7456
XIII.
Item, dico fore NX (rectioris itidem reflexi concursûs
incidentiæque punctis interjectam partem) minorem quartâ parte
totius N Π. Etenim fiat ang. HR Π = ang. N Π R; quapropter
erit HR = H Π; adeóque 2 H Π = HR + H Π & gt; R Π & gt; N Π.
item quoniam ang. RHN = 2 ang. HR Π = ang. XRH; eſt
XH = XR & gt; XN. quum itaque ſit H Π major ſemiſſe totius N Π;
& XH major ſemiſſe reſidui NH; liquet totam X Π majorem eſſe
triplâ X N; ſeu totam N Π majorem eſſe quadruplâ N X: Q. E. D.
incidentiæque punctis interjectam partem) minorem quartâ parte
totius N Π. Etenim fiat ang. HR Π = ang. N Π R; quapropter
erit HR = H Π; adeóque 2 H Π = HR + H Π & gt; R Π & gt; N Π.
item quoniam ang. RHN = 2 ang. HR Π = ang. XRH; eſt
XH = XR & gt; XN. quum itaque ſit H Π major ſemiſſe totius N Π;
& XH major ſemiſſe reſidui NH; liquet totam X Π majorem eſſe
triplâ X N; ſeu totam N Π majorem eſſe quadruplâ N X: Q. E. D.
XIV.
Hinc perſpicuum eſt, ſi fuerit NZ reflexi N Π quadrans,
quòd nullus alter hujuſmodi reflexus punctum Z permeabit. Etenim
alterius cujuſvis reflexus permeare dicatur; erit igitur, ſi obliquior
is fuerit, NZ & lt; {1/4} N Π; ſin rectior fuerit, erit NZ & gt; {1/4} N Π (ni-
mirum è proximè demonſtratis hæc conſequuntur) quæ repugnant
hypotheſi.
quòd nullus alter hujuſmodi reflexus punctum Z permeabit. Etenim
alterius cujuſvis reflexus permeare dicatur; erit igitur, ſi obliquior
is fuerit, NZ & lt; {1/4} N Π; ſin rectior fuerit, erit NZ & gt; {1/4} N Π (ni-
mirum è proximè demonſtratis hæc conſequuntur) quæ repugnant
hypotheſi.
XV.
Quinetiam ipſi N Π propiùs adjacentium occurſus puncto Z vi-
11Fig. 77. ciniores ſunt, hinc indè. Secent. inquam, radiorum LM, RS reflexi
L μ, R σ ipſam N Π punctis Y, X; iſtæ quidem (rectior) in Y, hic
(obliquior) in X; erit ZY & lt; ZX. Nam connectantur R Π,
L Π; & fiat ang. Π LH = ang. N Π L; ducantúrque rectæ RH,
RY. eſtque RH & gt; LH = H Π; adeóque ang. H Π; R & gt; ang.
HR Π; & proinde ang. NHR & lt; 2 ang. H Π R. item YR & gt; YL
= YH; proindéque rurſus ang. NYR & lt; 2 ang. YHR. quare
multo minor eſt ang. NY R quadruplo N Π R eſt autem ang. NXR
quadruplus anguli N Π R; igitur ang. NXR & gt; ang. NYR. pona-
tur jam, ſi fieri poteſt, punctum X ipſis Y, Z interjacere. erit igitur
angulus externus NYR interno NXR major; atqui minor oſtenſus
eſt. quæ repugnant. itaque potiùs eſt ZY & lt; ZX: Q. E. D.
11Fig. 77. ciniores ſunt, hinc indè. Secent. inquam, radiorum LM, RS reflexi
L μ, R σ ipſam N Π punctis Y, X; iſtæ quidem (rectior) in Y, hic
(obliquior) in X; erit ZY & lt; ZX. Nam connectantur R Π,
L Π; & fiat ang. Π LH = ang. N Π L; ducantúrque rectæ RH,
RY. eſtque RH & gt; LH = H Π; adeóque ang. H Π; R & gt; ang.
HR Π; & proinde ang. NHR & lt; 2 ang. H Π R. item YR & gt; YL
= YH; proindéque rurſus ang. NYR & lt; 2 ang. YHR. quare
multo minor eſt ang. NY R quadruplo N Π R eſt autem ang. NXR
quadruplus anguli N Π R; igitur ang. NXR & gt; ang. NYR. pona-
tur jam, ſi fieri poteſt, punctum X ipſis Y, Z interjacere. erit igitur
angulus externus NYR interno NXR major; atqui minor oſtenſus
eſt. quæ repugnant. itaque potiùs eſt ZY & lt; ZX: Q. E. D.
XVI.
Hinc obiter patet ad eaſdem partes incidentium reflexos ſeſe
priùs (velut ad φ) quàm ipſum N Π decuſſare.
priùs (velut ad φ) quàm ipſum N Π decuſſare.
XVII.
Quinimò rurſus hinc conſtat ad eaſdem axis partes plures
duobus in uno puncto reflexos non concurrere.
duobus in uno puncto reflexos non concurrere.
XVIII.
Demùm (utaliquando tandem deſtinatum attingamus