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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THEOR. ARITH.
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0101
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xlink:href
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<
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xml:space
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<
num
value
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132
">CXXXII</
num
>
.</
head
>
<
p
>
<
s
xml:id
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xml:space
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preserve
">SED quia aliquis poſſet in dubium reuocare, an poſſibile ſit inuenire tertium
<
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terminum rationalem, ſeu communicantem duobus datis terminis inter ſe com
<
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municantibus in tali proportionalitate, hoc eſt harmonica. </
s
>
<
s
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echoid-s1157
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xml:space
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s
>
</
p
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<
p
>
<
s
xml:id
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echoid-s1158
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xml:space
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preserve
">Sint duo termini dati
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>.a.o.</
var
>
et
<
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>.a.e.</
var
>
inter ſe communicantes, tertius verò inuentus
<
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/>
ſit
<
var
>.a.c.</
var
>
qui maximus, primò, ſit in ea proportionalitate, quem dico communicantem
<
lb
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eſſe cum primis datis.</
s
>
</
p
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<
p
>
<
s
xml:id
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echoid-s1159
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xml:space
="
preserve
">Nam ex conditionibus huiuſmodi proportionalitatis, habebimus primum ean-
<
lb
/>
dem proportionem eſſe
<
var
>.a.c.</
var
>
ad
<
var
>.a.o.</
var
>
quæ eſt
<
var
>.e.c.</
var
>
ad
<
var
>.e.o.</
var
>
vnde permutando ita erit
<
var
>.a.
<
lb
/>
c.</
var
>
ad
<
var
>.e.c.</
var
>
vt
<
var
>.a.o.</
var
>
ad
<
var
>.o.e.</
var
>
& quia ex .9. decimi Euclid
<
var
>.a.o.</
var
>
communicat cum
<
var
>.o.e.</
var
>
</
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>
<
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xml:id
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xml:space
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">quare
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/>
ex .10. eiuſdem
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var
>.a.c.</
var
>
communicabit cum
<
var
>.e.c.</
var
>
& per .9. cum
<
var
>.a.e.</
var
>
et per .8. cum
<
var
>.a.o.</
var
>
<
lb
/>
quod
<
unsure
/>
eſt propoſitum.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s1161
"
xml:space
="
preserve
">Sed ſi datus fuerit maximus
<
var
>.a.c.</
var
>
cum medio
<
var
>.a.e.</
var
>
interſe communicantes mini-
<
lb
/>
mum verò
<
var
>.a.o.</
var
>
probabo
<
reg
norm
="
communicantem
"
type
="
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">cõmunicantem</
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>
cum illis eſſe. </
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<
s
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xml:space
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">Cogitemus ergo
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var
>.c.f.</
var
>
æqua-
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lb
/>
jem eſſe differentiæ
<
var
>.c.e.</
var
>
cognitæ, vnde habebimus proportionem,
<
var
>a.c.</
var
>
ad
<
var
>.c.f.</
var
>
vt
<
var
>.a.o.</
var
>
<
lb
/>
ad
<
var
>.o.e.</
var
>
& componendo
<
var
>.a.f.</
var
>
ad
<
var
>.f.c.</
var
>
vt
<
var
>.a.e.</
var
>
ad
<
var
>.e.o.</
var
>
& quia (ex ſuppoſito).
<
var
>a.c.</
var
>
commu-
<
lb
/>
nicat cum
<
var
>.e.c.</
var
>
hoc eſt cum
<
var
>.c.f.</
var
>
</
s
>
<
s
xml:id
="
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xml:space
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preserve
">quare
<
lb
/>
ex eadem .9. dicti decimi
<
var
>.a.f.</
var
>
et
<
var
>.f.c.</
var
>
<
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type
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<
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<
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xlink:label
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fig-0101-01
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xlink:href
="
fig-0101-01a
"
number
="
138
">
<
image
file
="
0101-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-01
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>
inter ſe communicantes, & per .10.
<
var
>a.e.</
var
>
<
lb
/>
communicabit cum
<
var
>.o.e.</
var
>
& per .9.
<
var
>a.e.</
var
>
cò
<
lb
/>
municabit cum
<
var
>.a.o.</
var
>
vnde per .8.
<
var
>a.o.</
var
>
communicabit cum
<
var
>.a.c.</
var
>
ſimiliter.</
s
>
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type
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<
head
xml:id
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xml:space
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">THEOREMA
<
num
value
="
133
">CXXXIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
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xml:space
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preserve
">SED ſi nobis duo extremi termini propoſiti fuerint, & medium inuenire deſide
<
lb
/>
remus in dicta proportionalitate, ita faciendum erit.</
s
>
</
p
>
<
p
>
<
s
xml:id
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xml:space
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preserve
">Sint, exempli gratia, duo termini dati
<
var
>.q.b.</
var
>
et
<
var
>.b.r.</
var
>
minor
<
var
>.b.r.</
var
>
ex maiori
<
var
>.b.q.</
var
>
de-
<
lb
/>
trahatur, reſiduum verò
<
var
>.q.x.</
var
>
multiplicetur per
<
var
>.b.r.</
var
>
productum poſteà diuidatur per
<
lb
/>
<
var
>q.r.</
var
>
vnde proueniet nobis
<
var
>.x.l.</
var
>
pro differentia minori, quæ addita cum
<
var
>.b.x.</
var
>
minimo
<
lb
/>
termino, dabit nobis
<
var
>.b.l.</
var
>
mcdium terminum harmonicum.</
s
>
</
p
>
<
p
>
<
s
xml:id
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xml:space
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preserve
">Pro cuius ratione cogitemus dictum medium terminum
<
var
>.b.l.</
var
>
iam inuentum eſſe,
<
lb
/>
vnde ita erit proportio
<
var
>.q.l.</
var
>
ad
<
var
>.l.x.</
var
>
vt
<
var
>.q.b.</
var
>
ad
<
var
>.b.r.</
var
>
ex forma huius proportionalitatis,
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s1167
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xml:space
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preserve
">quare coniunctim ita erit
<
var
>.q.r.</
var
>
ad
<
var
>.r.b.</
var
>
vt
<
lb
/>
<
var
>q.x.</
var
>
ad
<
var
>.x.l.</
var
>
& proptereà ex .20. ſeptimi
<
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<
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xlink:label
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fig-0101-02
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fig-0101-02a
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number
="
139
">
<
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="
0101-02
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xlink:href
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productum, quod fit ex
<
var
>.q.r.</
var
>
in
<
var
>.x.l.</
var
>
æqua-
<
lb
/>
le erit producto
<
var
>.q.x.</
var
>
in
<
var
>.b.r</
var
>
. </
s
>
<
s
xml:id
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xml:space
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">Rectè igitur
<
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fit cum diuiditur hoc productum per
<
var
>.q.r.</
var
>
vt proueniat nobis
<
var
>.x.l.</
var
>
differentia minor.</
s
>
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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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<
num
value
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134
">CXXXIIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
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xml:space
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">POſſumus etiam harmonicè diuidere vnam datam proportionem abſque aliqua
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diuiſione productorum, ne nobis fractiones proueniant, hoc modo videlicet.
<
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</
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>
<
s
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xml:space
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">Nobis propoſitum ſit diuidere harmonicè ſeſquialteram
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>
inuenian-
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tur primo minimi termini huius proportionis ut putà .3. et .2. quarum ſumma, hoc
<
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/>
eſt quinque, multiplicetur per minorem ideſt .2. vnde proueniet nobis .10. qui qui-
<
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dem erit minor terminus trium quæſitorum, quorum maximus erit productum ſum </
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