergo PG. P _a_ : : (I. R : : ) *PH. PB. eſt autem PG
55*_Hypotb_. major quàm PH. ergo P _a_ major erit quam PB. quod planè
fieri nequit. Ergò AB non refringetur in medium ipſi EF ſub-
jectum.

VIII. Angulus incidentiæ major ad angulum ſuum refractum ma-
jorem habet rationem, quam angulus incidentiæ minor ad refra-
ctum fuum.

Erit ſcilicet (in figura numeri Sexti, cujus huc apparatus transfe-
66Fig. 27, 22. ratur) ang. GBP. _a_ BP. & gt; ang. HBP. δ BP. Nam

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