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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17
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rhead
="
THEOREM. ARITH.
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29
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file
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0029
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0029
"/>
numerorum, proueniat numerus æqualis numero producti duorum primorum nu-
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m erorum ſimul.</
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<
s
xml:id
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echoid-s251
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xml:space
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preserve
">Sint exempli gratia propoſiti numeri .2. et .8. qui mutuo diuiſi in primis dent pro
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uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-
<
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cta dabunt ſummam quatuor integrorum et quartæ partis vnius, ſumma autem qua
<
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/>
dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar
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tam partem vnius diuiſus dabit .16. pro proueniente, quæ .16. æqualia erunt pro
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ducto binarii in octonarium.</
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</
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<
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<
s
xml:id
="
echoid-s252
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xml:space
="
preserve
">Cuius rei hæc erit ſpeculatio, ſint duæ lineæ
<
var
>.o.e.</
var
>
et
<
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>.o.n.</
var
>
quæ duos numeros pro-
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poſitos ſignificent, inuicem ad angulum rectum
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>.o.</
var
>
coniunctæ, quarum quadrata
<
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ſint
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>.o.a.</
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>
et
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>.o.p.</
var
>
ipſorum productum ſit
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var
>.n.e.</
var
>
tum
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>.o.t.</
var
>
ſit proueniens ex diuiſione
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var
>.o.e.</
var
>
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/>
per
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>.o.n</
var
>
. </
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>
<
s
xml:id
="
echoid-s253
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xml:space
="
preserve
">Hęc ſingulatim conſideremus (
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norm
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nam
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type
="
context
">nã</
reg
>
ſi in partibus ſimplicibus quod dicimus ac
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ciderit, id ipſum in compoſitis conſequenter eueniet) quamobrem ex definitione di
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uiſionis dabitur eadem proportio
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var
>.o.e.</
var
>
ad
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var
>.o.t.</
var
>
quæ eft
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var
>.o.n.</
var
>
ad vnitatem, quæ ſit
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var
>.o.
<
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x</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">Nunc cogitemus
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ſuperficiem
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type
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reg
>
<
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norm
="
rectangulam
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type
="
context
">rectangulã</
reg
>
<
var
>.o.c.</
var
>
<
reg
norm
="
æqualem
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type
="
context
">æqualẽ</
reg
>
quadrato
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var
>.o.a</
var
>
. </
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>
<
s
xml:id
="
echoid-s255
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xml:space
="
preserve
">tunc numerus
<
var
>.
<
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c.t.</
var
>
proueniens erit, ut patet, ex diuiſione numeri quadrati
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var
>.o.a.</
var
>
per
<
reg
norm
="
numerum
"
type
="
context
">numerũ</
reg
>
<
var
>.o.t.</
var
>
<
reg
norm
="
eritque
"
type
="
simple
">eritq́</
reg
>
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lb
/>
<
reg
norm
="
eadem
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type
="
context
">eadẽ</
reg
>
proportio
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var
>.c.t.</
var
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ad
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quæ eſt
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>.o.e.</
var
>
ad
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>.o.t.</
var
>
ex ſecunda parte quintæ decimæ ſexti,
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aut .20. ſeptimi. </
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<
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Iam
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type
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">Iã</
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>
<
reg
norm
="
autem
"
type
="
context
">autẽ</
reg
>
dictum eſt
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var
>.o.e.</
var
>
ad
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>.o.t.</
var
>
ſic ſe habere ſicut
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>.o.n.</
var
>
ad
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var
>.o.x</
var
>
. </
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>
<
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xml:id
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<
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norm
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Itaque
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type
="
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">Itaq;</
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>
ex .
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11. quinti ſic ſe habebit
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>.c.t.</
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>
ad
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>.o.e.</
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>
ſicut
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>.o.n.</
var
>
ad
<
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>.o.x</
var
>
. </
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>
<
s
xml:id
="
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xml:space
="
preserve
">Sed ex prima ſexti, aut .18. vel .
