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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92
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61
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rhead
="
THEOR. ARITH.
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n
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73
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file
="
0073
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0073
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n.k.</
var
>
ipſius quadratum numerorum integrorum cognoſcetur, cui addito gnomone
<
var
type
="
gnomon
">.
<
lb
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n.o.K.</
var
>
cognoſcemus numerum
<
var
>.u.i.</
var
>
quæſitum.</
s
>
</
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<
p
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<
s
xml:id
="
echoid-s809
"
xml:space
="
preserve
">Sed cum nobis hæc via, tenenda propoſitum non fuit, hoc eſt primo loco inue
<
lb
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niendi quadrati minoris
<
var
>.n.K.</
var
>
ideo ſupereſt probandum gnomonem
<
var
>.t.o.c.</
var
>
vnitati
<
reg
norm
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ae- qualem
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type
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simple
">ę-
<
lb
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qualem</
reg
>
eſſe, nempe quadratulo
<
var
>.m.a.</
var
>
quod patebit, ſi conſideremus nos ſumpſiſſe
<
lb
/>
rectangulum
<
var
>.r.c.</
var
>
pro dimidio gnomonis
<
var
>.n.o.K</
var
>
. </
s
>
<
s
xml:id
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echoid-s810
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xml:space
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preserve
">etenim ſi ſupplemento etiam
<
var
>.n.r.</
var
>
qua
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/>
dratulum æquale
<
var
>.m.a.</
var
>
adderetur, pateret gnomonem
<
var
>.n.a.K.</
var
>
cum dicto quadratulo
<
lb
/>
collectum, æqualem eſſe gnomoni
<
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>.n.o.K</
var
>
: cum duo ſupplementa
<
var
>.m.t.</
var
>
et
<
var
>.m.c.</
var
>
inter ſe
<
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fint æqualia. </
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>
<
s
xml:id
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echoid-s811
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xml:space
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">Quamobrem inuento quadrato
<
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>.t.c.</
var
>
ex dimidio gnomonis cognito,
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lb
/>
additur vnitas, gnomon ſcilicet
<
var
>.t.o.c.</
var
>
ex quo cognoſcitur numerus
<
var
>.u.i.</
var
>
quæſitus.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s812
"
xml:space
="
preserve
">Quod autem quadratum
<
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>.g.p.</
var
>
numeris integris conſtet, hac ratione probatur viſum
<
lb
/>
enim fuit ſupra quadratum
<
var
>.n.K.</
var
>
verè quadratum eſſe, & numeris integris conſtare,
<
lb
/>
pariter etiam
<
var
>.t.c.</
var
>
<
reg
norm
="
ſeque
"
type
="
simple
">ſeq́;</
reg
>
mutuo conſequi (nam
<
var
>.K.c.</
var
>
eſt vnitas linearis) ex quo gnomon
<
lb
/>
<
var
>n.a.K.</
var
>
numero diſpari conſtabit, ex ijs quæ .90. theoremate probata fuerunt. </
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>
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<
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norm
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Itaque
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type
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">Itaq;</
reg
>
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/>
ex eodem theoremate neceſſe eſt gnomonem
<
var
>.t.d.c.</
var
>
etiam numero diſpari conſtare,
<
lb
/>
ita vt à numero
<
var
>.n.a.K.</
var
>
non niſi duabus vnitatibus differat, nempe vt
<
var
>.c.p.</
var
>
ſit vnitas li-
<
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/>
nearis, ſed ita reuera eſt, numerus enim
<
var
>.u.d.i.</
var
>
ex præſuppoſito par eſt, </
s
>
<
s
xml:id
="
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xml:space
="
preserve
">quare nume
<
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/>
rus
<
var
>.t.d.c.</
var
>
diſpar erit, cum alterum vnitate ſuperet, videlicet gnomone
<
var
>.t.o.c.</
var
>
vnita
<
lb
/>
ri æquali, tum
<
var
>.n.a.K.</
var
>
minor eſt
<
var
>.n.o.K.</
var
>
ex eodem gnomone
<
var
>.t.o.c.</
var
>
unitati æquali. </
s
>
<
s
xml:id
="
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xml:space
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preserve
">Ita
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/>
que
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>.n.a.K.</
var
>
minor erit
<
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>.u.d.i.</
var
>
per vnitatem, & minor
<
var
>.t.d.c.</
var
>
per duas unitates, ex quo ſe-
<
lb
/>
quitur
<
var
>.g.p.</
var
>
eſſe quadratum
<
reg
norm
="
integrorum
"
type
="
context
">integrorũ</
reg
>
ex dicto theoremate ac con ſequens quadrato
<
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/>
