8870
V.
Horum caſuum primus ad unum duntaxat ab una axis parte radi-
um pertinet, qui reliquos aliis caſibus convenientes medius diſterminat.
11Fig. 101. de poſteribus itaque duobus ſeparatim paullò diſpiciamus, Sit jam
itaque primò AC = AG = A γ; unde quilibet incidens cavo GB γ
radius (ut AN) major erit quam AC; hujus itaque reflexus axem
ſecet puncto K; dico, ſi ſemidiameter CB dividatur in Z; ut ſit CZ.
ZB: : AC. AB; fore CK & gt; CZ. etenim ob angulum ANK
22Fig. 100. biſectum, erit AC. CK: : AN. NK. vel permutando AC. AN
: : CK. NK. eſt autem AC. AB & lt; AC. AN ergo AC. AB
& lt; CK. NK & lt; CK. BK. ergo cùm ſit, ex hypotheſi, CZ. ZB
: : AC. AB; erit CZ. ZB & lt; CK. BK. componendóque CB.
ZB. & lt; CB. KB. unde ZB & gt; KB; ſeu CZ & lt; CK; Q.
E. D.
um pertinet, qui reliquos aliis caſibus convenientes medius diſterminat.
11Fig. 101. de poſteribus itaque duobus ſeparatim paullò diſpiciamus, Sit jam
itaque primò AC = AG = A γ; unde quilibet incidens cavo GB γ
radius (ut AN) major erit quam AC; hujus itaque reflexus axem
ſecet puncto K; dico, ſi ſemidiameter CB dividatur in Z; ut ſit CZ.
ZB: : AC. AB; fore CK & gt; CZ. etenim ob angulum ANK
22Fig. 100. biſectum, erit AC. CK: : AN. NK. vel permutando AC. AN
: : CK. NK. eſt autem AC. AB & lt; AC. AN ergo AC. AB
& lt; CK. NK & lt; CK. BK. ergo cùm ſit, ex hypotheſi, CZ. ZB
: : AC. AB; erit CZ. ZB & lt; CK. BK. componendóque CB.
ZB. & lt; CB. KB. unde ZB & gt; KB; ſeu CZ & lt; CK; Q.
E. D.
VI.
Hinc punctum Z eſt limes infra quem, Verſus centrum, nullus
reflexus axem interſecat.
reflexus axem interſecat.
_Coroll._
Hinc ſi puncta Z, ζ ſint limites punctorum A, _a_ (quorum
A remotius) erit CZ & gt; C ζ.
A remotius) erit CZ & gt; C ζ.
Nam BC.
AC &
lt;
BC.
_a_ C.
componendóque AB.
AC &
lt;
_a_ B. _a_ C. hoc eſt ZB. ZC & lt; ζ B. ζ C. vel compoſitè CB. ZC
& lt; CB. ζ C. ergò ZC & gt; ζ C.
_a_ B. _a_ C. hoc eſt ZB. ZC & lt; ζ B. ζ C. vel compoſitè CB. ZC
& lt; CB. ζ C. ergò ZC & gt; ζ C.
VII.
Quinetiam erit in hoc caſu;
ANq - ACq.
CNq:
:
AC. CK. Nam ducatur KH ad CN parallela, protractæ AN
occurrens in H; & connectatur CP; & eodem planè modo quo ſu-
periùs (in iis quæ circa convexas partes attigimus) oſtendetur fore
AN x NP. CNq: : AK. CK. unde diviſim erit AN x NP -
CNq. CNq: : AC. CK. eſt autem AN x NP = ANq -
AN x AP = ANq -: ACq - CNq = ANq - ACq +
CNq; adeóque AN x NP - CNq = ANq - ACq. ergò
demum erit ANq - ACq. CNq: : AC. CK: Q. E. D.
AC. CK. Nam ducatur KH ad CN parallela, protractæ AN
occurrens in H; & connectatur CP; & eodem planè modo quo ſu-
periùs (in iis quæ circa convexas partes attigimus) oſtendetur fore
AN x NP. CNq: : AK. CK. unde diviſim erit AN x NP -
CNq. CNq: : AC. CK. eſt autem AN x NP = ANq -
AN x AP = ANq -: ACq - CNq = ANq - ACq +
CNq; adeóque AN x NP - CNq = ANq - ACq. ergò
demum erit ANq - ACq. CNq: : AC. CK: Q. E. D.
Notetur;
ſi fuerit AC minor ſemiſſe ſemidiametri circuli re-
flectentis, quòd punctum A duos focos habebit ad eaſdem centri par-
tes, quorum alter ad partes D, alter ad B pertinebit; ſin AC major
fuerit iſtâ Semiſſe, focis qui ad diverſos vertices B, & D pertinent,
centrum C interjacebit.
flectentis, quòd punctum A duos focos habebit ad eaſdem centri par-
tes, quorum alter ad partes D, alter ad B pertinebit; ſin AC major
fuerit iſtâ Semiſſe, focis qui ad diverſos vertices B, & D pertinent,
centrum C interjacebit.
VIII.
Etiam hoc interſeram _Theorema_, præmiſſis conforme:
Si
fiat 2 CK. CN: : CN. F; itémque 2 CA. CN: : CN. E;
fiat 2 CK. CN: : CN. F; itémque 2 CA. CN: : CN. E;