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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90
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IO. BAPT. BENED.
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102
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file
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0102
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0102
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mæ iam dictæ in maiorem eorum, hoc eſt quod fit ex quinque in .3. quod erit .15. </
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<
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xml:space
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autem medium terminum harmonicum inter iſtos habeamus, accipiatur
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duplum
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type
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pro-
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ducti, quod fit ex primis minimis terminis, quod erit .12.</
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</
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<
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<
s
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xml:space
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">Cuius rei ſpeculatio eſt iſta: </
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>
<
s
xml:id
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xml:space
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">ſignificentur duo termini datæ proportionis ab
<
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>.q.b.</
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>
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et
<
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>.b.r.</
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quorum ſumma erit
<
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>.q.r.</
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>
cuius quadratum ſit
<
var
>.q.o.</
var
>
ſit etiam imaginata
<
var
>.b.e.</
var
>
<
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parallela ad
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>.o.r</
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>
. </
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>
<
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xml:space
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<
reg
norm
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Sitque
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type
="
simple
">Sitq́;</
reg
>
<
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>.b.x.</
var
>
æqualis
<
var
>.b.r.</
var
>
et
<
var
>.q.u.</
var
>
ſimiliter, & ducatur
<
var
>.x.y.</
var
>
parallela ad
<
lb
/>
<
var
>r.o.</
var
>
et
<
var
>.u.l.</
var
>
ad
<
var
>.q.x</
var
>
. </
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>
<
s
xml:id
="
echoid-s1175
"
xml:space
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preserve
">Tunc habebimus
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var
>.b.o.</
var
>
æquale ei producto, quod fit ex
<
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>.q.r.</
var
>
in
<
var
>.b.r.</
var
>
<
lb
/>
et
<
var
>.b.y.</
var
>
eidem etiam æquale, et
<
var
>.q.e.</
var
>
pro producto, quod fit ex
<
var
>.q.r.</
var
>
in
<
var
>.q.b.</
var
>
et
<
var
>.q.l.</
var
>
pro
<
lb
/>
eo, quod fit ex
<
var
>.q.x.</
var
>
in
<
var
>.b.r</
var
>
. </
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Vnde
<
var
>.q.l.</
var
>
cum
<
var
>.b.y.</
var
>
æquale fiet duplo ei, quod fit ex
<
var
>.q.b.</
var
>
<
lb
/>
in
<
var
>.b.r</
var
>
. </
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>
<
s
xml:id
="
echoid-s1177
"
xml:space
="
preserve
">Dico nunc
<
var
>.b.o.</
var
>
eſſe minimum terminum eorum, quos quærimus, et
<
var
>.y.b.</
var
>
cum
<
var
>.
<
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/>
x.u.</
var
>
medium
<
var
>.q.e.</
var
>
verò maximum huiuſmodi proportionalitatis.</
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>
</
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<
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>
<
s
xml:id
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xml:space
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">Primum ergo certi ſcimus ex prima ſexti vel .18. ſeptimi eandem exiſtere pro-
<
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portionem
<
var
>.q.e.</
var
>
ad
<
var
>.b.o.</
var
>
ſeu ad
<
var
>.b.y.</
var
>
quæ
<
var
>.q.b.</
var
>
ad
<
var
>.b.r</
var
>
: ſed
<
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>.u.y.</
var
>
ad
<
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>.u.x.</
var
>
eſt vt
<
var
>.y.l.</
var
>
ad
<
var
>.l.x.</
var
>
<
lb
/>
hoc eſt vt
<
var
>.q.b.</
var
>
ad
<
var
>.b.r.</
var
>
ideſt vt
<
var
>.q.e.</
var
>
ad
<
var
>.b.o.</
var
>
& ſumma
<
var
>.u.y.</
var
>
cum
<
var
>.u.x.</
var
>
ideſt
<
var
>.q.y.</
var
>
minor eſt
<
lb
/>
quam
<
var
>.q.e.</
var
>
maximus terminus per
<
var
>.b.y.</
var
>
minimum ter-
<
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/>
minum. </
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>
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xml:id
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xml:space
="
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">&
<
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type
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reg
>
<
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>.q.y.</
var
>
ad
<
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>.q.l.</
var
>
vt
<
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>.y.x.</
var
>
ad
<
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>.x.l.</
var
>
hoc eſt
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/>
<
figure
xlink:label
="
fig-0102-01
"
xlink:href
="
fig-0102-01a
"
number
="
140
">
<
image
file
="
0102-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-01
"/>
</
figure
>
vt
<
var
>.q.r.</
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>
ad
<
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>.r.b</
var
>
. </
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<
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xml:space
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">Vnde ex ſpeculatione
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præcedentis
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type
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">præcedẽtis</
