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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EPISTOL AE.
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325
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0325
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faciemus, quod diameter
<
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>.a.b.</
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>
dictæ ſphæræ ita ſe habcat ad
<
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>.e.f.</
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>
ex .10. ſexti, quæ
<
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<
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>e.f.</
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erit reliqua axis quæſita. </
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<
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xml:space
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">Vnde conſtituta cum fuerit ellipſis
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ex dictis axi-
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bus, </
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<
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xml:space
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">deinde circumuertendo ellipſim circa maiorem axem, conſtituemus ſphæroi-
<
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dem oblongam, ſi autem circumuertemus ipſam circa minorem axim conſtituemus
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ſphæroidem prolatam.</
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</
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<
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<
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xml:space
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">Quod autem talis operatio rationalis ſit, nulli dubium erit, que
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/>
tieſcunque co-
<
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/>
gnoſcet conum rectum
<
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>.e.u.f.</
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æqualem eſſe cono recto
<
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>.a.c.b.</
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>
ex .2. parte .12. duodeci
<
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mi Euclid. </
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<
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xml:space
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">& quod cum conus
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>.e.d.f.</
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>
duplus ſit cono
<
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>.e.u.f.</
var
>
ex lemmate collecto ab
<
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/>
11. duodecimi, conus
<
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>.e.d.f.</
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>
duplus exiſtit etiam cono
<
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>.a.c.b.</
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>
ex .7. quinti. </
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<
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xml:space
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inde ex .32. primi lib. de ſphæra, & cyllindro ſphæra
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>.a.c.b.q.</
var
>
quadrupla ſit cono
<
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>.a.
<
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c.b.</
var
>
ipſa conſequenter dupla erit cono
<
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>.e.d.f.</
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>
ſed ex .29. primi de conoidalibus, dimi
<
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/>
dium ſphæroidis
<
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>.e.d.f.t.</
var
>
hoc eſt
<
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>.e.d.f.</
var
>
dupla eſt cono
<
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>.e.d.f</
var
>
. </
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<
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xml:space
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">Quare talis medietas
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æqualis eſt ſphæræ propoſitæ, totaq́ue ſphæroides dupla erit ſphærę datæ. </
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<
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xml:space
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autem dico de proportione dupla, idem infero de qualibet alia, ſumendo
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ita pro
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portionatam ad
<
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>.d.x.</
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>
vt proponitur.</
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<
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xml:space
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">Sphęram autem inuenire quæ dimidia ſit ſphæroidis propoſitæ nullius erit nego-
<
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tij, quotieſcunque inuentus fuerit modus diuidendi vnam datam proportionem in
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tres æquales partes.</
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</
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<
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<
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xml:space
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">Sit propoſita ſphæroides
<
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>.e.f.d.t.</
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>
cuius axes ex conſequentia dantur
<
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>.e.f.</
var
>
et
<
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>.d.t.</
var
>
quę
<
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/>
quidem ſphæroides ſit primo oblonga, et
<
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>.u.x.</
var
>
ſit dimidium axis maioris. </
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>
<
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xml:id
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xml:space
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">imagine-
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tur etiam conus
<
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>
vt ſupra. </
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<
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xml:space
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">Imaginetur etiam factum eſſe, quod proponitur, hoc
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eſt, vt ſphæra
<
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>.a.b.c.q.</
var
>
ſit dimidium ipſius ſphæroidis, vnde conus
<
var
>.a.c.b.</
var
>
æqualis erit
<
lb
/>
cono
<
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>.e.u.x.</
var
>
vt ſupra demonſtratum eſt, & ſit
<
var
>.g.h.</
var
>
media proportionalis inter
<
var
>.u.x.</
var
>
et
<
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/>
<
var
>o.c</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">Iam viſum ſuperius fuit, quod eadem proportio erat ipſius
<
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var
>
ad
<
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>.g.h.</
var
>
quæ
<
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>.a.b.</
var
>
<
lb
/>
ad
<
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>.e.f.</
var
>
</
s
>
<
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xml:id
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xml:space
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preserve
">quare eadem quæ
<
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>.o.b.</
var
>
ad
<
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>.e.x.</
var
>
ſed
<
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>.u.x.</
var
>
et
<
var
>.e.x.</
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>
dantur. </
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<
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xml:id
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xml:space
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">inter quas
<
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>
et
<
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var
>
vel
<
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/>
<
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>o.c.</
var
>
(nam
<
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>.o.c.</
var
>
æqualis eſt
<
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>.o.b.</
var
>
) medię proportionales ſunt, eo quod cum
<
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>.g.h.</
var
>
media
<
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/>
proportionalis ſit inter
<
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>.u.x.</
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>
et
<
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>.o.c.</
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>
& proportio
<
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>.o.b.</
var
>
ad
<
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>.e.x.</
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>
æqualis ſit ei, quæ
<
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>.u.x.</
var
>
<
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/>
ad
<
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>.g.h.</
var
>
hoc eſt ei quæ
<
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>.g.h.</
var
>
ad
<
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>.o.c.</
var
>
vel. ad
<
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>.o.b.</
var
>
</
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<
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xml:space
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">quare quotieſcunque inuentæ fuerint
<
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g.h.</
var
>
et
<
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>.o.c.</
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>
vel
<
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>.o.b.</
var
>
mediæ proportionales inter
<
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>.d.x.</
var
>
et
<
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>.x.e.</
var
>
ipſa
<
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>.o.c.</
var
>
vel
<
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>.o.b.</
var
>
erit ſemi
<
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diameter ſphæræ quæſitę. </
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<
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xml:space
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">eodem modo faciendum erit ſi ſphęroides fuerit prolata.</
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