5335
tervallum IK minus ipſo KL ;
ſeu generalius efferendo, libere ſum-
ptis ipſis MN, NO ; erit IK. KL & lt; MN. NO. hoc verò non
aliter, opinor, elegantius quam ex adjunctis uno, vel altero Theore-
11Fig. 43. mate conſtabit.
ptis ipſis MN, NO ; erit IK. KL & lt; MN. NO. hoc verò non
aliter, opinor, elegantius quam ex adjunctis uno, vel altero Theore-
11Fig. 43. mate conſtabit.
XVI.
In primo caſu;
ſit (ut antehac) ZB.
AB :
: I.
R;
ſuper-
que diametro ZB conſtituatur ſemicirculus; cui à puncto B adapte-
tur BD = BA ; & per puncta Z, D ducta recta refringenti occur-
rat in Y ; tum ad ſemiaxes BZ, BY (centro nempe B, vertice Z) de-
ſcribatur Hyperbole HZG; in hac autem ſumpto quolibet puncto S
ducantur SN ad AB, & SK ad EF parallelæ. Denique ducantur
22Fig. 44. A N, KN _a_ erit KM _a_ incidentis AN refractus.
que diametro ZB conſtituatur ſemicirculus; cui à puncto B adapte-
tur BD = BA ; & per puncta Z, D ducta recta refringenti occur-
rat in Y ; tum ad ſemiaxes BZ, BY (centro nempe B, vertice Z) de-
ſcribatur Hyperbole HZG; in hac autem ſumpto quolibet puncto S
ducantur SN ad AB, & SK ad EF parallelæ. Denique ducantur
22Fig. 44. A N, KN _a_ erit KM _a_ incidentis AN refractus.
Nam ex _Hyperbolæ_ natura eſt KBq - ZBq.
BNq :
: BZq.
BYq : : ZDq. BDq (hoc eſt) : : ZBq - ABq. ABq. quare
componendo KBq - ZBq + BNq. BNq : : ZBq. ABq hoc
eſt KNq - ZBq. BNq : : ZBq. ABq. permutandóque KNq
- ZBq. ZBq : : BNq. ABq rurſuſque componendo KNq.
ZBq : : ANq . ABq. denuóque permutando KNq. ANq : :
ZBq. ABq : : Iq . Rq. quare KN. AN : : I. R. ergo KN ip-
ſius AN reſractus erit: Q. E. D.
BYq : : ZDq. BDq (hoc eſt) : : ZBq - ABq. ABq. quare
componendo KBq - ZBq + BNq. BNq : : ZBq. ABq hoc
eſt KNq - ZBq. BNq : : ZBq. ABq. permutandóque KNq
- ZBq. ZBq : : BNq. ABq rurſuſque componendo KNq.
ZBq : : ANq . ABq. denuóque permutando KNq. ANq : :
ZBq. ABq : : Iq . Rq. quare KN. AN : : I. R. ergo KN ip-
ſius AN reſractus erit: Q. E. D.
XVII.
Hinc refractorum cum axe concurſus (puta I, K, L) à ſe
diſtant intervallis ordinatim applicatarum ad _Hyperbolam_, puta recta-
rum, BZ, MR, NS, OT ; vel ipſarum O, ZI, ZK, ZL. Hæ ve-
rò (ceu paſſim notum, & à nobis aliquando generatim circa cunctas
hujuſmodi curvas oſtenſum eſt) in majori ratione creſcunt, quam ipſæ
BM, BN, BO ; nempe ZL . ZK & gt; LT . KS. & ZK. ZI & gt; KS.
IR. quare ſatìs liquet propoſitum. Enimverò prope verticem Z
ordinatarum differentiæ perquam exiguæ ſunt; ut bene multorum
perpendiculari AB adjacentium radiorum refracti velut è puncto
Z manare videantur ; utcunque circa ipſum præcipuè conſtipantur.
diſtant intervallis ordinatim applicatarum ad _Hyperbolam_, puta recta-
rum, BZ, MR, NS, OT ; vel ipſarum O, ZI, ZK, ZL. Hæ ve-
rò (ceu paſſim notum, & à nobis aliquando generatim circa cunctas
hujuſmodi curvas oſtenſum eſt) in majori ratione creſcunt, quam ipſæ
BM, BN, BO ; nempe ZL . ZK & gt; LT . KS. & ZK. ZI & gt; KS.
IR. quare ſatìs liquet propoſitum. Enimverò prope verticem Z
ordinatarum differentiæ perquam exiguæ ſunt; ut bene multorum
perpendiculari AB adjacentium radiorum refracti velut è puncto
Z manare videantur ; utcunque circa ipſum præcipuè conſtipantur.
XVIII.
Haud abſimiliter, in ſecundo caſu, ſuper ipſa AB deſcriba-
tur ſemicirculus; & huic accommodetur BD = BZ; & connexa
protractáque AD refringenti occurrat ad Y ; tum centro B ſemiaxi-
bus BZ, BY deſcribatur ellipſis HZG; & in hac accepto quocunque
33Fig. 45. puncto S ducantur SN ad ZB, & SK ad EF parallela; connectan-
tur denique rectæ AN , KN; erit KN incidentis AN refractus.
Etenim ex ellipſis natura eſt KSq. ZBq - SNq : : BYq. BZq
: : BYq. BDq : : BAq. ADq: : BAq. BAq - BZq. & per
tur ſemicirculus; & huic accommodetur BD = BZ; & connexa
protractáque AD refringenti occurrat ad Y ; tum centro B ſemiaxi-
bus BZ, BY deſcribatur ellipſis HZG; & in hac accepto quocunque
33Fig. 45. puncto S ducantur SN ad ZB, & SK ad EF parallela; connectan-
tur denique rectæ AN , KN; erit KN incidentis AN refractus.
Etenim ex ellipſis natura eſt KSq. ZBq - SNq : : BYq. BZq
: : BYq. BDq : : BAq. ADq: : BAq. BAq - BZq. & per