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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140 - 149
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IO. BABPT. BENED.
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366
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file
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0366
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0366
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<
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style
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xml:space
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">COROLLARIVM.</
head
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<
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>
<
s
xml:id
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xml:space
="
preserve
">Proportio maioris portionis ad minorem ſemper erit ſeſquialtera proportioni
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ipſius
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>.b.g.</
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ad
<
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>.a.b.</
var
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eo quod cum ſit proportio totalis portionis ad partialem vt trian-
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lb
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guli
<
var
>.b.g.e.</
var
>
ad
<
var
>.b.a.d.</
var
>
& hæc ſeſquialtera proportioni ipſius
<
var
>.g.e.</
var
>
ad
<
var
>.a.o.</
var
>
hoc eſt vt ip-
<
lb
/>
ſius
<
var
>.b.g.</
var
>
ad
<
var
>.b.a.</
var
>
ideo proportio ipſarum portionum erit ſimiliter ſeſquialtera pro-
<
lb
/>
portioni diametrorum.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4257
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xml:space
="
preserve
">Deinde ſi protractæ fuerint
<
var
>.b.d.</
var
>
et
<
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>.g.e.</
var
>
quouſque conueniant in puncto
<
var
>.z.</
var
>
habe
<
lb
/>
bis inter
<
var
>.g.z.</
var
>
et
<
var
>.a.o.</
var
>
duas
<
var
>.g.e.</
var
>
et
<
var
>.a.d.</
var
>
medias proportionales in proportionalitate con
<
lb
/>
tinua, eo quod cum (ex ijs quæ ſupra diximus.).
<
var
>a.d.</
var
>
media proportionalis ſit inter
<
var
>.
<
lb
/>
g.e.</
var
>
et
<
var
>.a.o.</
var
>
& proportio
<
var
>.g.z.</
var
>
ad
<
var
>.g.e.</
var
>
vt ipſius
<
var
>.a.d.</
var
>
ad
<
var
>.a.o.</
var
>
eo quodipſius
<
var
>.g.z.</
var
>
ad
<
var
>.a.d.</
var
>
<
lb
/>
& ipſius
<
var
>.g.e.</
var
>
ad
<
var
>.a.o.</
var
>
eſt vt ipſius
<
var
>.b.g.</
var
>
ad
<
var
>.b.a.</
var
>
ex ſimilitudine triangulorum, ideo di-
<
lb
/>
ctæ
<
reg
norm
="
proportiones
"
type
="
simple
">ꝓportiones</
reg
>
erunt
<
reg
norm
="
inuicem
"
type
="
context
">inuicẽ</
reg
>
æquales. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Vnde permutatim ita erit ipſius
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>.g.z.</
var
>
ad
<
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>.g.e.</
var
>
<
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/>
vt ipſius
<
var
>.a.d.</
var
>
ad
<
var
>.a.o.</
var
>
& ut ipſius
<
var
>.g.e.</
var
>
ad
<
var
>.a.d</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
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xml:space
="
preserve
">Amplius etiam dico, quod proportio pa
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/>
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xlink:label
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fig-0366-01
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xlink:href
="
fig-0366-01a
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number
="
403
">
<
image
file
="
0366-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0366-01
"/>
</
figure
>
rabolæ totalis ad partialem, eadem eſt, quę
<
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/>
cubi ipſius
<
var
>.g.e.</
var
>
ad cubum ipſius
<
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>.a.d.</
var
>
& ex
<
reg
norm
="
con
"
type
="
context
">cõ</
reg
>
<
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/>
ſequenti, vt cuborum earundem baſium, eo
<
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/>
quod cum ſit, ex .36. vndecimi Euclid. pro-
<
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/>
portio cubi ipſius
<
var
>.g.e.</
var
>
ad cubum ipſius
<
var
>.a.d.</
var
>
<
lb
/>
tripla ei quæ ipſius
<
var
>.g.e.</
var
>
ad
<
var
>.a.d.</
var
>
ideo æqualis
<
lb
/>
erit ei quę trianguli
<
var
>.b.g.e.</
var
>
ad triangulum
<
var
>.b.
