137119
AV.
TV :
: CV.
CD;
&
conſequentes ſubduplandò, dividendó-
que AK. KV : : KV. KD : : KV. CK. vel AK. KH : : KH. CK;
hoc eſt HN (KX). NG : : KH. CK; quare CK x KX = KH
x NG. eſt autem XZq = CZq - CXq = XGq - CXq =
NGq + KHq - 2 NG x KH: - KXq - CKq + 2 CK
x KX = NGq + KHq - KXq - CKq. Sumatur K ζ = KX,
diſcurſúmque ſimilem adhibendo liquebit fore ξζ = XZ; & ideo
Dζ = CZ. unde Cζ - Dζ (DZ - CZ) = Cζ - CZ =
ξγ - XG = 2 KH = TV. quare manifeſtum eſt _cnrvas_ TZ,
Vζ eſſe _Hyperbolas,_ quarum axis TV, foci C,D.
que AK. KV : : KV. KD : : KV. CK. vel AK. KH : : KH. CK;
hoc eſt HN (KX). NG : : KH. CK; quare CK x KX = KH
x NG. eſt autem XZq = CZq - CXq = XGq - CXq =
NGq + KHq - 2 NG x KH: - KXq - CKq + 2 CK
x KX = NGq + KHq - KXq - CKq. Sumatur K ζ = KX,
diſcurſúmque ſimilem adhibendo liquebit fore ξζ = XZ; & ideo
Dζ = CZ. unde Cζ - Dζ (DZ - CZ) = Cζ - CZ =
ξγ - XG = 2 KH = TV. quare manifeſtum eſt _cnrvas_ TZ,
Vζ eſſe _Hyperbolas,_ quarum axis TV, foci C,D.
IV.
Tertiò demùm, ſit angulus CAE ſemirectus (vel CA =
11Fig. 192. CE) erit tum punctum Z ad parabolam; quæ itidem ita determina-
tur. Fiat angulus ACP ſemirectus, & ab ipſarum AE, CP in-
terſectione R ducatur RT ad CE parallela; erit T_Vertex_, atque C
_Focus Parabolæ._ id quod ex bene nota ſectionis hujus proprietate con-
ſtat; qua ſcilicet eſt TA = TR = TC (ob angulos TAR,
TCR ſemirectos) & AX = XG = CZ.
11Fig. 192. CE) erit tum punctum Z ad parabolam; quæ itidem ita determina-
tur. Fiat angulus ACP ſemirectus, & ab ipſarum AE, CP in-
terſectione R ducatur RT ad CE parallela; erit T_Vertex_, atque C
_Focus Parabolæ._ id quod ex bene nota ſectionis hujus proprietate con-
ſtat; qua ſcilicet eſt TA = TR = TC (ob angulos TAR,
TCR ſemirectos) & AX = XG = CZ.
V.
Manifeſtum eſt verò rectam AE ſectiones has ad E contingere.
quia nempe perpetuò major eſt CZ (vel XG) ordinatâ XZ; adeó-
que puncta G extra cuŕvas unaquæque jacent hoc eſt tota AG extra
illas cadit.
quia nempe perpetuò major eſt CZ (vel XG) ordinatâ XZ; adeó-
que puncta G extra cuŕvas unaquæque jacent hoc eſt tota AG extra
illas cadit.
VI.
Hiſce præſtratis:
_Eſto Circulare ſpeculum_ MBND, cen-
22Fig. 193. trum habens C; cui exponatur recta quæpiam F α G; & huic per-
pendicularis ſit recta C α; quam ad parte@ averſas ſumpta CA, ad-
æquet. Sit etiam CE ad CA perpendicularis, ac æqualis qua-
dranti diametri BD; connexáque recta AE producatur utcunque.
ſumpto jam in recta F α G puncto quolibet F, connectatur FC, &
radiationis ab F in ipſa FC limes, ſeu _focus_, ſit Z; ac per Z du-
catur ZX ad AC perpendicularis, ipſi AE occurrens in H; dico
fore XH parem ipſi CZ.
22Fig. 193. trum habens C; cui exponatur recta quæpiam F α G; & huic per-
pendicularis ſit recta C α; quam ad parte@ averſas ſumpta CA, ad-
æquet. Sit etiam CE ad CA perpendicularis, ac æqualis qua-
dranti diametri BD; connexáque recta AE producatur utcunque.
ſumpto jam in recta F α G puncto quolibet F, connectatur FC, &
radiationis ab F in ipſa FC limes, ſeu _focus_, ſit Z; ac per Z du-
catur ZX ad AC perpendicularis, ipſi AE occurrens in H; dico
fore XH parem ipſi CZ.
Nam (è jam antè monſtratis) eſt FC.
CZ :
: FM.
MZ (hoc eſt)
: : FC - CB. CB - CZ. hinc erit α C. CX (AC. CX) : :
FC - CB. CB - CZ. quare (ducendo in ſe extrema, ac media)
erit AC x CB - AC x CZ = CX x FC - CX x CB. hoc
eſt (ipſi CX x FC ſubſtituendo AC x CZ, propter α C. CX : :
FC. CZ) erit AC x CB - AC x CZ = AC x CZ - CX x
CB. tranſponendóque AC x CB + CX x CB = 2 AC x CZ.
: : FC - CB. CB - CZ. hinc erit α C. CX (AC. CX) : :
FC - CB. CB - CZ. quare (ducendo in ſe extrema, ac media)
erit AC x CB - AC x CZ = CX x FC - CX x CB. hoc
eſt (ipſi CX x FC ſubſtituendo AC x CZ, propter α C. CX : :
FC. CZ) erit AC x CB - AC x CZ = AC x CZ - CX x
CB. tranſponendóque AC x CB + CX x CB = 2 AC x CZ.