Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]

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266254IO. BABPT. BENED. cundam verò ex .37. et .38. eiuſdem, propterea quod in .37. probat mediante maiori
diametro ipſius hyperbolis & defectionis, In .38. autem mediante minori diametro
Tertia autem paſſio, non niſi circulo conuenit; pace ipſius Cardani dictum ſit.
Quapropter ſit circulus .q.o.b. cuius diameter ſit .q.b. contingentes vero ab extre
mitate diametri ſint .d.b. et .q.g. per punctum autem .o. quoduis, ipſius circunferentiæ,
tranſeant .b.o.g. et .q.o.d.
tunc dico productum .q.o. in .q.d. vel .b.o. in .b.g. ęquale eſ-
ſe quadrato .q.b. quod ita probo.
Nam angulus .q.b.d. ſeu .b.q.g. rectus eſt ex .17. tertij Eucli. et .b.o.q. ſimiliter re-
ctus ex .30. ipſius lib. angulus verò .b.q.d. ſeu .q.b.g. communis eſt.
quare .b.q. media
proportionalis erit inter dictas lineas .q.d. et .q.o. & inter .b.g. et .b.o.
Vnde ſequetur
propoſitum ex .16.6. Eucli.
Sed ſi circa diametrum .q.b. mente fingamus aliquam elipſim, quætangat ipſum
circulum duobus punctis me-
diantibus .q. et .b. (nam pluribus
eſſet impoſſibile, ex .27. quarti
Pergei) clarè patebit, quod pum
ctus .o. erit extra circunferentiam
ipſius defectionis,
quare ipſa cir
cunferentia ſecabit .b.g. vel .q.
d.
in alio puncto, vnde ipſi non
occurret id quod probauimus
de circulo.
Cardanum dicere hyperbolem
ita vocari, eo quod angulus con
tentus ab axe ipſius figuræ, & à
latere trigoni in hyperbole ma-
ior ſit quam in parabole, quod
eriam confirmat paulo inferius,
nam hoc verum non eſt, imo fal
ſiſſimum.
Talis enim ſectio ita
nominata fuit, hoc eſt hyperbo
les, ſimili ratione, qua elipſis ſeu
defectio etiam vocata fuit, nam
tum ordinatę .l.m. minor eſt pro
ducto lineæ .e.m. in .e.t. per figu
ram ſimilcm producto .d.e. in .e.
t.
quæ eandem obtineat altitu-
dinem
ipſius .e.m. vt ipſe Pergeus
monſtrat in .13. primi lib. ita in