<
lb
/>
a.d.</
var
>
cum proportio horum duorum triangu
<
lb
/>
lorum compoſita ſit (vt ſupra vidimus) ex
<
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/>
ea quæ
<
var
>.g.e.</
var
>
ad
<
var
>.a.o.</
var
>
& ex ea quæ
<
var
>.g.e.</
var
>
ad
<
var
>.a.d.</
var
>
<
lb
/>
& hæc medietas illius, ſed trianguli ita ſe in
<
lb
/>
uicem habenr, vt parabolę, </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">quare ipſæ para-
<
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/>
bolæ ſeinuicem habebunt, vt cubi ipſarum
<
lb
/>
baſium.</
s
>
</
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</
div
>
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xml:id
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type
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<
head
xml:id
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style
="
it
"
xml:space
="
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">Cubum fabricare æqualem pyramidi propoſitæ.</
head
>
<
head
xml:id
="
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xml:space
="
preserve
">AD EVNDEM.</
head
>
<
p
>
<
s
xml:id
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xml:space
="
preserve
">CVbum fabricare æqualem propoſitæ pyramidi quadrilateræ, nullius erit diffi-
<
lb
/>
cultatis, ſuppoſita tamen pro reperta diuiſione cuiuſuis datæ proportionis in
<
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/>
tres partes æquales. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Nam ex .6. duodecimi Eucli. patet omne corpus ſerratile d-ui
<
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/>
ſibile eſſe in tres pyramides quadrilateras æquales, ſcimus etiam quod cuilibet py-
<
lb
/>
ramidi quadrilateræ poteſt reperiri ſuum ſerratile. </
s
>
<
s
xml:id
="
echoid-s4263
"
xml:space
="
preserve
">Sit igitur propoſita pyramis qua
<
lb
/>
drilatera
<
var
>.m.g.f.h.</
var
>
cuius ſerratile ita inueniemus, ducendo primum
<
var
>.h.i.</
var
>
parallelam
<
lb
/>
ipſi
<
var
>.g.f.</
var
>
et
<
var
>.f.i.</
var
>
ipſi
<
var
>.g.h.</
var
>
in ſuperficie trianguli
<
var
>.f.g.h.</
var
>
et
<
var
>.m.K.</
var
>
ipſi
<
var
>.g.h.</
var
>
in ſuperficie
<
lb
/>
trianguli
<
var
>.m.g.h.</
var
>
& æqualem dictæ
<
var
>.g.h.</
var
>
ducetur poſtea
<
var
>.K.h.</
var
>
et
<
var
>.K.i.</
var
>
& habebimus cor
<
lb
/>
pus
<
var
>.f.K.g.</
var
>
ſerratile, & triplum pyramidi propoſitæ. </
s
>
<
s
xml:id
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"
xml:space
="
preserve
">Nunc duplicemus ipſum, du-
<
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/>
cendo
<
var
>.K.x.</
var
>
in ſuperficie trianguli
<
var
>.i.k.h.</
var
>
parallelam,
<
reg
norm
="
æqualemque
"
type
="
simple
">æqualemq́;</
reg
>
ipſi
<
var
>.i.h.</
var
>
et
<
var
>.m.y.</
var
>
<
lb
/>
in ſuperficie trianguli
<
var
>.f.m.g.</
var
>
parallelam, ę
<
reg
norm
="
qualemque
"
type
="
simple
">qualemq́;</
reg
>
ipſi
<
var
>.f.g.</
var
>
ducatur poſtea
<
var
>.g.y.</
var
>
et
<
var
>.h.
<
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/>
x.</
var
>
quarum
<
reg
norm
="
vnaquæque
"
type
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">vnaquæq;</
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>
æqualis erit ipſi
<
var
>.f.m.</
var
>
vnde habebimus corpus
<
var
>.f.x.</
var
>
parallelepe-
<
lb
/>
pidum, & ſexcuplum ipſi pyramidi propoſitæ.</
s
>
</
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