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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53
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THEOREM. ARITH.
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65
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file
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0065
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0065
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proportione diuidentium, quamuis ex aduerſo.</
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<
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xml:space
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">Cuius ratio ex .15. ſexti aut .20. ſeptimi dependet. </
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cillimè deprehendi poteſt.</
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<
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xml:space
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">THEOREMA
<
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value
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81
">LXXXI</
num
>
.</
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<
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<
s
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xml:space
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">CVR quantitate in tres continuas partes proportionales ſecta, & per ſingulas
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ipſarum diuiſa, ſumma trium prouenientium quadrato medij prouenientis
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æqualis eſt.</
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</
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<
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<
s
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xml:space
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">Exempli gratia, proponitur .14. diuidendus in tres continuas partes proportio-
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nales, nempe .8. 4. 2.
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ipſeque
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type
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numerus .14. per ſingulas diuiditur, ex quo tria proue-
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nientia oriuntur, nempe ex prima parte .8.
<
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norm
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proueniens
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type
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">proueniẽs</
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>
erit .1. cum tribus quartis par
<
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/>
tibus ex ſecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-
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nient .7. integri, qui in ſummam collecti dant .12. integros & vnam quartam par-
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tem tantumdem, videlicet quantum quadratum prouenientis medij, nempe .3.
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cum dimidio.</
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</
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<
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<
s
xml:id
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xml:space
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preserve
">Cuius ſpeculationis gratia, totalis numerus ſignificetur linea
<
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>.n.c.</
var
>
qui in tres par-
<
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tes diuidatur
<
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>.n.a</
var
>
:
<
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>a.e.</
var
>
et
<
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>.e.c.</
var
>
quæ ſint continuæ proportionales, quarum ſingulis,
<
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/>
numerum
<
var
>.n.c.</
var
>
diuiſum eſſe cogitemus, proueniens autem ex diuiſione
<
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>.n.c.</
var
>
per
<
var
>.n.
<
lb
/>
a.</
var
>
ſit
<
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>.i.d.</
var
>
quod verò prouenit ex diuiſione
<
var
>.n.c.</
var
>
per
<
var
>.a.e.</
var
>
ſit
<
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>.d.u.</
var
>
proueniens quoque ex
<
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/>
diuiſione
<
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>.n.c.</
var
>
per
<
var
>.e.c.</
var
>
ſit
<
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>.u.o.</
var
>
quorum ſumma ſit
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var
>.i.o.</
var
>
quæ aſſeritur eſſe numeri æqua-
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lis numero quadrati
<
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>.d.u</
var
>
. </
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>
<
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xml:id
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xml:space
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">Quod hac ratione probabo, producatur linea
<
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donec
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>.
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o.p.</
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>
æqualis ſit
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>.o.u.</
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>
<
reg
norm
="
erigaturque
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type
="
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">erigaturq́;</
reg
>
<
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>.o.m.</
var
>
æqualis
<
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>.d.i.</
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>
perpendiculariter
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>.o.p.</
var
>
in puncto
<
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>.o.</
var
>
<
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/>
quæ producatur donec
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>.o.q.</
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>
vnitati ſit æqualis,
<
reg
norm
="
terminenturque
"
type
="
simple
">terminenturq́;</
reg
>
duo rectangula
<
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>.m.p.</
var
>
<
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/>
et
<
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>.q.i.</
var
>
ex quo habebimus rectangulum, aut productum
<
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>.m.p.</
var
>
æquale quadrato
<
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>.d.u.</
var
>
<
lb
/>
ex .16 ſexti aut .20. ſeptimi, quandoquidem tria prouenientia
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>.o.u</
var
>
:
<
var
>u.d.</
var
>
et
<
var
>.d.i.</
var
>
ex
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pręcedenti theoremate ſunt inter ſe continua proportionalia, proportionalitate qua
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partes
<
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>.n.c</
var
>
. </
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>
<
s
xml:id
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xml:space
="
preserve
">Iam verò ſi probauero
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>.q.i.</
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>
productum, producto
<
var
>.m.p.</
var
>
æquale eſſe, pro-
<
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poſitum quoque probatum erit. </
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>
<
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xml:id
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xml:space
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preserve
">Numerus enim producti
<
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>.q.i.</
var
>
æqualis eſt numero.
