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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78
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<
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52
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IO. BAPT. BENED.
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n
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64
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file
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0064
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0064
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<
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<
s
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xml:space
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preserve
">Sint exempli gratia .4. quantitates
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:
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:
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>e.f</
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: et
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>.g.h</
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: inuicem proportionales in
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proportionalitate arithmetica. </
s
>
<
s
xml:id
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xml:space
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preserve
">Hoc eſt vt quæ proportio (licet impropriè dicta)
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eſt ipſius
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>.a.b.</
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ad
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>.c.d.</
var
>
<
reg
norm
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eadem
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type
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">eadẽ</
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>
ſit ipſius
<
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>.e.f.</
var
>
ad
<
var
>.g.h</
var
>
. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Tunc permutando dico eandem pro
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portionem fore ipſius
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>.a.b.</
var
>
ad
<
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>.e.f.</
var
>
quæ ipſius
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var
>.c.d.</
var
>
ad
<
var
>.g.h</
var
>
.</
s
>
</
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>
<
p
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<
s
xml:id
="
echoid-s691
"
xml:space
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preserve
">Nam, ex hypotheſi, differentia qua
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var
>.a.b.</
var
>
ſuperat
<
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>.c.d.</
var
>
(quæ ſit
<
var
>.m.b.</
var
>
) æqualis eſt
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lb
/>
differentiæ qua
<
var
>.e.f.</
var
>
ſuperat
<
var
>.g.h.</
var
>
(quæ ſit
<
var
>.i.f.</
var
>
) vnde
<
var
>.a.m.</
var
>
reſiduum ex
<
var
>.a.b.</
var
>
æquale erit
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lb
/>
<
var
>c.d.</
var
>
& reſiduum
<
var
>.e.i.</
var
>
æquale
<
var
>.g.h</
var
>
. </
s
>
<
s
xml:id
="
echoid-s692
"
xml:space
="
preserve
">Sit igitur exempli gratia
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>.c.d.</
var
>
maior
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>.g.h.</
var
>
per
<
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>.c.n.</
var
>
<
lb
/>
vnde
<
var
>.n.d.</
var
>
æqualis erit
<
var
>.g.h.</
var
>
</
s
>
<
s
xml:id
="
echoid-s693
"
xml:space
="
preserve
">quare
<
var
>.a.m.</
var
>
maior erit
<
var
>.e.i.</
var
>
per
<
var
>.a.K.</
var
>
æqualem
<
var
>.c.n.</
var
>
ex com-
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muni ſcientia. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Vnde
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var
>.K.m.</
var
>
æqualis erit
<
var
>.n.d.</
var
>
hoc eſt ipſi
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var
>.g.h.</
var
>
hoc eſt ipſi
<
var
>e.i</
var
>
. </
s
>
<
s
xml:id
="
echoid-s695
"
xml:space
="
preserve
">Quare ex
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/>
communi conceptu
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>.b.K.</
var
>
æqualis erit ipſi
<
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>.f.e.</
var
>
ſed
<
var
>.n.d.</
var
>
æqualis eſt
<
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>.g.h.</
var
>
vt dictum eſt.
