746346

Liber

67~

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PROPOSITIO XLIV.

Theorema.

Raditu

refra~us

dua~ff~

fi1perficieb1u parallelU,

inc1de11t• reftmmur parallelU1.

Supponunmr fuperficies

AR,

CD, medii den–

lioris, vcrbi gratia virri, aur aqua: , effe paralle.

lz , in quas

acre radius

incidat , qui poíl:

duplicem refcall:ionem propagetur per lmeam

IM ,

hanc dico i:adio EF paral\clarn

elle.

Ducan–

rur perpendiculares N F G, PIK qux parallela:

crunr, curn fupc1ficies

CD fupponaurnr pa–

xallcla:.

Dernoníl:rario. Sinus anglili inclinarionis EFN

ad linum anguli

refcall:i.GF!

,Íeu

altem~.

P, '.e

liaber ut rria ad duo, 1n mgreffo ab acre ad

vl–

rrum. Sed in egreffo viui in"acrem angulus in–

clinarionis FIP, aut illi oppoficus KIL/e habet ad

refrall:um KIM,ut

ad;; ergoanguli KIM,GFO

funr a:quales,& cum KIL,FI feu oppofims FIP_,&

IFG altcrnus lint zquales,ernnr anguli OFI,MIL

zquales ', oppoliti ex eadem pane ; ergo (

per

i9.

E,,c1.)

linez FO, IM parnllelz funr.

~are

pro uno eodernque radio fere fumi pof–

funr , cum difforenria lir multo rninor craffitie

lcntis.Probavi in

18,

quando duz fopcrficics con–

cava , & convexa erant parallela: radium refl:itui

patallelum.

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PROPOSITIO XLV.

Thorema.

/n plano.convexú

""!

plano-concavis

~ad'.i

verticem i11cidentt1, remitt1m,t1ir paralltl"

Radius DB,in

fpecilli phno·convexi,

aut

pla·

Tonr.

111~

no·concavi. verticem ll incidnr ; dico poít dupli–

ccm refraéboncm , rad1urn rcfraél:um ipíi refpon–

drntcm ,

effe

e1dem parallelum. lntelligacur pct

p1mél:um ll planum tnngens lll-1.

Demonílratio.

~efraél:iones,licur

reAexioncs,

eodcm modo acc1dunt refpcél:u fuperficici curvz ,

refpeéht fu

P.edic1.et

pl~~a:

contigenris curvam

111 punll:o 111ctdent1a: radn de quo agitur

ícd

inciderct radius Dll in fuperficiem p\a1fam lll-1,

parallelam mique fuperficiei planz, radius refra–

ll:us

radio incidcnti Dll effer parallelus

(per

pr.teedmtem

)

ergo etiam in hoc caíu , eidem

Pª'

rallelus erit;

~od

demoníl:randmn erar.

Ha:c dcmoníl:rario fuam vim haber, in lencibus

plano concavis.

tfili<i'3-·E@--

N>l'E"13@i·m~,H&m-~

PRO

P O

L V l.

Thoerema.

Vide

figitrJm

pr:rccdentcm.

ll.~dii

oblique

incidentes

prop~

verticem

ientem convexo-convexam au.t co11ca110-

co11cavam, remitrnnwr

pa~alleli.

Radins

tranfcat per cenrrum magnitudi•

nis lentis convexo-convexa:, fitque cadius inci•

dens NO cujus refrall:us in lente fit

01 ,

dico

hunc radium NO , poít duplicem refraéHonem,

, habcrc refrall:um IL , libi parallelnm.

Demoníl:ratio. Cum radius O

rranfcat pee

cenuum magnimdinis lenris , erunt arcus

IH limiles, íi nempe fpha:ricitates lint zqua–

lium fphzrarum fegmenta , quod íi linr ina:qua•

lium , dabitur femper aliquis radius intra lentem;

qui hinc inde abícinclar arcm limiles, arque adco

eandem inclinationcm efficiar ; ergo cui tom in

ingre!fo 'c¡uam in egreffu

refpondea~

a:quales an–

gul i refrall:ionis arque adeo reliqui NOI,

OIL

altcrni erunt; ergo radii NO IL paralleli erunr,

Q.uod erat demoníl:randum.

Addo infuper quod daris multis radiis inciden–

tibus parallelis , fcmper invenietur aliquis cuí

refrall:us refpondcns parallelus cric. Ur

den•

tur

radii parallcli CD , FG; ncmpe CD in vertí•

ccm

incidens,

CF, cujus ccfrall:us in lente

GH,

pee verticcm oppofitum cranfeat , cum to•

intervallum imcr

DE , GH

plenum lit ra.• '

diis ref.aél:is, aliquis neceffari¿, per'cencrurn

ma–

gnitudinis lenris tronfibi1, am abfcindet, in op–

poÍtris fuperficicbus curvis arcus hinc inde

(i~

miles.

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PROPOSITIO XLVII.

Theorema.

Radii paralleli

obliq11ei11cide111es in

fl1ptrftcíem

pla

am fpecilli plRno-convexi

1mi1111111~

ad tliftantiam diametri,

Haél-enus conlideravimus tantum radiorum

.Q.qq

•:Ü