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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394
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rhead
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IO. BAPT. BENED.
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406
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file
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0406
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0406
"/>
iam dicti ad cubum inſius
<
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ex .11. quinti erit vt dupli
<
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cum
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type
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ſimplo
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ad
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.</
s
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</
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<
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">δ</
note
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<
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<
s
xml:id
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echoid-s4666
"
xml:space
="
preserve
">Superius autem vbi. β. demonſtratum fuit ita eſſe ipſius
<
var
>.m.n.</
var
>
ad
<
var
>.n.t.</
var
>
vt cubi
<
var
>.m.n.</
var
>
<
lb
/>
ad cubum
<
var
>.x.n.</
var
>
& inter. α et. β probatum fuit ita eſſe cubi
<
var
>.a.f.</
var
>
ad cubum
<
var
>.d.g.</
var
>
vt
<
lb
/>
cubi
<
var
>.m.n.</
var
>
ad cubum
<
var
>.x.n</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4667
"
xml:space
="
preserve
">Vnde ex .11. quinti
<
var
>.m.n.</
var
>
ad
<
var
>.n.t.</
var
>
erit vt cubi
<
var
>.a.f.</
var
>
ad cubum
<
lb
/>
<
var
>d.g</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4668
"
xml:space
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preserve
">Dicit poſtea quod eadem proportio erit inter cubum
<
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>.d.g.</
var
>
& corpus illud quod
<
lb
/>
pro baſi habeat quadratum inſius
<
var
>.d.g.</
var
>
altitudinem verò vt dictum eſt, quæ eſt inter
<
lb
/>
<
var
>d.g.</
var
>
& compoſitum ex duplo
<
var
>.a.f.</
var
>
cum ſimplo
<
var
>.d.g.</
var
>
quod compoſitum eſt altitudo di
<
lb
/>
cta, &
<
reg
norm
="
verum
"
type
="
context
">verũ</
reg
>
dicit ex ratione ſuperius allegata pro reliquo corpore & cubo ipſius
<
var
>.a.f</
var
>
.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s4669
"
xml:space
="
preserve
">Quare etiam quemadmodum
<
var
>.t.n.</
var
>
ſe habet ad duplum ipſius
<
var
>.o.n.</
var
>
cum ſimplo
<
var
>.t.n.</
var
>
<
lb
/>
ex ijſdem rationibus ſupradictis, vbiloquuti ſumus de
<
var
>.x.n.</
var
>
cum
<
var
>.m.n</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4670
"
xml:space
="
preserve
">Diſponantur
<
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norm
="
nunc
"
type
="
context
">nũc</
reg
>
omnia tali ordine, ita vt
<
var
>.u.</
var
>
primum ſit corpus quod pro ſua ba
<
lb
/>
ſi habeat quadratum ipſius
<
var
>.a.f.</
var
>
& c.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4671
"
xml:space
="
preserve
">Et
<
var
>.y.</
var
>
ſit cubus ipſius
<
var
>.a.f.</
var
>
et
<
var
>.s.</
var
>
ſit cubus ipſius
<
var
>.d.g.</
var
>
et
<
var
>.z.</
var
>
ſit corpus quod baſim ha-
<
lb
/>
bet quadratum ipſius
<
var
>.d.g.</
var
>
altitudinem verò vt ſupradictum eſt, et
<
var
>.p.</
var
>
ſit compoſitum
<
lb
/>
dupli
<
var
>.n.x.</
var
>
cum ſimplo
<
var
>.m.n.</
var
>
et
<
var
>.l.</
var
>
ſit compoſitum dupli ipſius
<
var
>.n.o.</
var
>
cum ſimplo
<
var
>.t.n.</
var
>
<
lb
/>
Sed
<
var
>.u.</
var
>
locata ſit è regione
<
var
>.p.</
var
>
et
<
var
>.y.</
var
>
è regione
<
var
>.m.n.</
var
>
et
<
var
>.s.</
var
>
è regione
<
var
>.n.t.</
var
>
et
<
var
>.z.</
var
>
è regione
<
var
>.l.</
var
>
<
lb
/>
& habebimus proportionem ipſius
<
var
>.u.</
var
>
ad
<
var
>.y.</
var
>
vt
<
var
>.y.</
var
>
ad
<
var
>.m.n.</
var
>
& ipſius
<
var
>.y.</
var
>
ad
<
var
>.s.</
var
>
vt
<
var
>.m.n.</
var
>
ad
<
var
>.