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>
theo
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rematis, ſequitur
<
var
>.u.y.</
var
>
eſſe differentiam inter
<
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norm
="
maximum
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type
="
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">maximũ</
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>
<
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& medium terminum, et
<
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>
eſſe differentiam inter
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medium & minimum dictæ proportionalitatis. </
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>
<
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xml:space
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">Nam
<
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eadem proportio eſt
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>.q.e.</
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>
maximi termini ad
<
var
>.b.o.</
var
>
mi-
<
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/>
nimi. quæ
<
var
>.u.y.</
var
>
(differentia inter
<
var
>.q.e.</
var
>
& gnomonem
<
var
>.
<
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/>
u.b.y.</
var
>
) ad
<
var
>.u.x.</
var
>
(differentia inter dictum
<
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>.u.b.y.</
var
>
et
<
var
>.b.y.</
var
>
<
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/>
minimum terminum, quia ſunt ambæ ut
<
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>.q.b.</
var
>
ad
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>.b.r.</
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>
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/>
vt diximus. </
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<
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xml:space
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">Quare
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>.b.y.</
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<
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type
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">coniunctũ</
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cum
<
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>.x.u.</
var
>
medius
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/>
terminus erit, qui quidem (vt dictum eſt) duplus eſt ei
<
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quod fit ex
<
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>.q.b.</
var
>
in
<
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>.b.r</
var
>
.</
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>
</
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</
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<
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xml:space
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">THEOREMA
<
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value
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>
.</
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<
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xml:space
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">ALIVM etiam modum ab antiquis traditum ad hoc problema perficiendum
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inueni, qui talis eſt. </
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>
<
s
xml:id
="
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xml:space
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">Inueniatur primo inter datos terminos extremos, me-
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dius terminus in arithmetica proportione, per
<
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quem
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type
="
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">quẽ</
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>
<
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/>
<
figure
xlink:label
="
fig-0102-02
"
xlink:href
="
fig-0102-02a
"
number
="
141
">
<
image
file
="
0102-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-02
"/>
</
figure
>
multiplicetur vnuſquiſque dictorum extremorum,
<
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deinde multiplicentur ipſi extremi interſe, vnde
<
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habebimus tria producta eadem proportione inui
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cem exiſtentia, vt quærebatur.</
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</
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<
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<
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xml:id
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xml:space
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">Exempli gratia, ponamus duos propoſitos ter-
<
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minos eſſe .3. et .2. quorum medius arithmeticè
<
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/>
eſſet .2. cum dimidia vnitate, per quem cum vnum
<
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quemque priorum multiplicauerimus,
<
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norm
="
emergent
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type
="
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">emergẽt</
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>
no-
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bis duo producta, quorum primum ideſt maius eſſet
<
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/>
7. cum dimidia vnitate, reliquum verò eſſet
<
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/>
quinque, productum poſteà quod ex ipſis extremis
<
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/>
prouenit, erit .6. quod quidem eſt harmonicè collo
<
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/>
catum inter .7. cum dimidia vnitate, & quinque.</
s
>
</
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>
<
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>
<
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xml:id
="
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xml:space
="
preserve
">Cuius rei ſpeculatio omnis à præcedenti theore-
<
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mate dependet. </
s
>
<
s
xml:id
="
echoid-s1187
"
xml:space
="
preserve
">Sint exempli gratia, duo termini </
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>
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