<
var
>t.c</
var
>
. </
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>
<
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xml:space
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">quare
<
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>.c.p.</
var
>
vnitas erit, & radices
<
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>.q.K.</
var
>
et
<
var
>.q.p.</
var
>
horum quadratorum numero bina-
<
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rio inter ſe different. </
s
>
<
s
xml:id
="
echoid-s817
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xml:space
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preserve
">Vnà etiam ſcienda eſt cauſa, cur numerus propoſitus neceſſa
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figure
xlink:label
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fig-0073-01
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xlink:href
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fig-0073-01a
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number
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102
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<
image
file
="
0073-01
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0073-01
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</
figure
>
riò binario maior eſſe debeat. </
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>
<
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xml:space
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">Etenim
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ipſe
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ſit futurus gnomon
<
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var
>
nec poſſit minor eſſe
<
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numero ternario, vt patet ex .90. theoremate,
<
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idcirco ſequitur neceſſariò maiorem eſſe bina-
<
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rio debere. </
s
>
<
s
xml:id
="
echoid-s819
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xml:space
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preserve
">Quòd ſi diſpar numerus propone-
<
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/>
retur, nec forma operis nec ſpeculationis
<
reg
norm
="
mutan- da
"
type
="
context
">mutã-
<
lb
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da</
reg
>
eſſet. </
s
>
<
s
xml:id
="
echoid-s820
"
xml:space
="
preserve
">Non erit tamen neceſſarium vt ipſa
<
lb
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quadrata
<
var
>.n.K.</
var
>
et
<
var
>.g.p.</
var
>
numeris integris conſta-
<
lb
/>
rent. </
s
>
<
s
xml:id
="
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"
xml:space
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preserve
">Sæpius enim fractis
<
reg
norm
="
componerentur
"
type
="
context
">cõponerentur</
reg
>
, quod
<
lb
/>
ex .90. theoremate facile erit ſpeculari nihilo-
<
lb
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minus fractis integris,
<
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="
ipſisque
"
type
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">ipſisq́;</
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>
collectis cum ſuis
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fractis ſummæ eſſent quadratæ.</
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<
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xml:space
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">THEOREMA
<
num
value
="
93
">XCIII</
num
>
.</
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>
<
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<
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xml:space
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">CVR propoſitis duobus numeris altero pari, altero verò diſpari, duplo primi
<
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minore per vnitatem, ſi alium inuenire numerum voluerimus, cui alterum iſto
<
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rum coniunctum proferat quadratum, & altero detracto, quadratum ſuperſit. </
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>
<
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"
xml:space
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">Re-
<
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ctè datos numeros in ſummam colligemus, quam ſummam in duas quam maximas
<
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/>
poterimus partes diuidemus, quarum vna pari, altera diſpari conſtet, tum vtran-
<
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/>
que in ſeipſam multiplicabimus, & quadrato minori, duorum numerorum propo-
<
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ſitorum quemuis ademus, ex quo cupimus nobis quadratum minus ſupereſſe, & pro
<
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ueniet nobis numerum quæſitum.</
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<
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<
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xml:space
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">Exempli gtatia, proponuntur numeri .11. et .6. quorum alter alicui numero ad- </
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