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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48
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32
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I.O. BAPT. BENED.
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n
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44
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file
="
0044
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xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0044
"/>
licet quanta ſumma eſt maioris cum proueniente.</
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<
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<
s
xml:id
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echoid-s426
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xml:space
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preserve
">Cuius ſpeculationis cauſa, maior numerus ſignificetur
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>.a.i.</
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et minor linea
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>.a.o.</
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ex
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quo ex præſupoſito
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>.o.i.</
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>
vnitas erit. </
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>
<
s
xml:id
="
echoid-s427
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xml:space
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preserve
">Sit autem proueniens ex diuiſione
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>.a.i.</
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per
<
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>.a.o.
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a.e</
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: </
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<
s
xml:id
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xml:space
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">quod
<
var
>.e.a.</
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directè coniungatur ipſi
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>.a.i.</
var
>
et productum
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>.a.i.</
var
>
in
<
var
>.a.e.</
var
>
ſit
<
var
>.u.i</
var
>
. </
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>
<
s
xml:id
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echoid-s429
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xml:space
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preserve
">Probabo
<
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numerum ſuperficialem
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>.u.i.</
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>
æqualem eſſe lineari
<
var
>.i.a.e</
var
>
. </
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>
<
s
xml:id
="
echoid-s430
"
xml:space
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preserve
">quare meminiſſe oportet,
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decimotertio theoremate probatum fuiſſe, quod ſi numerus diuiſibilis per pro-
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ueniens diuidatur, proueniens futurus ſit numerus diuidens, </
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>
<
s
xml:id
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xml:space
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preserve
">quare
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>.a.o.</
var
>
erit pro-
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ueniens ex diuiſione
<
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>.a.i.</
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>
per
<
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>.a.e.</
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>
& ex deſinitione diuiſionis ita ſe habebit
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>.e.a.</
var
>
ad
<
var
>.
<
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/>
a.i.</
var
>
ſicut
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>.o.i.</
var
>
ad
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>.o.a.</
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>
& componondo ita
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>.e.i.</
var
>
ad
<
var
>.a.i.</
var
>
ſicut
<
var
>.i.a.</
var
>
ad
<
var
>.o.a.</
var
>
</
s
>
<
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xml:id
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xml:space
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preserve
">quare
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var
>.a.i.</
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>
erit me-
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/>
dia pportionalis inter
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>.e.i.</
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>
et
<
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>.a.o.</
var
>
ſed
<
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>.a.i.</
var
>
non modò diuiſa
<
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norm
="
nunc
"
type
="
context
">nũc</
reg
>
cogitatur ab
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>.e.a.</
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>
ex
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/>
quo ſit proueniens
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>.a.o.</
var
>
ſed etiam per eandem
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>.e.a.</
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>
multiplicata, ex quo produ-
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ctum oriatur
<
var
>.u.i</
var
>
. </
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Itaque
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type
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">Itaq;</
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ex .25. theobema-
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te
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>
media eſt proportionalis inter
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>.u.</
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>
<
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<
figure
xlink:label
="
fig-0044-01
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xlink:href
="
fig-0044-01a
"
number
="
60
">
<
image
file
="
0044-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-01
"/>
</
figure
>
i. et
<
var
>.a.o</
var
>
. </
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>
<
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xml:space
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preserve
">Quare. ex .11. quinti. eadem erit
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proportio
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>.u.i.</
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>
ad
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>.a.i.</
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>
ſicut
<
var
>.e.i.</
var
>
ad eandem
<
var
>.
