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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147
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99
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THEOREM. ARIT.
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111
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file
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0111
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0111
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ni
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>.o.</
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ad
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>.c</
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. </
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<
s
xml:id
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echoid-s1292
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xml:space
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preserve
">Idem dico de reliquis proportionibus ſuperparticularibus.</
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<
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<
s
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xml:space
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">Sed ſi data proportio numerorum fuerit ex ſuper partientibus, vt exempli gra-
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tia de quinque ad tria, efficiemus, vt
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et
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>.e.</
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ſint prima relata ipſius
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>.o.</
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et
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>.c.</
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vnde
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proportio
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>.a.</
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ad
<
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>.e.</
var
>
ita ſe habe-
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bit ad proportionem
<
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>.o.</
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ad
<
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>.c.</
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<
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<
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xlink:label
="
fig-0111-01
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xlink:href
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number
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153
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<
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file
="
0111-01
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xlink:href
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vt quinque ad
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norm
="
vnum
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type
="
context
">vnũ</
reg
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& propor-
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tio
<
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>.i.</
var
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ad
<
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>.c.</
var
>
ut tria ad
<
reg
norm
="
vnum
"
type
="
context
">vnũ</
reg
>
. </
s
>
<
s
xml:id
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xml:space
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">Qua-
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re proportio
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>.a.</
var
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ad
<
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>.e.</
var
>
ad pro-
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portionem
<
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>.i.</
var
>
ad
<
var
>.c.</
var
>
ſe habebit,
<
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/>
vt quinque ad tria, & ſic de reliquis.</
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>
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<
p
>
<
s
xml:id
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xml:space
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">Pro alijs, eundem ordinem ſeruando, obtinebimus quod volumus.</
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</
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</
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>
<
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xml:id
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type
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level
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n
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148
">
<
head
xml:id
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xml:space
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">THEOREMA
<
num
value
="
148
">CXLVIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s1296
"
xml:space
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preserve
">QVamuis in .16. ſexti et .20. ſeptimi manifeſtè pateat ratio, quare rectè fiatac
<
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cipiendam radicem quadratam illius producti, quod fit ex duobus datis
<
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terminis, vt medium proportionale geometricè inter ipſos habeamus: </
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>
<
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xml:id
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xml:space
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preserve
">nihilomi-
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nus, quia per aliam methodum hoc idem ſcire poſſumus, inconueniens non erit a-
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liquid circa hoc dicere.</
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>
</
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>
<
p
>
<
s
xml:id
="
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xml:space
="
preserve
">Cogitemus igitur exempli gratia, tres numeros continuè proportionales geo-
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metricè
<
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>.a.b</
var
>
:
<
var
>c.d.</
var
>
et
<
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>.e.f.</
var
>
quorum
<
var
>.a.b.</
var
>
et
<
var
>.e.f.</
var
>
tantummodo nobis cogniti ſint, imagine-
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/>
mur etiam
<
var
>.g.a.</
var
>
eſſe productum quod fit ex
<
var
>.a.b.</
var
>
in
<
var
>.e.f.</
var
>
et
<
var
>.d.k.</
var
>
quadratum
<
var
>.c.d.</
var
>
et
<
var
>.a.h.</
var
>
<
lb
/>
id quod fit ex
<
var
>.a.b.</
var
>
vnde eandem proportionem habebimus
<
var
>.a.h.</
var
>
ad
<
var
>.a.g.</
var
>
quæ eſt
<
var
>.h.b.</
var
>
<
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/>
ad
<
var
>.b.g.</
var
>
ex prima .6. aut .18. vel .19. ſepti-
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/>
mi, ſed per .11. octaui ita eſt quadrati
<
var
>.a.</
var
>
<
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/>
<
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xlink:label
="
fig-0111-02
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xlink:href
="
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"
number
="
154
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<
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file
="
0111-02
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xlink:href
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</
figure
>
h. ad quadratum
<
var
>.k.d.</
var
>
vt
<
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>.a.b.</
var
>
ad
<
var
>.e.f.</
var
>
hoc
<
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eſt vt
<
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>.h.b.</
var
>
ad
<
var
>.b.g.</
var
>
ergo per .11. quinti ita
<
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erit
<
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>.a.h.</
var
>
ad
<
var
>.a.g.</
var
>
vt ad
<
var
>.k.d.</
var
>
vnde
<
var
>.a.g.</
var
>
æqua
<
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/>
le erit
<
var
>.k.d.</
var
>
per .9. quinti. </
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>
<
s
xml:id
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xml:space
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preserve
">Rectè ergo erit
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accipere radicem quadratam
<
var
>.a.g.</
var
>
pro
<
var
>.c.
<
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d.</
var
>
quod etiam eſt diuidere vnam datam
<
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<
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norm
="
proportionem
"
type
="
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">proportionẽ</
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>
per æqualia, hoc eſt in duas
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æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poſteriori-
<
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bus Theorematibus ad demonſtrandum priora illis, ſed hoc .148. dictum ſit luden
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di loco.</
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</
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</
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<
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n
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<
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xml:id
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xml:space
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">THEOREMA
<
num
value
="
149
">CXLIX</
num
>
.</
head
>
<
p
>
<
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xml:id
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xml:space
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<
emph
style
="
sc
">Vnde</
emph
>
fiat
<
reg
norm
="
quod
"
type
="
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">ꝙ</
reg
>
ſi quis inuenire voluerit ſecundum terminum ex quatuor nume
<
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ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-
<
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/>
modo nobis cogniti ſint, rectè factum ſit quadrare primum eorum, & hoc quadra-
<
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tum poſteà per alium terminum cognitum multiplicare, cuius producti demum ac-
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cipere radicem cubam pro ſecundo termino quæſito, hocloco videbimus.</
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<
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<
s
xml:id
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xml:space
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">Imaginemur quatuor terminos continuè proportionales, vt dictum eſt, eſſe.</
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