<
lb
/>
n.t.</
var
>
quod ſuperius iam demonſtratum fuit, vbi, δ. et
<
var
>.s.</
var
>
ad
<
var
>.z.</
var
>
ita ſe habebit vt
<
var
>.n.t.</
var
>
ad
<
var
>.
<
lb
/>
l.</
var
>
vt vltimò probatum fuit. </
s
>
<
s
xml:id
="
echoid-s4672
"
xml:space
="
preserve
">Quare ex .22. quinti ita ſe habebit
<
var
>.u.</
var
>
ad
<
var
>.z.</
var
>
vt
<
var
>.p.</
var
>
ad
<
var
>.l.</
var
>
<
lb
/>
quemadmodum dicit Archi.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4673
"
xml:space
="
preserve
">Et quia vt ſe habet
<
var
>.u.</
var
>
ad
<
var
>.z.</
var
>
ita facta fuit
<
var
>.h.i.</
var
>
ad
<
var
>.i.K.</
var
>
vbi
<
var
>.R.</
var
>
ideo ex .11. quinti vt ſe
<
lb
/>
habet
<
var
>.h.i.</
var
>
ad
<
var
>.i.K.</
var
>
ita ſe habebit
<
var
>.p.</
var
>
ad
<
var
>.l.</
var
>
vt ipſe dicit: </
s
>
<
s
xml:id
="
echoid-s4674
"
xml:space
="
preserve
">Et ex .18. quinti ita erit
<
var
>.h.K.</
var
>
<
lb
/>
ad
<
var
>.K.i.</
var
>
vt
<
var
>.p.l.</
var
>
ad
<
var
>.l.</
var
>
& ex communi conceptu
<
var
>.g.f.</
var
>
ſe habebit ad
<
var
>.h.K.</
var
>
vt quintuplum
<
lb
/>
ipſius
<
var
>.p.l.</
var
>
ad
<
var
>.p.l.</
var
>
& ex .22. eiuſdem ita ſe habebit
<
var
>.f.g.</
var
>
ad
<
var
>.i.k.</
var
>
vt quintuplum ipſius
<
var
>.p.
<
lb
/>
l.</
var
>
ad
<
var
>.l.</
var
>
quintuplum autem ipſius
<
var
>.p.l.</
var
>
compoſitum eſt ex quintuplo ipſius
<
var
>.n.m.</
var
>
cum
<
lb
/>
decuplo ipſius
<
var
>.n.x.</
var
>
cum quintuplo ipſius
<
var
>.n.t.</
var
>
cum decuplo ipſius
<
var
>.n.o.</
var
>
vt à te facilè
<
lb
/>
computare potes.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4675
"
xml:space
="
preserve
">Verum etiam erit ex communi ſcientia quod
<
var
>.g.f.</
var
>
ad
<
var
>.f.k.</
var
>
eſt ut quintuplum ipſius
<
lb
/>
<
var
>p.l.</
var
>
ad duplum ipſius
<
var
>.p.l.</
var
>
eo quod ſuperius ſuppoſitum fuit
<
var
>.h.K.</
var
>
eſſe
<
reg
norm
="
quintam
"
type
="
context
">quintã</
reg
>
mediam,
<
lb
/>
vnde
<
var
>.k.f.</
var
>
relinquebatur pro duabus quintis inferioribus, duplum autem
<
var
>.p.l.</
var
>
com-
<
lb
/>
poſitum eſt ex duplo ipſius
<
var
>.m.n.</
var
>
cum duplo ipſius
<
var
>.n.t.</
var
>
cum quadruplo ipſius
<
var
>.n.x.</
var
>
&
<
lb
/>
cum quadruplo ipſius
<
var
>.x.o</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4676
"
xml:space
="
preserve
">Ex conuerſa proportionalitate deinde ita ſe habet,
<
var
>i.K.</
var
>
ad
<
var
>.i.k.</
var
>
ad
<
var
>.f.g.</
var
>
vt
<
var
>.l.</
var
>
ad quin-
<
lb
/>
tuplum ipſius
<
var
>.p.l.</
var
>
et
<
var
>.k.f.</
var
>
ad
<
var
>.f.g.</
var
>
vt duplum ipſius
<
var
>.p.l.</
var
>
ad quintuplum ipſius
<
var
>.p.l</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4677