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</
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>
<
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xml:id
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xml:space
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">Cum ergo
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>.b.K.</
var
>
æqualis ſit
<
var
>.e.f.</
var
>
et
<
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>.d.n.</
var
>
ipſi
<
var
>.g.h.</
var
>
et
<
var
>.a.b.</
var
>
maior ſit ipſa
<
var
>.K.b.</
var
>
per
<
var
>.a.K.</
var
>
æqua-
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/>
lem ipſi
<
var
>.c.n.</
var
>
per quam
<
var
>c.n</
var
>
:
<
var
>d.c.</
var
>
maior eſt ipſa
<
var
>.d.n.</
var
>
ſequitur verum eſſe
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reg
norm
="
propoſitum
"
type
="
context
">propoſitũ</
reg
>
hoc
<
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/>
eſt, quod eadem proportio ſit ipſius
<
var
>.a.b.</
var
>
ad
<
var
>.e.f.</
var
>
quæ
<
var
>.c.d.</
var
>
ad
<
var
>.g.h.</
var
>
arithmetice ſcilicet.</
s
>
</
p
>
<
figure
position
="
here
"
number
="
87
">
<
image
file
="
0064-01
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xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-01
"/>
</
figure
>
</
div
>
<
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xml:id
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type
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<
head
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xml:space
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">THEOREMA
<
num
value
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79
">LXXIX</
num
>
.</
head
>
<
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>
<
s
xml:id
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xml:space
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preserve
">CVR prouenientia duorum numerorum diuidentium eiuſdem numeri diuiſi-
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bilis, geometricè
<
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norm
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eandem
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type
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">eandẽ</
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inter ſe
<
reg
norm
="
proportionem
"
type
="
context
">proportionẽ</
reg
>
ſeruant,
<
reg
norm
="
quam
"
type
="
context
">quã</
reg
>
ipſimet
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reg
norm
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diuidentes
"
type
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context
">diuidẽtes</
reg
>
.</
s
>
</
p
>
<
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>
<
s
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xml:space
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preserve
">Exempli gratia ſi per ſenarium & octonarium numerus vigintiquatuor diuida-
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tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</
s
>
</
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<
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<
s
xml:id
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xml:space
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preserve
">Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis
<
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>.u.x.</
var
>
et
<
var
>.n.e.</
var
>
diuidentes
<
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/>
autem ſint
<
var
>.u.o.</
var
>
et
<
var
>.e.o.</
var
>
</
s
>
<
s
xml:id
="
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xml:space
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preserve
">quare ex ijs, quæ .10.
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<
figure
xlink:label
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fig-0064-02
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xlink:href
="
fig-0064-02a
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number
="
88
">
<
image
file
="
0064-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-02
"/>
</
figure
>
theoremate dicta fuerunt
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>.u.x.</
var
>
per
<
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>.u.o.</
var
>
diui-
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ſo dabit
<
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>.x.o.</
var
>
& diuiſo
<
var
>.n.e.</
var
>
per
<
var
>.e.o.</
var
>
dabit
<
var
>.o.
<
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/>
n</
var
>
. </
s
>
<
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xml:id
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xml:space
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">Dicimus itaque
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eandem
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type
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">eandẽ</
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>
eſſe
<
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norm
="
proportionem
"
type
="
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">proportionẽ</
reg
>
<
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/>
<
var
>o.x.</
var
>
ad
<
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>.o.n.</
var
>
quæ
<
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>.e.o.</
var
>
ad
<
var
>.o.u.</
var
>
quod patet ſub
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/>
ſcriptam figuram conſiderantibus, in qua,
<
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/>
ex .15. ſexti aut .20. ſeptimi, eadem propor-
<
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/>
tio cernitur
<
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>.o.x.</
var
>
ad
<
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>.o.n.</
var
>
quæ
<
var
>.o.e.</
var
>
ad
<
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>.o.u</
var
>
.</
s
>
</
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>
</
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>
<
div
xml:id
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type
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"
level
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n
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">
<
head
xml:id
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"
xml:space
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preserve
">THEOREMA
<
num
value
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">LXXX</
num
>
.</
head
>
<
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>
<
s
xml:id
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xml:space
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preserve
">CVR quauis quantitate, tribus
<
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<
figure
xlink:label
="
fig-0064-03
"
xlink:href
="
fig-0064-03a
"
number
="
89
">
<
image
file
="
0064-03
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-03
"/>
</
figure
>
aut quatuor aut etiam pro libi-
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to pluribus diuidentibus numeris di-
<
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/>
uifa, prouenientia eandem prorſus
<
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/>
inter ſe proportionem ſeruabunt,
<
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/>
quam ipſi diuidentes habere compe
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riuntur.</
s
>
</
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>
<
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>
<
s
xml:id
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xml:space
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preserve
">Exempli gratia, proponitur nu-
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merus .60. quinque numeris diuiden
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dus, vtpotè .30. 20. 15. 12. 10. pro-
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uenientia erunt .2. 3. 4. 5. 6. eadem </
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