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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31
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rhead
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THEOREM. ARITH.
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43
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file
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0043
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0043
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ad vnitatem
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ad
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ſicut
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ad
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& componendo
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ad
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ſicut
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ad
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: & euerſim
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ad
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vt
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ad
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. </
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<
s
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xml:space
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">Quare, ex .20. ſepti
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mi, recte vtimur regula de tribus. </
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<
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xml:space
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">Idem & de altera parte dico, quamuis qui vnam
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teneat, alteram quo que habiturus ſit. </
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<
s
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echoid-s410
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xml:space
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">Non mirum tamen ſi huiuſmodi problema
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ab antiquis definitum non fuerit, qui hanc vltimam partem non cognouerunt.</
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<
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xml:space
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47
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num
>
.</
head
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<
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<
s
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xml:space
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">CVR duobus numeris mutuó diuiſis, ſi per ſummam prouenientium, produ-
<
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ctum vnius in alterum multiplicetur, vltimum productum, ſummæ quadra-
<
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tn
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extent
="
2
"/>
m duorum numerorum æquale futurum ſit.</
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<
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<
s
xml:id
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echoid-s412
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xml:space
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">Exempli gratia, propoſitis .16. et .4. mutuò diuiſis, ſumma prouenientium erit
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4
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4.</
num
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integrorum cum quarta parte, qua ſumma multiplicata cum producto
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norm
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primorum
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type
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context
">primorũ</
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>
<
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numerorum, nempe .64. dabuntur .272. integri ſuperficiales, qui ſummæ quadra-
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torum duorum numerorum æquantur.</
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<
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<
s
xml:id
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echoid-s413
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xml:space
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preserve
">Hoc vt conſideremus, duo numeri partibus
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et
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in linea
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ſignificentur,
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quorum productum ſit
<
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&
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quadratum
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type
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ipſius
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ſit
<
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>.e.p</
var
>
: ipſius verò
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>
ſit
<
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>.e.q.</
var
>
pro-
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ueniens
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autem
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type
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ex diuiſione
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>.e.i.</
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>
per
<
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>.a.e.</
var
>
ſit
<
var
>.o.u.</
var
>
proueniens
<
reg
norm
="
autem
"
type
="
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<
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>.a.e.</
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>
per
<
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>.e.i.</
var
>
ſit
<
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>.o.t.</
var
>
quo-
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rum ſumma ſit
<
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>.o.u.t.</
var
>
tum productum
<
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>.e.d</
var
>
: linea
<
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>.u.n.</
var
>
ſignificetur ad angulum
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rectum
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type
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>
<
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coniuncta in puncto
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>.u.</
var
>
extremo ipſius
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>.o.u.t.</
var
>
productum
<
reg
norm
="
autem
"
type
="
wordlist
">aũt</
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<
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>.u.o.t.</
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>
in
<
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>.u.n.</
var
>
ſit
<
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>.n.t</
var
>
. </
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>
<
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xml:id
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xml:space
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">Iam
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probandum nobis eſt
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>
æqualem eſſe ſummæ duorum quadratorum
<
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>.q.e.p</
var
>
. </
s
>
<
s
xml:id
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echoid-s415
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xml:space
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">Quod
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ſingillatim probo, & aſſero productum
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var
>
æquale eſſe quadrato
<
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>.q.e.</
var
>
&
<
reg
norm
="
productum
"
type
="
context
">productũ</
reg
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<
var
>.
<
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s.t.</
var
>
quadrato
<
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>.e.p</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">Nam ex .35. theoremate patet numerum
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medium eſſe
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pro- portionalem
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type
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portionalẽ</
reg
>
inter
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>.e.d.</
var
>
et
<
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>.o.u</
var
>
: cum numerus
<
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>.e.i.</
var
>
ex præſuppoſito ab
<
var
>.e.a.</
var
>
multiplicetur
<
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& diuidatur, cuius multiplicationis produ-
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/>
ctum eſt
<
var
>.d.e</
var
>
: nempe
<
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>.u.n.</
var
>
& proueniens ex
<
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/>
<
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fig-0043-01
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number
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<
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file
="
0043-01
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xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0043-01
"/>
</
figure
>
diuiſione eſt
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var
>
: </
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>
<
s
xml:id
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xml:space
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">quare ex dicto theorema-
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te
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media proportionalis eſt inter
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>
et
<
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>.
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u.o</
var
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. </
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<
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<
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Itaque
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productum
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æquale eſt qua-
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drato
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ex .16. ſexti vel .20. ſeptimi. </
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<
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xml:id
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xml:space
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">Idem
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dico de producto
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<
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nempe
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type
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">nẽpe</
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>
æquale eſſe qua-
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drato
<
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>.e.p.</
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>
quandoquidem numerus
<
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>.a.e.</
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>
ab
<
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<
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>e.i.</
var
>
multiplicatur ac diuiditur, cuius multi-
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plicationis productum eſt
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>.d.e.</
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>
nempe
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>
&
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proueniens ex diuiſione
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>
: </
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>
<
s
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xml:space
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">inter quæ ex .
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35. theoremate
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media proportionalis
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eſt. </
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<
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xml:space
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">Quare ex allatis propoſitionibus
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<
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>.s.t.</
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>
æquale eſt quadrato
<
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>.e.p.</
var
>
ſed
<
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<
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productum
<
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ſumma eſt duorum productorum
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>
et
<
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>.s.t.</
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>
ex prima ſecundi Eucli.
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</
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<
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xml:id
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xml:space
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">Itaque verum eſſe quod dictum eſt, conſequitur.</
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<
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xml:space
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">THEOREMA
<
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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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">CVR ſi quis maiorem duorum numerorum ſola vnitate inter ſe differentium,
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per minorem diuidat,
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type
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">maioremq́;</
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per proueniens multiplicet, productum,
<
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<
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summæ
"
type
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>
ipſius maioris cum eodem proueniente æquale erit.</
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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
">Exempli gratia .10 per .9. diuiſo, datur vnum cum nona parte, quo multiplica-
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to per proueniens, ipſo nempe .10: </
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>
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xml:space
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">datur productum .11. cum nona parte, tantum ſci </
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