<
lb
/>
a.i</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">Igitur ex .9. prædicti numerus
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>.u.i.</
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>
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æqualis erit numero
<
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>.e.i.</
var
>
quod erat propoſitum.</
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</
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</
div
>
<
div
xml:id
="
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type
="
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"
level
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3
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n
="
49
">
<
head
xml:id
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xml:space
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preserve
">THEOREMA
<
num
value
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49
">XLIX</
num
>
.</
head
>
<
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>
<
s
xml:id
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xml:space
="
preserve
">IDipſtim etiam alia ratione conſiderari poteſt.</
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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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">Linea
<
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>.u.a.</
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>
ſecetur in puncto
<
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>.t.</
var
>
ita vt
<
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>.a.t.</
var
>
æqualis ſit vnitati
<
var
>.o.i.</
var
>
& media paral
<
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/>
lela
<
var
>.t.n.</
var
>
terminetur productum
<
var
>.t.i.</
var
>
quod conſtabit æquali numero, quamuis ſuperfi-
<
lb
/>
ciali, numero
<
var
>.a.i.</
var
>
tametſi lineari. </
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Tumparallela ducatur à puncto
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>.o.</
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>
ipſi
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>.a.u.</
var
>
termi
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norm
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neturque
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type
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">neturq́;</
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>
productum
<
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>.o.u.</
var
>
ex quo bina producta dabuntur
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>.u.o.</
var
>
et
<
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>.t.i.</
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>
inter ſe æqualia
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ex .15. ſexti aut .20. ſeptimi cum ita ſe habeat
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ad
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>
ſicut
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>.a.o.</
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>
ad
<
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>.a.t.</
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>
ſed
<
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>.a.i.</
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>
ad
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>.
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a.o.</
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>
permutando ſic ſe habet ſicut
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>.a.u.</
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>
ad
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>.a.t.</
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>
& ex prima ſexti aut .18. vel .19. ſepti-
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mi ſic ſe habet
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>.u.i.</
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>
ad
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>.u.o.</
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ſicut
<
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>.a.i.</
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>
ad
<
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>.a.</
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>
<
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/>
<
figure
xlink:label
="
fig-0044-02
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xlink:href
="
fig-0044-02a
"
number
="
61
">
<
image
file
="
0044-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-02
"/>
</
figure
>
o. hoc eſt
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ad
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>
ope .11. quinti. </
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<
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">Iam
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ex definitione diuiſionis ita ſe habet
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>.a.e.</
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>
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ad
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>
ſicut
<
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>.o.i.</
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>
ad
<
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>.o.a.</
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>
& componendo
<
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>.
<
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e.i.</
var
>
ad
<
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>.a.i.</
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>
ſicut
<
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>.i.a.</
var
>
ad
<
var
>.o.a</
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>
. </
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>
<
s
xml:id
="
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xml:space
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preserve
">Itaque ex præ-
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dicta .11. ſic ſe habebit
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>.e.i.</
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>
ad
<
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>.i.a.</
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>
ſicut
<
var
>.u.
<
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i.</
var
>
ad
<
var
>.t.i.</
var
>
ſed
<
var
>.t.i.</
var
>
numero conſtat æquali
<
var
>.a.
<
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/>
i</
var
>
. </
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>
<
s
xml:id
="
echoid-s441
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xml:space
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">quare ex .9. quinti numerus
<
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>.u.i.</
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>
numero
<
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>.e.i.</
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>
æqualis erit.</
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>
</
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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
="
50
">
<
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xml:id
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xml:space
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">THEOREMA
<
num
value
="
50
">L</
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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
">CVR diuidentes numerum propoſitum in duas eiuſmodi partes, vt
<
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="
productum
"
type
="
context
">productũ</
reg
>
<
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/>
vnius in alteram cum i pſarum differentia in ſummam collectum, æquale ſit
<
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/>
alicui alteri numero maiori primo. </
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>
<
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xml:id
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xml:space
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">Rectè primum ex ſecundo detrahunt, reſiduum
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verò conſeruant, tum ex primo ſemper binarium deſumunt,
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dimidiumque
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type
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">dimidiumq́;</
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>
conſer-
<
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uant, alterum verò dimidium in ſeipſo multiplicant, & ex quadrato numerum con
<
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/>
ſeruatum eruunt,
<
reg
norm
="
reſiduique
"
type
="
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">reſiduiq́;</
reg
>
radicem ex dimidio conſeruato, quod vltimum reſi-
<
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/>
duum propoſiti numeri quæſita pars minor eſt.</
s
>
</
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>
<
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>
<
s
xml:id
="
echoid-s444
"
xml:space
="
preserve
">Exempli gratia, ſi proponatur numerus .20. ita
<
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norm
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diuidendus
"
type
="
context
">diuidẽdus</
reg
>
, vt
<
reg
norm
="
productum
"
type
="
context
">productũ</
reg
>
vnius partis
<
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/>
in alteram, cum partium differentia collectum in ſummam, æquale ſit propoſito </
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>
</
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</
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