# Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

## 49.Notæ in Propoſitionem IX. & X.

AT in hyper-
bola, & el-
lipſi educamus G
F ad a ex A D, & H N ad s ex F G,
& I S ad T ex C
G, ſi educta oc-
A, & M Q poſita
ad m ex a, F G,
& X in I T, & ex
m, S X, m y, x n,
S Z inter N S, M
inconcinna exponi-
tur vniuerſa con-
ſtructio buius pro-
poſitionis, ideo cu-
raui eam reddere
clariorem, dicendo; Educamus rectas lineas G F quidem ſec antem A D in a, & c.

### 49.1.

g

Quadratum igitur I H eſt æquale triangulo I H S, & c. Qaia nimirum. Quadratum I H eſt æquale duplo iſoſcelei, & rectanguli trianguli I H S.

### 49.1.

h

Et ſimiliter quadratum I Q æquale eſt duplo trianguli I Q X, & c. Sci-
licet duplo trapezĳ I S m Q cum duplo trianguli S m X.

### 49.1.

i

Et hoc quidem propter ſimilitudinem triangulorum, at componendo
proportionem in hyperbola, tum inuertendo, & reflectendo in ellipſi
fit, & c. Huiuſmodi verba inepta ad concluſionem inferendam commutaui di-
eorum differentias in ellipſi fit, & c. Quæ quidem expeditè (vt in primo præce-
cedentium Lemmatum oſtenſum eſt) progreſſum declarant.

### 49.1.

k
l

Vt proportio inclinati, ſiue tranſuerſæ ad latitudinem figuræ compara-
tæ; igitur planum m n eſt exemplar, & c. Subiungo: nam, vt dictum eſt in
quinta, & ſexta huius, poteſt hìc demonſtrari, quod figura m n ſimilis eſt ei,
quæ continetur latere tranſuerſo E C, & ſumma in hyperbola, & differentia in
ellipſi laterum tranſuerſi, & recti iuxta definitiones octauam, & nonam.

Quadratum R I æquale eſt duplo trianguli R V I, & quadratum O R in
hyperbola æquale eſt duplo trapezij R G, & in ellipſi æquale eſt duplo
trapezij R K, & c. Legendum puto quadratum R I æquale eſt duplo trianguli
R V I, & quadratum O R æquale eſt duplo trapezĳ R G, at in ellipſi quando
O R cadit infra centrum F æquale eſt duplo trapezĳ R K, & c. Deindè
quum triangulum R V I ſimile ſit triangulo I H S propter parallelas V R, S
H; ideò triangulum R V I erit quoque iſoſceleum, & rectangulum. Poſtea qua-

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