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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67
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rhead
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THEOREM. ARIT.
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79
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file
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0079
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0079
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r.f.</
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hoc eſt
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ad
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ex .11. quinti. </
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<
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xml:space
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bit
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: cum
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æqualis ſit
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. </
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æqualis ſit
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<
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ad
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vt
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permutando
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type
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ſic
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ſe habebit
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vt
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& compon endo ita
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vt
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& permutando ſic
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vt. de
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nempe vt
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& permutan
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do ita
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vt
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& componendo ita
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ad
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vt
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et
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& permutando ſic
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et
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nempe ad
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: vt
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n.b.</
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hoc eſt. ut
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ad
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quod erat propoſitum.</
s
>
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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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<
head
xml:id
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xml:space
="
preserve
">THEOREMA
<
num
value
="
105
">CV</
num
>
.</
head
>
<
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<
s
xml:id
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xml:space
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preserve
">CVR deſideranti ſummam quorumcunque terminorum progreſſionis conti-
<
lb
/>
nuæ geometricæ cognoſcere. </
s
>
<
s
xml:id
="
echoid-s907
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xml:space
="
preserve
">Rectè minimus terminus ex maximo detrahen
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/>
dus eſt,
<
reg
norm
="
reſiduumque
"
type
="
simple
">reſiduumq́;</
reg
>
per denominantem progreſſionis dempta vnitate diuidendum,
<
lb
/>
<
reg
norm
="
prouenientique
"
type
="
simple
">prouenientiq́;</
reg
>
maximum terminum addendum, ex quo oritur ſumma quæſita.</
s
>
</
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<
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<
s
xml:id
="
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xml:space
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preserve
">Exempli gratia, ſi darentur quatuor termini continui proportionales .8. 12. 18.
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27. primum hoc eſt minimum .8. ex vltimo .27. detraheremus: </
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<
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type
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.19. qui
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per denominantem progreſſionis, dempta vnitate, diuideretur. </
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<
s
xml:id
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xml:space
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preserve
">Quo loco animad
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uertendum eſt, quamlibet
<
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norm
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denominationem
"
type
="
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">denominationẽ</
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>
cuiuſcunque proportionis numerorum
<
lb
/>
ſupra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eſt, &
<
lb
/>
idipſum de ſuperparticularibus, & ſuperpartientibus eſt intelligendum, vt in præ-
<
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ſenti proportio ſeſquialtera inter duos terminos cogitanda eſt, nempe inter vnum
<
lb
/>
& dimidium, atque vnum. </
s
>
<
s
xml:id
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xml:space
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preserve
">Seſquitertia autem inter vnum & tertiam partem,
<
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& vnum. </
s
>
<
s
xml:id
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xml:space
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">Seſquiquinta inter vnum cum quinta parte, & vnum. </
s
>
<
s
xml:id
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xml:space
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preserve
">De ſuperpartien
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/>
tibus idem aſſero quod de proportione
<
reg
norm
="
ſuperbipartiente
"
type
="
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">ſuperbipartiẽte</
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>
tertias appellata, vt .5.
<
lb
/>
ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, ſuperbipartiens quar-
<
lb
/>
tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans ſcilicet, ſem
<
lb
/>
per ſit vnitas, alter verò denominans. </
s
>
<
s
xml:id
="
echoid-s914
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xml:space
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">Idem de cæteris. </
s
>
<
s
xml:id
="
echoid-s915
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xml:space
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preserve
">Quare in præſenti exem
<
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/>
plo, detracta vnitate ex denominante progreſſionis, ſupererit tantummodo dimi-
<
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/>
dium, quo diuiſo .19. proueniet .38. qui numerus æqualis erit ſummæ
<
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norm
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reliquorum
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type
="
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">reliquorũ</
reg
>
<
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omnium terminorum, cui coniuncto vltimo termino .27. dabitur ſumma quæſita .65.</
s
>
</
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<
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<
s
xml:id
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xml:space
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">Pro cuius ſpeculatione, quatuor termini ſignificentur, quatuor lineis
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:
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:
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b.i.</
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>
primus autem terminus
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ex vltimo
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>.b.i.</
var
>
detrahatur,
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type
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ſit
<
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>.n.i.</
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>
& ex
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ſecundo
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>.f.r.</
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>
cuius reſiduum ſit
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var
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proportio verò progreſſionis ea ſit, quæ
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ad
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y.</
var
>
quo vnitas repræſentatur (ex quo ſic ſe habebit
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>
ad
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>
vt
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ad
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) qua
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var
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var
>
de
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tracta ex
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var
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ſuperſit
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var
>
. </
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>
<
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xml:id
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xml:space
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">Tum erecta
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cogitetur linea
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var
>
indefinita per
<
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<
figure
xlink:label
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fig-0079-01
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xlink:href
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fig-0079-01a
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number
="
107
">
<
image
file
="
0079-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0079-01
"/>
</
figure
>
pendicularis
<
var
>.b.i.</
var
>
à puncto
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var
>.n.</
var
>
quę diui
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datur in puncto
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>
ita vt
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var
>
æqualis
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ſit vnitati
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var
>
& in puncto
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var
>
ita. vt
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u.</
var
>
æqualis ſit
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var
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>
ex quo eadem erit
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proportio
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>
ad
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>
vt
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>
ad
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<
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nem- pe
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type
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pe</
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>
<
var
>.o.r.</
var
>
ad
<
var
>.m.s</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
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preserve
">Nam cú ſic ſe habeat
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var
>.
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f.r.</
var
>
ad
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var
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>
hoc eſt ad
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var
>.f.o.</
var
>
vt
<
var
>.g.h.</
var
>
ad
<
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>
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hoc eſt ad
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var
>.g.</
var
>
permutando
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norm
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quoque
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type
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>
ſic
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ſe habebit
<
var
>.f.r.</
var
>
ad
<
var
>.g.h.</
var
>
vt
<
var
>.f.o.</
var
>
ad
<
var
>.g</
var
>
. </
s
>
<
s
xml:id
="
echoid-s919
"
xml:space
="
preserve
">Ita
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que ex .19. quinti
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var
>
ad
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var
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var
>
vt
<
var
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var
>
ad
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var
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var
>
ex quo ex .11. eiuſdem
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var
>
ad
<
var
>.h.</
var
>
vt
<
var
>.f.o.</
var
>
ad </
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>
</
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