Apollonius <Pergaeus>, Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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[131.] Notæ in Propoſit. XVI. XVII. XVIII.
Page: 152 (114)
[132.] SECTIO DECIMASEPTIMA Continens XIX. XX. XXI. XXII. XXIII. XXIV. & XXV. Propoſ. Apollonij. PROPOSITIO XIX.
Page: 154 (116)
[133.] PROPOSITIO XX. XXI. & XXII.
Page: 155 (117)
[134.] PROPOSITIO XXIII. & XXIV.
Page: 156 (118)
[135.] PROPOSITIO XXV.
Page: 157 (119)
[136.] Notæ in Propoſit. XIX.
Page: 157 (119)
[137.] Notæ in Propoſit. XX. XXI. XXII.
Page: 158 (120)
[138.] Notæ in Propoſ. XXIII. XXIV.
Page: 161 (123)
[139.] Notæ in Propoſ. XXXV.
Page: 161 (123)
[140.] SECTIO DECIMAOCTAVA Continens XXXII. XXXIII. XXXIV. XXXV. XXXVI. XXXVII. XXXVIII. XXXIX. XXXX. XXXXVII. XXXXVIII. Propoſit. Apollonij. PROPOSITIO XXXII.
Page: 162 (124)
[141.] PROPOSITIO XXXIII. XXXIV.
Page: 163 (125)
[142.] PROPOSITIO XXXV.
Page: 163 (125)
[143.] PROPOSITIO XXXVI.
Page: 164 (126)
[144.] PROPOSITIO XXXVII. XLVI.
Page: 164 (126)
[145.] PROPOSITIO XXXVIII.
Page: 165 (127)
[146.] PR OPOSITIO XXXIX.
Page: 166 (128)
[147.] PROPOSITIO XXXX.
Page: 166 (128)
[148.] PROPOSITIO XXXXVII.
Page: 166 (128)
[149.] PROPOSITIO XXXXVIII.
Page: 167 (129)
[150.] Notæ in Propoſit. XXXII.
Page: 167 (129)
[151.] Notæ in Propoſit. XXXIII. XXXIV.
Page: 167 (129)
[152.] Notæ in Propoſit. XXXV.
Page: 168 (130)
[153.] Notæ in Prop. XXXVI.
Page: 169 (131)
[154.] Notæ in Prop. XXXVIII.
Page: 169 (131)
[155.] Notæ in Propoſit. XXXIX.
Page: 170 (132)
[156.] Notæ in Propoſit. XXXXVIII.
Page: 170 (132)
[157.] LIBRI QVINTI FINIS.
Page: 170 (132)
[158.] APOLLONII PERGAEI CONICORVM LIB VI. DEFINITIONES. I.
Page: 171 (133)
[159.] II.
Page: 171 (133)
[160.] III.
Page: 171 (133)
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164126Apollonij Pergæi
PROPOSITIO XXXVI.
IN ſectione elliptica quatuor lineæ
158[Figure 158] breuiſſimæ, vt B D, F I, G K,
H L, non conueniunt omnes in vno
puncto.
Alioquin ſit occurſus in E, & prius ſit
B D perpendicularis ſuper A C, tranſi-
ens per D centrum ſectionis; & quia E
eſt occurſus duarum breuiſſimarum B D,
1135. huius. F I, & B E tranſit per centrum; igitur
159[Figure 159] G K non eſt linea breuiſſima, quod eſt
contra hypotheſim. Si vero nullus eorũ
tranſit per centrum, educamus per cen-
trum D O perpendicularem ad A C; qua-
re duæ breuiſſimæ F I, G K conueniunt
intra angulum A D O (34. ex 5.) ſimi-
liter H L, M N breuiſſimæ occurrunt in-
tra angulum C D O (34. ex 5.) ſed cõ-
ueniunt in E, quod eſt abſurdum; igitur
quatuor lineæ breuiſſimæ non cõueniunt in vno puncto; quod erat oſten-
dendum.
PROPOSITIO XXXVII. XLVI.
IN coniſectione A B, cuius centrum D duci non poſſunt-duæ
lineæ maximæ in ellipſi, neque duæbreuiſſimæ in omnibus
ſectionibus, vt A E, A F ad vnum punctum A circumferentiæ
ſectionis terminatæ.
Educamus A G perpendicularem ad axim B E. Si itaque ſectio fue-
rit parabole, fiet E G æqualis F G, quia quælibet earum eſt æqualis di-
midio erecti (13. ex 5.) ſi vero fuerit hyperbole, aut ellipſis, fiet D G
ad G E, vt D G ad G F; quia quælibet earum eſt, vt proportio figuræ
(14. 15. ex 5.) igitur G F æqualis eſt G E, quod eſt abſurdum. Simi-
liter ſi B G fuerit minor duarum axium ellipſis, & fuerint A E, A F
rami maximi oſtendetur, quod G F æqualis ſit G E (23. ex 5.) Patet
igitur, vt dictum eſt, quod ex vno puncto ſectionis educi non poſſunt
ad axim illius duæ lineæ maximæ, neque breuiſſimæ, & hoc erat oſten-
dendum.

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