Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

biſecat angulum ANK; erit AN. NK : : AT. TK. & lt; AB. BK. quare BZ. AB + AN. NK & lt; BZ. AB + AB. BK
(communem adſciſcendo rationem BZ ad AB). eſt autem BZ. AB + AN. NK = CZ. AC + AC. CK = CZ. CK & BZ. AB + AB. BK = BZ. BK. ergo CZ. CK & lt; BZ. BK. permutandóque CZ. BZ & lt; CK. BK. quin & componendo CB. BZ & lt; CB. BK. ideóque BZ & gt; BK; quare punctum Z centro
propinquius eſt, quàm ipſum K: Q. E. D.

_Coroll_. Hinc ſi puncta Z, ζ fuerint limites punctorum radiantium
A, _a_ (quorum A fit à fpeculo remotiùs, quàm _a_) erit CZ & lt; C ζ. Nam eſt BC. AB & lt; BC, _a_ B. adeóque compoſitè AC. AB & lt; _a_ C. _a_ B. hoc eſt CZ. BZ & lt; C ζ. B ζ. quare componendo
BC. BZ & lt; BC. B ζ. & indè BZ & gt; B ζ.

IX. Porrò, conſectatur è præmiſſis, quòd ſi duorum quorumvis
incidentium AN, AR reflexi GN, HR axem interſecent punctis
K, L; erit CL, CK : : ACq - ANq. ACq - ARq. ‖ Nam
quoniam eſt AC. CK : : ACq - ANq. CBq. itémque CL. AC : : CBq. ACq - ARq. erit ex æquo perturbatè CL. CK : : ACq - ANq. ACq - ARq.

X. Simili planè diſcurſu, ſi fuerit AC. AB : : CZ. ZB. erit CZ. CK : : ACq - ANq. ACq - ABq. & CL. CZ : :
ACq - ARq. ACq - ABq.

XI. Hinc perſpicuum eſt obliquioris reflexi concurſum à centro
magìs elongari quam rectioris; quod nempe ſit CL & gt; CK. Cùm
enim ſit ACq - ANq & gt; ACq - ARq; erit CL & gt; CK.

XII. Hinc neceſſariò duo quilibet ad eaſdem axis partes incidentium
reflexi (quales NK, RL) ſeſe priùs quàm axem interſecabunt, puta
ad X. quo poſito.

XIII. Adnotari poteſt angulum GXH vel KXL (à reflexis oc-
currentibus incluſum) æquari angulo NCR unà cum differentia angu-
lorum incidentiæ; vel, duplo angulo NCR unà cum ang. NAR. ‖ Etenim
ang. KXL = ang. ALR - AKN = ang. ACR + CRL -: ang. ACN + CNK = ang. ACR - ACN + : ang. CRL -
CNK = ang NCR +: ang CRS - CNP. ‖ Quinetiam ang. CRS - CNP = ang. RCA + CAR -: ang. NCA +

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