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</
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<
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xml:id
="
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xml:space
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">ſummæ
<
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>.i.o</
var
>
. </
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>
<
s
xml:id
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xml:space
="
preserve
">Habemus autem ex definitione diuiſionis ita ſe habere
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>.n.c.</
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>
ad
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>.i.d.</
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>
ſicut
<
var
>.
<
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n.a.</
var
>
ad
<
var
>.o.q</
var
>
. </
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>
<
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xml:id
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xml:space
="
preserve
">Itaque permutando ſic ſe habebit
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ad
<
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>.n.a.</
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>
ſicut
<
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>.d.i.</
var
>
hoc eſt
<
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>.m.o.</
var
>
ad
<
var
>.
<
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o.q.</
var
>
ſed ſicut ſe habet
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>.n.c.</
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>
ad
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var
>.n.a.</
var
>
ita pariter ſe habet
<
var
>.i.o.</
var
>
ad
<
var
>.o.u.</
var
>
hoc eſt ad
<
var
>.o.p</
var
>
. </
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>
<
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xml:id
="
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xml:space
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">Ita-
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que
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>.i.o.</
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ad
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>.o.p.</
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>
ſic ſe habebit ſicut
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var
>.m.o.</
var
>
ad
<
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>.o.q.</
var
>
ex quo ex .15. ſexti aut .20. ſeptimi
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var
>.
<
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q.i.</
var
>
æqualis erit
<
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>.m.p.</
var
>
& conſequenter quadrato
<
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>.d.u</
var
>
. </
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>
<
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xml:id
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xml:space
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">Vt autem lector minori labo-
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re cognoſcere queat
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>
ad
<
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>.o.u.</
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>
ſic ſe habere, vt
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>.n.c.</
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>
ad
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>.n.a.</
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>
ſciendum eſt quòd, ſic
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ſe habet
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>.i.d.</
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ad
<
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>.d.u.</
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>
ut
<
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>.c.e.</
var
>
ad
<
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>.e.a.</
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>
ex quo componendo ſic ſe habebit
<
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>.i.u.</
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>
ad
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>.d.u.</
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>
ſi-
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cut
<
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>.c.a.</
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>
ad
<
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>.a.e.</
var
>
& permutando ita
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>.i.u.</
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>
<
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<
figure
xlink:label
="
fig-0065-01
"
xlink:href
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fig-0065-01a
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number
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90
">
<
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file
="
0065-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0065-01
"/>
</
figure
>
ad
<
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>.c.a.</
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>
vt
<
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>.d.u.</
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>
ad
<
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>.e.a.</
var
>
ſed cum ex
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type
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cedẽti</
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theoremate ſic ſe habeat
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>.d.u.</
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>
<
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ad
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>.u.o.</
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>
ſicut
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>.e.a.</
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>
ad
<
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>.a.n.</
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>
permutando
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ſic ſe habebit
<
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>.d.u.</
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>
ad
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>.a.e.</
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>
ſicut
<
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>.u.o.</
var
>
ad
<
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/>
<
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>a.n.</
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>
ex quo ex .11. quinti ſic ſe habe-
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bit
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>.i.u.</
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>
ad
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>.c.a.</
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>
prout
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>.o.u.</
var
>
ad
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var
>.a.n.</
var
>
per-
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mutandoq́ue
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>.i.u.</
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>
ad
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>.u.o.</
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>
vt
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>.c.a.</
var
>
ad
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>.a.n.</
var
>
& componendo, ita
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>.i.o.</
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>
ad
<
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>.u.o.</
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>
ſicut
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>.c.n.</
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>
<
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ad
<
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>.a.n</
var
>
.</
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>
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