272234Apollonij Pergæi
319[Figure 319]
cireuli G C O B, &
G Q P L ſe ſe contingentes in communi puncto G rectæ li-
neæ G O ducaturque diameter L b Q æquidiſtans ipſi B C: & vt latus rectum
ad tranſuer ſum ſectionis X, ita fiat quadratum G d ad quadratum d A; &
coniungantur rectæ lineæ A G, & A O, ducaturque ex puncto P recta linea P
N parallela ipſi O A occurrens G A in N, atque A, & N fiant vertices duorum
conorum A B C, & N L Q, & ſecetur D d æqualis ſemiſſi potentis figuram
ſectionis X; ducaturque per punctum D planum E M F æquidiſtans plano com-
muni A G O per axes ducto, efficiens in conicis ſuperficiebus ſectiones H I K, &
T V c; Dico eas eſſe hyperbolas quæſitas. Quoniam propter parallelas A O, N
P eſt A G ad G O, vt N G ad G P, & ad ſemißes conſequentium, ſcilicet A G
ad G d, atque N G ad G b proportionales erunt, ideoque A d, N b erunt pa-
rallelæ, & A d ad d G, ſeu ad d C eſt vt N b ad b G, ſeu ad b Q; eſtque
d C etiam parallela b Q; ergo plana A B C, & N L Q parallela ſunt, &
anguli A d C, & N b Q æquales ſunt, atque triangula A d C, & N b Q
ſimilia crunt inter ſe; ideoque circa angulos æquales C, & Q erit A C ad C d,
vt N Q ad Q b, & ad conſequentium duplas, ſcilicet A C ad C B, atq; N Q
ad Q L proportionales erunt; & propterea triangula A B C, & N L Q ſimilia
exunt, & ſimiliter poſita, & inter ſe parallela; ergo efficient in duobus planis A O
G, & M E F inter ſe æquidiſtantibus ſectionũ diametros I D, & V a parallelas
conorũ axibus A d, & N b, & inter ſe; quare conſtituent cum ſectionũ
neæ G O ducaturque diameter L b Q æquidiſtans ipſi B C: & vt latus rectum
ad tranſuer ſum ſectionis X, ita fiat quadratum G d ad quadratum d A; &
coniungantur rectæ lineæ A G, & A O, ducaturque ex puncto P recta linea P
N parallela ipſi O A occurrens G A in N, atque A, & N fiant vertices duorum
conorum A B C, & N L Q, & ſecetur D d æqualis ſemiſſi potentis figuram
ſectionis X; ducaturque per punctum D planum E M F æquidiſtans plano com-
muni A G O per axes ducto, efficiens in conicis ſuperficiebus ſectiones H I K, &
T V c; Dico eas eſſe hyperbolas quæſitas. Quoniam propter parallelas A O, N
P eſt A G ad G O, vt N G ad G P, & ad ſemißes conſequentium, ſcilicet A G
ad G d, atque N G ad G b proportionales erunt, ideoque A d, N b erunt pa-
rallelæ, & A d ad d G, ſeu ad d C eſt vt N b ad b G, ſeu ad b Q; eſtque
d C etiam parallela b Q; ergo plana A B C, & N L Q parallela ſunt, &
anguli A d C, & N b Q æquales ſunt, atque triangula A d C, & N b Q
ſimilia crunt inter ſe; ideoque circa angulos æquales C, & Q erit A C ad C d,
vt N Q ad Q b, & ad conſequentium duplas, ſcilicet A C ad C B, atq; N Q
ad Q L proportionales erunt; & propterea triangula A B C, & N L Q ſimilia
exunt, & ſimiliter poſita, & inter ſe parallela; ergo efficient in duobus planis A O
G, & M E F inter ſe æquidiſtantibus ſectionũ diametros I D, & V a parallelas
conorũ axibus A d, & N b, & inter ſe; quare conſtituent cum ſectionũ