jam aliquoties inſinuatâ; ſcilicet ut ſit AB. YB : : √ Iq -Rq. I; & deſignetur quilibet refractus KN; tum continuetur ratio YB ad
BN; ut ſit ad has proportione quarta BP; & per punctum P du-
catur recta PZ ad AB parallela; refracto KN occurrens in Z; dico
nullum alium refractum per Z traſire. Nam ſi ſieri poteſt tranſeat
alius ZR; & per Y traducantur rectæ NYG, RYS ; è præmon-
ſtratis apparet quòd ſit RS = NG. item è prædictis manifeſtum eſt quò RS & gt; N G. quæ repugnant.

XVI. Non diſpari ratione, quoad caſum ſecundum, deſignetur quilibet
refractus KN; & fiat KB . GB : : √ Rq - Iq. R; tum adnexâ
GN, ad ipſas NG, GB ſumatur tertia proportionalis V; & ſiat
NG. V : : BN. NP; & per punctum P ducatur PY ad BA pa-
rallela refractum NK decuſſans in Z; dico nullum alium refractum
per ipſum Z meare. Nam, ſi neges, tranſeat alius ZR; & per
Y trajiciatur RY S; & quoniam ZP . YP : : KB. GB : : √ Rq
- Iq. R. ex * antedictis apparet fore RS = NG. quinetiam ob
NGq . GBq : : NG. V : : BN . NP . erit dividendo NBq. GBq : : BP . NP . hoc eſt NPq. PYq : : BP . NP; inde facile
deducitur eſſe BP quartam proportionalem in ratione YP ad PN ; conſequentérque fore RS minimâ NG majorem . quod adver-
ſatur oſtenſis . itaque potiùs per Z nullus alius tranſit reſractus: Q. E. D.

XV I. Prætereà, ſi refractum NKZ interſecet alius quilibet M I,
ad rectiorem pertinens incidentem (hoc eſt ut incidentiæ punctum M
inter B, & N jaceat) interſectio X ſolitario puncto Z citerior erit
(ſeu perpendiculari KB propinquior) . Nam ab X demittatur per-
pendicularis XQ. ipſam NG ſecans in γ ; & (in primo caſu) per
M, Y traducatur recta MY H. ergò MH = Nγ. quare minima earum
quæ per Y angulo XQF interſeri poſſuntinter puncta M, N cadet
(utì nuper admonitum, & adſtructum). puta ad φ. ergò quum ſit
BP quarta proportionalis in ratione YB ad BN ; & BQ quarta
proportionalis in ratione YB ad B φ, erit PB & gt; QB; adeóque
recta XQ rectis ZP, KB interjacet : Q. E . D.

In ſecundo caſu, per γ trajiciatur recta M γ H. ergò cùm ſit
Q X. Q γ : : PZ. PY : : √ Rq - Iq. R. erit HM = GN. ergò
minima per γ ducibilium angulo AB F intercipienda punctis M, N
intercider; puta ad φ. quare QB quarta proportionalis erit in ratione
γ Qad Q φ; & eſt γ Q. Qφ & gt; (γ Q. QN.) : : YP. PN . &