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19. ſeptimi, ſic ſe habet
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productum
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type
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>
<
var
>.n.e.</
var
>
ad
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>.e.x.</
var
>
ſicut
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var
>.o.n.</
var
>
ad
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var
>.o.x</
var
>
. </
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>
<
s
xml:id
="
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xml:space
="
preserve
">quare denuo ſic ſe ha-
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bebit numerus
<
var
>.c.t.</
var
>
ad numerum
<
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>.o.e.</
var
>
ſicut nume-
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/>
rus
<
var
>.n.e.</
var
>
ad numerum
<
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>.x.e</
var
>
. </
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>
<
s
xml:id
="
echoid-s260
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xml:space
="
preserve
">Sed numerus
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>.o.e.</
var
>
cum
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/>
<
figure
xlink:label
="
fig-0029-01
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xlink:href
="
fig-0029-01a
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number
="
38
">
<
image
file
="
0029-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0029-01
"/>
</
figure
>
numero
<
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>.x.e.</
var
>
ſpecie idem eſt, igitur ex .9. quinti nu
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merus
<
var
>.c.t.</
var
>
numero
<
var
>.n.e.</
var
>
æqualis erit.</
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>
</
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>
<
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>
<
s
xml:id
="
echoid-s261
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xml:space
="
preserve
">Id ipſum de quadrato ipſius
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>.o.n.</
var
>
videlicet
<
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>.p.o.</
var
>
<
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/>
dico. </
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>
<
s
xml:id
="
echoid-s262
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xml:space
="
preserve
">Nam ſi proueniens
<
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>.o.n.</
var
>
diuiſo per
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>.o.e.</
var
>
ideſt
<
var
>.
<
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/>
o.i.</
var
>
proportionale reſpondens ad
<
var
>.o.t.</
var
>
cum
<
var
>.o.t.</
var
>
<
lb
/>
<
reg
norm
="
coniunctum
"
type
="
context
">coniunctũ</
reg
>
fuerit, et per
<
reg
norm
="
hanc
"
type
="
context
">hãc</
reg
>
ſummam diuiſa ſumma
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/>
quadratorum
<
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>.o.a.</
var
>
et
<
var
>.o.p.</
var
>
patet per ſe proueniens
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lb
/>
futurum eiuſdem numeri
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var
>.c.t.</
var
>
<
reg
norm
="
ipſumque
"
type
="
simple
">ipſumq́</
reg
>
<
var
>.c.t.</
var
>
proue-
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niens ſemper ſuturum.</
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>
</
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<
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<
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xml:space
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">Quo autem lucidius res hæc innoteſcat. </
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>
<
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xml:space
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">Cogi
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temus proueniens quadrati
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>.o.p.</
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>
diuiſi ab
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>.o.i.</
var
>
re-
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<
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="
ſpondentisque
"
type
="
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">ſpondentisq;</
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>
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>.o.t.</
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>
eſſe
<
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>.i.u.</
var
>
quod via prædicta inue-
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nitur æqualis eſſe numero
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>.n.e.</
var
>
ex quo conſe-
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/>
quenter æquale
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>.c.t</
var
>
: cogitato deinde rectangu-
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lo
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>.o.u.</
var
>
æquali
<
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>.o.p.</
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>
coniuncto
<
var
>.o.c</
var
>
:totum
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var
>.t.u.</
var
>
æqua-
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/>
le erit compoſito duorum quadratorum
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var
>.o.a.</
var
>
et
<
var
>.o.
<
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/>
p.</
var
>
cum in nullo numerus
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>.c.t.</
var
>
mutetur, tam ex com-
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poſito
<
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>.t.u.</
var
>
<
reg
norm
="
quam
"
type
="
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">quã</
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>
ex ſimplici
<
var
>.o.c.</
var
>
ex quo propoſiti ſe
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/>
ſe ueritas profert.</
s
>
</
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</
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<
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xml:space
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">THEOREMA
<
num
value
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">XXVII</
num
>
.</
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>
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<
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">PRoposvervnt</
emph
>
veteres nobile quidem problema, ſed quod tamen citra al-
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gebraticam effectionem, aut neſcierunt, aut noluerunt diſſoluere, quod nihi-
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lominus facillimum eſt.</
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>
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