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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34
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IO. BAPT. BENED.
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34
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file
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0034
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0034
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35
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<
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xml:space
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">THEOREMA
<
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35
">XXXV</
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>
.</
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<
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echoid-s309
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<
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style
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">QVivis</
emph
>
numerus per alterum multiplicatus, & diuiſus, medius eſt propor-
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tionalis inter productum multiplicationis, & proueniens diaiſionis.</
s
>
</
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<
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<
s
xml:id
="
echoid-s310
"
xml:space
="
preserve
">Exempli gratia, ſi .20.
<
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="
multiplicentur
"
type
="
context
">multiplicẽtur</
reg
>
per quinque & inde per quinque diuidantur
<
lb
/>
productum erit .100. proueniens .4. inter quos numeros .20. medius eſt propor-
<
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tionalis.</
s
>
</
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<
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>
<
s
xml:id
="
echoid-s311
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xml:space
="
preserve
">Hoc vt ſpeculemur, proponatur numerus multiplicandus & diuidendus, qui ſi-
<
lb
/>
gnificetur linea
<
var
>.u.e.</
var
>
multiplicans autem & diuidens linea
<
var
>.a.u.</
var
>
multiplicationis
<
lb
/>
productum ſit
<
var
>.e.a.</
var
>
proueniens ex diuiſione ſit
<
var
>.o.e</
var
>
. </
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>
<
s
xml:id
="
echoid-s312
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xml:space
="
preserve
">Nunc proueniens
<
var
>.e.o.</
var
>
per
<
reg
norm
="
nu- merum
"
type
="
context
">nu-
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merũ</
reg
>
<
var
>.a.u.</
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>
diuidentem multiplicetur, cuius multiplicationis productum ſit
<
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>.e.i.</
var
>
<
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/>
quare, eadem erit proportio numeri
<
var
>.a.e.</
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>
<
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/>
ad numerum
<
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>.e.i.</
var
>
quæ eſt numeri
<
var
>.u.e.</
var
>
ad
<
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/>
<
figure
xlink:label
="
fig-0034-01
"
xlink:href
="
fig-0034-01a
"
number
="
47
">
<
image
file
="
0034-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-01
"/>
</
figure
>
numerum
<
var
>.e.o.</
var
>
ex prima ſextiaut .18. vel
<
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19. ſeptimi. </
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>
<
s
xml:id
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xml:space
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preserve
">Sed cum numerus
<
var
>.u.e.</
var
>
ex
<
ref
id
="
ref-0010
">.11. theoremate præſentis libri</
ref
>
, numero
<
var
>.e.
<
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i.</
var
>
æqualis ſit. </
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>
<
s
xml:id
="
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"
xml:space
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preserve
">verum eſſe, quod propoſi-
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tum fuit conſequetur.</
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</
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>
<
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xml:id
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type
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"
level
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n
="
36
">
<
head
xml:id
="
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"
xml:space
="
preserve
">THEOREMA
<
num
value
="
36
">XXXVI</
num
>
.</
head
>
<
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>
<
s
xml:id
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xml:space
="
preserve
">CVR ij, qui propoſitum numerum ita multiplicare & diuidere cupiunt, vt pro
<
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/>
ductum multiplicationis, tam ſit multiplex prouenienti ex diuiſione, quam
<
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/>
quæritur, rectè ſumant aliquem numerum pro multiplicante & diuidente, qui ſit ra
<
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/>
dix quadrata denominantis quęſitę multiplicitatis.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s316
"
xml:space
="
preserve
