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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rhead
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IO. BAPT. BENED.
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n
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14
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file
="
0014
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0014
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quis, qua ratione fractus numerus
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minor ſit in ſuo integro
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fracto
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in
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ſuo integro
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aut fracto
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in ſuo integro
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conſideret is quo pacto pro-
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portio
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>.c.i.</
var
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ad
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>.d.b.</
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minor ſit proportione
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var
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ad
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>.a.b.</
var
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et
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>.a.c.</
var
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ad
<
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>.a.d.</
var
>
hac ratione. </
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>
<
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xml:space
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">Ma-
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nifeſtum eſt ex
<
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">prima ſexti de quantitate
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continua</
ref
>
, aut
<
ref
id
="
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">.18. ſeptimi Euclidis</
ref
>
de diſcre
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<
figure
xlink:label
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fig-0014-01
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xlink:href
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fig-0014-01a
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number
="
2
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<
image
file
="
0014-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-01
"/>
</
figure
>
ta, proportionem ipſius
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ad
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eſſe ſi-
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cut
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>.a.i.</
var
>
ad
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>.a.b.</
var
>
& cum
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var
>
minor ſit
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>.d.i.</
var
>
<
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velut pars ſuo toto, proportio,
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>c.i.</
var
>
ad
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>.d.b.</
var
>
<
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minor erit proportione
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var
>
ad
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>.d.b.</
var
>
ex .8.
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quinti, </
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>
<
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xml:id
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xml:space
="
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">quare minor erit pariter proportio-
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ne
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>.a.i.</
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>
ad
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>.a.b.</
var
>
ex
<
ref
id
="
ref-0005
">.12.
<
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norm
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eiuſdem
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type
="
context
">eiuſdẽ</
reg
>
</
ref
>
vnà etiam pro-
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/>
portio
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>.c.i.</
var
>
ad
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var
>.d.b.</
var
>
minor erit
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>.a.c.</
var
>
ad
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var
>.a.d.</
var
>
<
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/>
ex eiſdem cauſis, medio
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>.c.b</
var
>
. </
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>
<
s
xml:id
="
echoid-s47
"
xml:space
="
preserve
">Ex quibus pa-
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tet ratio, cur fracti diuerſarum denomina-
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/>
tionum ad vnicam reducantur. </
s
>
<
s
xml:id
="
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xml:space
="
preserve
">Cur etiam
<
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/>
numeros integros in partes fractis ſimiles
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/>
frangere liceat, quæ omnia ex ſubſequenti
<
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/>
figura facilè cognoſci poſſunt.</
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>
</
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>
</
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>
<
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type
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<
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xml:space
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">THEOREMA
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.</
head
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<
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style
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">QVae</
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>
ſit ratio, cur hi, qui numeros, fractos diuerſarum denominationum col-
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ligere volunt, & in ſummam redigere, multiplicent vnum ex numerantibus
<
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/>
per denominatorem alterius, & poſtmodum denominatores adinuicem, quorum
<
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/>
vltimum productum, commune eſt denominans duorum priorum productorum,
<
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quæ collecta in ſummam efficiunt quod quærebatur.</
s
>
</
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>
<
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Qua in re ſciendum eſt, denominantes conſiderari tanquam partes vnius
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norm
="
eiuſdem- q́ue
"
type
="
context
">eiuſdẽ-
<
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q́ue</
reg
>
magnitudinis quantitatis continuæ, linearum (verbigratia)
<
var
>a.b.</
var
>
et
<
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>.a.d.</
var
>
<
reg
norm
="
æqualium
"
type
="
context
">æqualiũ</
reg
>
<
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/>
in longitudine,
<
reg
norm
="
quarum
"
type
="
context
">quarũ</
reg
>
<
var
>.a.b.</
var
>
in quatuor partes diuidatur, et
<
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>.a.d.</
var
>
in tres. </
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>
<
s
xml:id
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xml:space
="
preserve
">Quare ſi colli-
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gere voluerimus duo tertia cum tribus quartis, multiplicabimus
<
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>.a.c.</
var
>
duo tertia,
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/>
cum
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var
>.a.b.</
var
>
diuiſa in 4. partes, produceturq́ue
<
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>.c.b.</
var
>
octo partium ſuperficialium, de-
<
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/>
hinc multiplicando
<
var
>.a.i.</
var
>
tres quartas cum
<
var
>.a.d.</
var
>
diuiſa in .3. partes producetur
<
var
>.i.d.</
var
>
pri
<
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/>
mis ſingulis æqualis, nouem partium ſuper
<
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/>
ficialium, multiplicata deinde
<
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>a.b.</
var
>
diui-
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/>
<
figure
xlink:label
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fig-0014-02
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xlink:href
="
fig-0014-02a
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number
="
3
">
<
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file
="
0014-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-02
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</
figure
>
ſa in .4. partes per
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>
in .3. diuiſa, produ-
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cetur quadratum
<
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>.d.b.</
var
>
in continuo, in 12.
<
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/>
partes diuiſum, quod erit totum commune
<
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/>
ſingulis productis, quorum primum erat
<
var
>.c.
<
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/>
b</
var
>
. </
s
>
<
s
xml:id
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xml:space
="
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">Quare
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>.c.b.</
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>
ita ſe habet ad totum
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>.d.b.</
var
>
ſi-
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cut
<
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>.a.c.</
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>
ad
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>.a.d.</
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>
ex prima ſexti in continuis,
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/>
aut .18. ſeptimi in diſcretis quantitatibus,
<
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et
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>.d.i.</
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>
ad
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>.d.b.</
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>
ſicut
<
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>.a.i.</
var
>
ad
<
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>.a.b.</
var
>
ex eiſdem
<
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propoſitionibus. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Collectis deinde parti-
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/>
bus producti
<
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>.c.b.</
var
>
cum partibus producti
<
var
>.
<
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/>
d.i.</
var
>
manifeſtè depræhendetur eiuſmodi
<
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/>
ſummam componi ex partibus vnius totius
<
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/>
communis ſingulis earum.</
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>
</
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