"
xml:space
="
preserve
">Vnde
<
lb
/>
ex .24. quinti
<
var
>.i.f.</
var
>
ſe habebit ad
<
var
>.f.g.</
var
>
vt
<
reg
norm
="
duplum
"
type
="
context
">duplũ</
reg
>
ipſius
<
var
>.p.l.</
var
>
cum ſimplo
<
var
>.l.</
var
>
ad quintuplum
<
lb
/>
ipſius
<
var
>.p.l</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4678
"
xml:space
="
preserve
">Deinde ex conuerſa proportionalitate quintuplum ipſius
<
var
>.p.l.</
var
>
ſe habebit
<
lb
/>
<
note
xlink:label
="
note-0406-02
"
xlink:href
="
note-0406-02a
"
position
="
left
"
xml:space
="
preserve
">θ</
note
>
ad duplum ipſius
<
var
>.p.l.</
var
>
cum ſimplo
<
var
>.l.</
var
>
vt
<
var
>.f.g.</
var
>
ad
<
var
>.f.i</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4679
"
xml:space
="
preserve
">Sed compoſitum dupli ipſius
<
var
>.p.l.</
var
>
<
lb
/>
cum ſimplo
<
var
>.l.</
var
>
æquale eſt duplo ipſius
<
var
>.m.n.</
var
>
cum quadruplo ipſius
<
var
>.x.n.</
var
>
cum ſexcuplo
<
lb
/>
ipſius
<
var
>.o.n.</
var
>
cum triplo ipſius
<
var
>.n.t.</
var
>
vt per te computare potes.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4680
"
xml:space
="
preserve
">Superius enim ſumpta fuit
<
var
>.i.r.</
var
>
ad quam ita ſe haberet
<
var
>.f.h.</
var
>
hoc eſt tres quintæ ip-
<
lb
/>
ſius
<
var
>.f.g.</
var
>
vt
<
var
>.m.t.</
var
>
ad
<
var
>.t.n</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4681
"
xml:space
="
preserve
">Quare ex conuerſa proportionalitate ita ſe habebit
<
var
>.i.r.</
var
>
ad tres
<
lb
/>
quintas ipſius
<
var
>.f.g.</
var
>
vt
<
var
>.t.n.</
var
>
ad
<
var
>.t.m</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4682
"
xml:space
="
preserve
">Et quia
<
var
>.o.n.</
var
>
ſumpta fuit æqualis ipſi
<
var
>.b.g.</
var
>
et
<
var
>.m.n.</
var
>
ipſi
<
lb
/>
<
var
>b.f.</
var
>
ideo
<
var
>.m.o.</
var
>
ex communi ſcientia æ qualis erit ipſi
<
var
>.g.f</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4683
"
xml:space
="
preserve
">Vnde proportio
<
var
>.r.i.</
var
>
ad tres
<
lb
/>
quintas ipſius
<
var
>.m.o.</
var
>
erit vt
<
var
>.n.t.</
var
>
ad
<
var
>.t.m.</
var
>
vt inquit Archi.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4684
"
xml:space
="
preserve
">Sed vbi. θ. iam probauimus ita ſe habere
<
var
>.i.f.</
var
>
ad
<
var
>.f.g.</
var
>
vt duplum
<
reg
norm
="
ipſius
"
type
="
simple
">ipſiꝰ</
reg
>
<
var
>.p.l.</
var
>
cum ſim-
<
lb
/>
plo
<
var
>.l.</
var
>
ſe habet ad quintuplum ipſius
<
var
>.p.l.</
var
>
hoc eſt
<
var
>.i.f.</
var
>
ad
<
var
>.m.o.</
var
>
vt duplum ipſius
<
var
>.p.l.</
var
>
cum
<
lb
/>
ſimplo
<
var
>.l.</
var
>
ad quintuplum ipſius
<
var
>.p.l</
var
>
.</
s
>
</
p
>
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