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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IO. BAPT. BENED.
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110
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file
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0110
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0110
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ſequenti</
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reſiduę proportionis; </
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<
s
xml:id
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xml:space
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">quæ quidem reſidua proportio eſſet vt .4. ad .3. hoc
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eſt ſeſquitertia, & ſic de cæteris.</
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<
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<
s
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xml:space
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">Pro cuius ratione, ſit proportio
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>
ad
<
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>.n.</
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ea quæ (exempli gratia) maior ſit, à
<
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qua volumus demere proportionem
<
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>.t.</
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>
ad
<
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>.u.</
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>
minorem ſcilicet. </
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>
<
s
xml:id
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echoid-s1278
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xml:space
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preserve
">Nunc autem
<
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productum
<
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>.x.</
var
>
in
<
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>.u.</
var
>
ſit
<
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>.a.g.</
var
>
illud verò
<
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>.t.</
var
>
in
<
var
>.
<
lb
/>
n.</
var
>
ſit
<
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>.a.d</
var
>
. </
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>
<
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xml:space
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">Tunc dico proportionem
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ad
<
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>.a.</
var
>
<
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<
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xlink:label
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0110-01
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xlink:href
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d. eſſe reſiduam quæſitam. </
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xml:space
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<
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>
productum
<
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u. in
<
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>.n.</
var
>
vnde eadem proportio erit producti
<
var
>.a.
<
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/>
g.</
var
>
ad productum
<
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>.a.b.</
var
>
quę
<
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>.x.</
var
>
ad
<
var
>.n.</
var
>
et
<
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>.a.d.</
var
>
ad
<
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>a.b.</
var
>
<
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/>
quæ
<
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>.t.</
var
>
ad
<
var
>.u.</
var
>
ex prima ſexti, ſeu .18. vel .19. ſe-
<
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/>
ptimi, ſed proportio
<
var
>.a.g.</
var
>
ad
<
var
>.a.b.</
var
>
hoc eſt
<
var
>.x.</
var
>
ad
<
var
>.
<
lb
/>
n.</
var
>
componitur ex ea, quæ eſt
<
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>.a.g.</
var
>
ad
<
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>.a.d.</
var
>
& ea,
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/>
quæ eſt
<
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>.a.d.</
var
>
ad
<
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>.a.b.</
var
>
hoc eſt
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>.t.</
var
>
ad
<
var
>.u.</
var
>
ergò ea, quę
<
lb
/>
eſt
<
var
>.a.g.</
var
>
ad
<
var
>.a.d.</
var
>
erit quàm quærebamus.</
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>
</
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</
div
>
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xml:id
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type
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<
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xml:id
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xml:space
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">THEOREMA
<
num
value
="
146
">CXLVI</
num
>
.</
head
>
<
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>
<
s
xml:id
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xml:space
="
preserve
">RATIO verò, quòd rectè fiat, quotieſcunque aliquam proportionem dupli-
<
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/>
care volentes, quadramus terminos ipſius proportionis, vel ſi eam triplicare
<
lb
/>
voluerimus, cubamus ipſos terminos, vel ſi eam quadruplicare voluerimus
<
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/>
inuenimus cenſicos cenſicos terminorum ipſius proportionis, & ſic de ſingulis, in
<
ref
id
="
ref-0016
">.17
<
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/>
Theo. huiuſmodi tractatus</
ref
>
manifeſta eſt.</
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>
</
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>
</
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>
<
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type
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n
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">
<
head
xml:id
="
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"
xml:space
="
preserve
">THEOREMA
<
num
value
="
147
">CXLVII</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">QVotieſcunque nobis propoſiti fuerint duo numeri ad libitum, deſideraremus
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/>
q́ue duas proportiones tali relatione inuicem refertas, quali ſunt hi duo pro
<
lb
/>
poſiti numeri inter ſe, ita faciendum erit.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s1283
"
xml:space
="
preserve
">Sciendum primo eſt proportionem maioris numeri propoſiti ad minorem ſem-
<
lb
/>
per eſſe alicuius ex quinque generum, hoc eſt aut erit generis multiplicis, aut ſu-
<
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/>
perparticularis, aut multiplicis ſuperparticularis, aut ſuper partientis, aut multi-
<
lb
/>
plicis ſuperpartientis.</
s
>
</
p
>
<
p
>
<
s
xml:id
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"
xml:space
="
preserve
">Nunc autem ſi erit ex genere multiplici, iam ab antiquis traditus eſt modus,
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>
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ſequi debemus. </
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>
<
s
xml:id
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xml:space
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">Cuius ſpeculatio à me inuenta patet .in .17. Theo. huius libri, vt
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in præcedenti dixi.</
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>
</
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<
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>
<
s
xml:id
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xml:space
="
preserve
">Sed ſi talis proportio datorum numerorum erit alicuius aliorum generum, ita
<
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/>
agemus, ſi fuerit ſuperparticularis.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Sit exempli gratia, ſeſquialtera, tunc ſumantur duo numeri inuicem inæquales,
<
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/>
quos à caſu volueris
<
var
>.o.</
var
>
et
<
var
>.c.</
var
>
qui quidem cubentur, & eorum cubi ſint
<
var
>.a.</
var
>
et
<
var
>.e</
var
>
. </
s
>
<
s
xml:id
="
echoid-s1288
"
xml:space
="
preserve
">Inuenia
<
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/>
tur poſteà. u. ita proportionatus ad
<
var
>.o.</
var
>
vt
<
var
>.o.</
var
>
eſt ad
<
var
>.c.</
var
>
ex regula de tribus, hoc eſt diui-
<
lb
/>
dendo quadratum ipſius
<
var
>.o.</
var
>
per
<
var
>.c.</
var
>
vnde nobis proueniat
<
var
>.u.</
var
>
& quia proportio
<
var
>.a.</
var
>
ad
<
var
>.e.</
var
>
<
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/>
tripla eſt proportioni
<
var
>.o.</
var
>
ad
<
var
>.c.</
var
>
& proportio
<
var
>.u.</
var
>
ad
<
var
>.c.</
var
>
dupla eſt
<
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norm
="
eidem
"
type
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>
, quæ
<
var
>.o.</
var
>
ad
<
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>.c.</
var
>
ideo
<
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/>
proportio
<
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>.a.</
var
>
ad
<
var
>.e.</
var
>
ſeſquialtera erit proportioni
<
var
>.u.</
var
>
ad
<
var
>.c</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Sed ſi proportio numerorum propoſitorum fuerit ſeſquitertia, faciemus
<
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>.a.</
var
>
et
<
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>.e.</
var
>
<
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/>
eſſe cenſica cenſica ipſius
<
var
>.o.</
var
>
et
<
var
>.c</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">tunc ſumemus
<
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>.u.</
var
>
conſequentem ad
<
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>.o.</
var
>
vt dictum eſt,
<
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/>
deinde inueniremus
<
var
>.i.</
var
>
conſequens ad
<
var
>.u.</
var
>
ita ut
<
var
>.u.</
var
>
conſequens ipſius
<
var
>.o</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">tunc habebi-
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mus proportionem
<
var
>.i.</
var
>
ad
<
var
>.c.</
var
>
triplam, & eam quæ eſt
<
var
>.a.</
var
>
ad
<
var
>.e.</
var
>
quadruplam proportio- </
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>
</
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