">Exempli gratia, proponuntur .20. multiplicanda atque diuidenda, ita vt pro-
<
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/>
ductum multiplicationis nonuplum ſit prouenienti ex diuiſione, nempè, vt pro-
<
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/>
ueniens, nona pars ſit eiuſmodi producti, </
s
>
<
s
xml:id
="
echoid-s317
"
xml:space
="
preserve
">quare quadratam radicem ipſorum no-
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uem, ideſt denominantis ſumunt, tria ſcilicet, multiplicant igitur & diuidunt
<
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/>
data .20. ex quo productum erit .60. proueniens autem .6. cum duabus tertijs. </
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>
<
s
xml:id
="
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xml:space
="
preserve
">&
<
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propoſitum ſequitur.</
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>
</
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<
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>
<
s
xml:id
="
echoid-s319
"
xml:space
="
preserve
">Cuius ſpeculationis cauſa, ſignificetur numerus propoſitus linea
<
var
>.u.e.</
var
>
multipli-
<
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/>
cans autem & diuidens linea
<
var
>.u.a.</
var
>
productum ſit
<
var
>.e.a.</
var
>
proueniens
<
var
>.e.o.</
var
>
quadratum
<
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/>
verò
<
var
>.a.u.</
var
>
ſit
<
var
>.x.a.</
var
>
erit igitur proportio
<
var
>.a.e.</
var
>
ad
<
var
>.e.o.</
var
>
dupla proportioni
<
var
>.a.e.</
var
>
ad nume
<
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/>
rum
<
var
>.u.e.</
var
>
ex præcedenti theoremate: </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Adhæc, cogitemus in linea
<
var
>.u.a.</
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>
vnitatem
<
var
>.
<
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u.i.</
var
>
<
reg
norm
="
terminenturque
"
type
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simple
">terminenturq́;</
reg
>
duo producta
<
var
>.e.i.</
var
>
et
<
var
>.x.i.</
var
>
</
s
>
<
s
xml:id
="
echoid-s321
"
xml:space
="
preserve
">quare eadem erit proportio
<
var
>.a.e.</
var
>
ad
<
var
>.e.i.</
var
>
<
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/>
quæ eſt
<
var
>.a.e.</
var
>
ad
<
var
>.u.e.</
var
>
numerus enim
<
var
>.e.i.</
var
>
(quamuis ſuperficialis) idem eſt cum nume-
<
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/>
ro lineari
<
var
>.u.e.</
var
>
ſed
<
var
>.a.e.</
var
>
ad
<
var
>.e.i.</
var
>
ſic ſe habet ſicut
<
var
>.a.u.</
var
>
ad
<
var
>.u.i.</
var
>
ex prima ſexti aut .18.
<
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/>
vel .19. ſeptimi, (quod ipſum dico de
<
var
>.a.x.</
var
>
ad
<
var
>.x.i.</
var
>
) </
s
>
<
s
xml:id
="
echoid-s322
"
xml:space
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preserve
">quare proportio
<
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>.a.x.</
var
>
ad
<
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>.x.i.</
var
>
hoc
<
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/>
eſt
<
var
>.x.u.</
var
>
ęqualis erit
<
reg
norm
="
proportioni
"
type
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">ꝓportioni</
reg
>
<
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>.a.e.</
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>
ad
<
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>.u.e.</
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>
at trigeſimotertio & trigeſimoquarto theo
<
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/>
remate probatum eſt proportionem numeri
<
var
>.a.x.</
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>
ad vnitatem, duplam eſſe propor-
<
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/>
tioni eiuſdem numeri
<
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>.a.x.</
var
>
ad
<
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>.u.x.</
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>
ſequitur
<
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igitur cum dimidia ſint æqualia, tota etiam
<
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æqualia eſſe: </
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>
<
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xml:id
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xml:space
="
preserve
">hoc eſt proportionem numeri
<
var
>.
<
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<
figure
xlink:label
="
fig-0034-02
"
xlink:href
="
fig-0034-02a
"
number
="
48
">
<
image
file
="
0034-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-02
"/>
</
figure
>
a.e.</
var
>
ad numerum
<
var
>.e.o.</
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>
æqualem eſſe propor
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tioni numeri
<
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>.a.x.</
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>
ad vnitatem. </
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Itaque rectè
<
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ſumitur numerus
<
var
>.a.u.</
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>
eiuſmodi vt
<
reg
norm
="
quadratum
"
type
="
context
">quadratũ</
reg
>
</
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>
</
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