5618Apollonij Pergæi
PROPOSITIO IX. & X.
AT in hyper-
11g
26[Figure 26]
bola (10.)
& ellipſi educa-
mus rectas lineas,
G F quidem ſecã-
tem A D in a, &
N H occurrẽtem
F G in S, & I S
ſecantem C G in
T, pariterque M
Q ſecantem F G
in m, & I T in X,
& ex punctis m, S,
x educamus inter
N S, M X rectas
m y, X n, S Z pa-
rallelas ipſi C I.
Et quia C F ad C
G, nempe F H ad
H S poſita eſt, vt
F H ad H I erit H I æqualis H S;
22h
27[Figure 27]
quadratum igitur I H eſt æquale
duplo trianguli I H S, & quadra-
tum N H ęquale eſt duplo trape-
zij H G; quare quadratum N I
33Prop. I. h. æquale eſt duplo trapezij I G;
ſimiliter quadratum I Q ęquale eſt
44i duplo trianguli I Q X, & quadra-
tum M Q eſt æquale duplo trape-
zij Q G; itaque quadratum ex I M
æquale eſt duplo trapezij I G cum
duplo trianguli m S X, quod eſt æ-
quale plano m n: Et C F ad C G,
nempe proportio figuræ eſt, vt S Z,
nempe Z X ad Z m (& hoc quidem
propter ſimilitudinem triangulorũ)
quare comparãdo priores ad ſum-
55Lem. 1. h. mas terminorum in hyperbola, &
66k ad eorundem differentias in ellipſi
fiet X Z (quæ eſt æqualis ipſi X n)
ad X m, vt proportio inclinati, ſiue
77l tranſuerſæ ad latitudinem figuræ
comparatæ; igitur planum m n eſt exemplar, eſtque applicatum ad X n,
88Def 9.
11g
& ellipſi educa-
mus rectas lineas,
G F quidem ſecã-
tem A D in a, &
N H occurrẽtem
F G in S, & I S
ſecantem C G in
T, pariterque M
Q ſecantem F G
in m, & I T in X,
& ex punctis m, S,
x educamus inter
N S, M X rectas
m y, X n, S Z pa-
rallelas ipſi C I.
Et quia C F ad C
G, nempe F H ad
H S poſita eſt, vt
F H ad H I erit H I æqualis H S;
22h
duplo trianguli I H S, & quadra-
tum N H ęquale eſt duplo trape-
zij H G; quare quadratum N I
33Prop. I. h. æquale eſt duplo trapezij I G;
ſimiliter quadratum I Q ęquale eſt
44i duplo trianguli I Q X, & quadra-
tum M Q eſt æquale duplo trape-
zij Q G; itaque quadratum ex I M
æquale eſt duplo trapezij I G cum
duplo trianguli m S X, quod eſt æ-
quale plano m n: Et C F ad C G,
nempe proportio figuræ eſt, vt S Z,
nempe Z X ad Z m (& hoc quidem
propter ſimilitudinem triangulorũ)
quare comparãdo priores ad ſum-
55Lem. 1. h. mas terminorum in hyperbola, &
66k ad eorundem differentias in ellipſi
fiet X Z (quæ eſt æqualis ipſi X n)
ad X m, vt proportio inclinati, ſiue
77l tranſuerſæ ad latitudinem figuræ
comparatæ; igitur planum m n eſt exemplar, eſtque applicatum ad X n,
88Def 9.